# Edge states for second order elliptic operators

David Gontier  CEREMADE, University of Paris-Dauphine, PSL University, 75016
Paris, France<EMAIL_ADDRESS>
###### Abstract.

We present a general framework to study edge states for second order elliptic
operators. We associate an integer valued index to some bulk materials, and we
prove that for any junction between two such materials, localised states must
appear at the boundary whenever the indices differ.

© 2020 by the author. This paper may be reproduced, in its entirety, for non-
commercial purposes.

##  1\. Introduction and statement of the main results

The bulk-edge correspondence states that one can associate an integer valued
index ${\mathcal{I}}\in{\mathbb{Z}}$ to some bulk materials (represented here
by Schrödinger (PDE) or Hill’s (ODE) operators). When the material is cut,
edge states appear at the boundary whenever ${\mathcal{I}}\neq 0$. In
addition, it is believed that any junction between a left and a right
materials having indices ${\mathcal{I}}_{L}$ and ${\mathcal{I}}_{R}$ must also
have edge states near the junction whenever
${\mathcal{I}}_{L}\neq{\mathcal{I}}_{R}$. We prove this fact in this paper.

Since the original works of Hatsugai [Hat93a, Hat93b], most studies on bulk-
edge correspondence focused on tight-binding models (e.g. [GP13, ASBVB13]),
set on half-spaces. In these tight-binding models, boundary conditions at the
cut are quite simple to describe, and it turns out that the index is
independent of these boundary conditions.

In the context of continuous models, it is unclear that one can define an
index which is indeed independent of the chosen boundary conditions. In
[Gon20], we proved that it was the case in a simple one-dimensional model for
dislocations. We extend this work here, and give a general framework to define
the edge index for different self-adjoint extensions of Schrödinger operators.

We consider two types of continuous models. In the first part, we study
families of Hill’s operator (ODE) set on ${\mathbb{C}}^{n}$, of the form

$h_{t}:=-\partial_{xx}^{2}+V_{t},\quad\text{acting on}\quad
L^{2}({\mathbb{R}},{\mathbb{C}}^{n}),$

where $t\mapsto V_{t}$ is a continuous periodic family of bounded potentials,
with values in the set of $n\times n$ hermitian matrices. On the second part
of the article, we study its PDE version, that is families of Schrödinger’s
operators of the form

$H_{t}:=-\Delta+V_{t},\quad\text{acting on}\quad
L^{2}\left({\mathbb{R}}\times(0,1)^{d-1},{\mathbb{C}}\right).$

Here, ${\mathbb{R}}\times(0,1)^{d-1}$ is a tube in ${\mathbb{R}}^{d}$.
Projecting $H_{t}$ in finite dimensional basis, we can think of the PDE case
as the $n=\infty$ version of the ODE one.

In these models, we interpret the bulk-edge index as the intersection of
Lagrangian planes on a boundary space ${\mathcal{H}}_{b}$. Roughly speaking,
this space contains the values $(\psi(0),\psi^{\prime}(0))$ of the admissible
wave-functions $\psi$. In the context of Hill’s operators, we will have
${\mathcal{H}}_{b}={\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$, while for
Schrödinger operators, ${\mathcal{H}}_{b}=H^{3/2}(\Gamma)\times
H^{1/2}(\Gamma)$, where $\Gamma=\\{0\\}\times(0,1)^{d-1}$ is the cut.

The link between edge states and Lagrangian planes was already mentioned e.g.
in [ASBVB13] and [GP13] for discrete models (tight-binding approximation).
Based on the recent developments on Lagrangian planes and second order
elliptic operator by Howard, Latushkin and Sukhtayev in a series on papers
[HS16, HLS17, HLS18], we extend the picture to the continuous cases of Hill’s
and Schrödinger operators, and show that the presence of edge modes comes from
the crossings of these planes. This framework also allows to treat both the
ODE and PDE setting, as noticed by the some of the previous authors in [LS18],
based on the seminal work of Booß-Bavnbek and Furutani on infinite dimensional
Lagrangian planes [BBF98, BBZ13, Fur04].

Our main result can be stated as follows (here for the Hill’s case). It
extends the one in the previous work [Gon20] and [Dro18]. Let
$n\in{\mathbb{N}}\setminus\\{0\\}$ be fixed, and let

$V_{t}(x):=V(t,x):{\mathbb{T}}^{1}\times{\mathbb{R}}\to{\mathcal{S}}_{n},$ (1)

be a periodic family of matrix-valued bounded potentials (which are not
necessarily periodic in $x$). Here, ${\mathbb{T}}^{1}\approx[0,1]$ is the one-
dimensional torus, and ${\mathcal{S}}_{n}$ denotes the set of $n\times n$
hermitian matrices. We assume that $t\mapsto V_{t}$ is continuous from
${\mathbb{T}}^{1}$ to $L^{\infty}({\mathbb{R}},{\mathcal{S}}_{n})$. We
consider the family of (bulk) Hill’s operators

$h_{t}:=-\partial_{xx}^{2}+V_{t}\quad\text{acting on}\quad
L^{2}({\mathbb{R}},{\mathbb{C}}^{n}).$

For $E\in{\mathbb{R}}$, we say that $E$ is in the gap of the family $(h_{t})$
if $E\notin\sigma(h_{t})$ for all $t\in{\mathbb{T}}^{1}$. We also consider the
family of (edge) Hill’s operators

$h^{\sharp}_{D,t}:=-\partial_{xx}^{2}+V_{t}\quad\text{acting on}\quad
L^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n}),$

with Dirichlet boundary conditions at $x=0$. While $E$ is not in the spectrum
of the bulk operator $\sigma(h_{t})$, it may belong to the spectrum of the
bulk operator $h^{\sharp}_{D,t}$. In this case, the corresponding eigenstate
is called an edge mode.

As $t$ runs through ${\mathbb{T}}^{1}\approx[0,1]$, a spectral flow may appear
for the family $h^{\sharp}_{D,t}$. We denote by ${\rm
Sf}(h^{\sharp}_{D,t},E,{\mathbb{T}}^{1})$ the net number of eigenvalues of
$h^{\sharp}_{D,t}$ going downwards in the gap where $E$ lives. We define the
bulk/edge index of $(h_{t})$ as this spectral flow:

${\mathcal{I}}(h_{t},E):={\rm
Sf}\left(h^{\sharp}_{D,t},E,{\mathbb{T}}^{1}\right).$

Our main theorem is the following (see Theorem 30 for the proof in the Hill’s
case, and Theorem 40 for the one in the Schrödinger case).

######  Theorem 1 (Bulk-edge correspondence for junctions).

Let $t\mapsto V_{R,t}$ and $t\mapsto V_{L,t}$ be two continuous periodic
families of bounded potentials on ${\mathbb{R}}$. Let $E\in{\mathbb{R}}$ be in
the gap of both corresponding (bulk) Hill’s operators $(h_{L,t})$ and
$(h_{R,t})$. Let $\chi:{\mathbb{R}}\to[0,1]$ be any switch function,
satisfying $\chi(x)=1$ for $x<-X$ and $\chi(x)=0$ for $x>X$ for some $X>0$
large enough, and let

$h^{\chi}_{t}:=-\partial_{xx}^{2}+V_{L,t}(x)\chi(x)+V_{R,t}(x)(1-\chi(x)).$

Then

${\rm
Sf}\left(h^{\chi}_{t},E,{\mathbb{T}}^{1}\right)={\mathcal{I}}(h_{R,t},E)-{\mathcal{I}}(h_{L,t},E).$

The operator $h^{\chi}_{t}$ is a domain wall operator. On the far left, we see
the potential $V_{L,t}$, while on the far right, we see $V_{R,t}$, so this
operator models a junction between a left potential and a right one.

The proofs in the Hill’s and Schrödinger cases are quite different. The Hill’s
case being finite dimensional, we are able to describe precisely the spectral
flow. We prove in particular that the bulk/edge index is independent of the
chosen boundary conditions. We interpret this spectral flow as a Maslov index
of some pair of Lagrangian planes, and as the winding number of some
unitaries. This somehow generalises some results in [GP13] or [ASBVB13] for
tight-binding models. The arguments we use are model independent, and can be
generalised in many ways.

###  1.1. Plan of the paper

In Section 2, we recall some basic facts on symplectic spaces and self-adjoint
extensions of operators. We then prove our results concerning Hill’s operators
in Section 3, and explain how to adapt the proofs for Schrödinger operators in
Section 4. We discuss the role of the boundary space
${\mathcal{H}}_{b}=H^{3/2}(\Gamma)\times H^{1/2}(\Gamma)$ in Appendix A.

###  1.2. Notation of the paper

We denote by ${\mathbb{N}}:=\\{1,2,3,\cdots\\}$ and by
${\mathbb{N}}_{0}:=\\{0,1,2,3,\cdots\\}$. For $\Omega\subset{\mathbb{R}}^{d}$
an open set, we denote by $L^{p}(\Omega,{\mathbb{C}})$ the usual Lebesgue
space, and by $H^{s}(\Omega,{\mathbb{C}})$ the Sobolev ones. The set
$H^{s}_{0}(\Omega,{\mathbb{C}})$ is the completion of
$C^{\infty}_{0}(\Omega,{\mathbb{C}})$ for the $H^{s}$ norm.

Let ${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$ be two Hilbert spaces. For a
bounded operator $A:{\mathcal{H}}_{1}\to{\mathcal{H}}_{2}$, its dual $A^{*}$
is the map from ${\mathcal{H}}_{2}\to{\mathcal{H}}_{1}$ so that

$\forall x_{1}\in{\mathcal{H}}_{1},\ \forall
x_{2}\in{\mathcal{H}}_{2},\quad\langle
x_{1},A^{*}x_{2}\rangle_{{\mathcal{H}}_{1}}=\langle
Ax_{1},x_{2}\rangle_{{\mathcal{H}}_{2}}.$

The operator $A$ is unitary if $A^{*}A={\mathbb{I}}_{{\mathcal{H}}_{1}}$ and
$AA^{*}={\mathbb{I}}_{{\mathcal{H}}_{2}}$.

For $E$ a Banach space, we say that a map $t\mapsto v(t)\in E$ is continuously
differentiable if $v^{\prime}(t)$ is well-defined in $E$ for all $t$ (that is
$\|v^{\prime}(t)\|_{E}<\infty$), and if $t\mapsto v^{\prime}(t)$ is
continuous.

###  1.3. Acknowledgements

This work was funded by the CNRS international cooperation tool PICS.

##  2\. First facts and notations

###  2.1. Lagrangian planes in complex Hilbert spaces

Let us first recall some basic facts on symplectic Banach/Hilbert spaces. In
the original work of Maslov [MBA72], popularised by Arnol’d [Arn67], the
authors consider real Banach spaces $E$. Following the recent developments, we
present the theory for complex Banach spaces.

####  2.1.1. Basics in symplectic spaces

Let $E$ be a complex Banach space. A symplectic form on $E$ is a non
degenerate continuous sesquilinear form $\omega:E\times E\to{\mathbb{C}}$ such
that

$\forall x,y\in E,\quad\omega(x,y)=-\overline{\omega(y,x)}.$

For $\ell$ a vectorial subspace of $E$, we denote by

$\ell^{\circ}:=\left\\{x\in E,\quad\forall
y\in\ell,\quad\omega(x,y)=0\right\\}.$

The space $\ell^{\circ}$ is always closed. Such a subspace is called isotropic
if $\ell\subset\ell^{\circ}$, co-isotropic if $\ell^{\circ}\subset\ell$, and
Lagrangian if $\ell=\ell^{\circ}$. We also say that $\ell$ is a Lagrangian
plane in the latter case. The set of all Lagrangian planes of $E$, sometime
called the Lagrangian-Grassmanian, is denoted by $\Lambda(E)$.

######  Example 2 (In ${\mathbb{R}}^{2n}$).

In the real Hilbert space $E={\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$, the
canonical symplectic form is given by (we write ${\mathbf{x}}=(x,x^{\prime})$,
${\mathbf{y}}=(y,y^{\prime})$, etc. the elements in
${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$)

$\forall{\mathbf{x}},{\mathbf{y}}\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n},\quad\omega({\mathbf{x}},{\mathbf{y}}):=\langle
x,y^{\prime}\rangle_{{\mathbb{R}}^{n}}-\langle
x^{\prime},y\rangle_{{\mathbb{R}}^{n}},$

When $n=1$, the Lagrangian planes are all the one-dimensional vectorial
subspaces of ${\mathbb{R}}^{2}$. Conversely, if $({\mathbb{R}}^{N},\omega)$ is
a symplectic space, then $N=2n$ is even, and all Lagrangian planes are of
dimension $n$.

######  Example 3 (In ${\mathbb{C}}^{2n}$).

Similarly, in the complex Hilbert space ${\mathbb{C}}^{2n}$, the canonical
symplectic form is given by

$\forall{\mathbf{z}}_{1},{\mathbf{z}}_{2}\in{\mathbb{C}}^{n}\times{\mathbb{C}}^{n},\quad\omega({\mathbf{z}}_{1},{\mathbf{z}}_{2}):=\langle
z_{1},z_{2}^{\prime}\rangle_{{\mathbb{C}}^{n}}-\langle
z_{1}^{\prime},z_{2}\rangle_{{\mathbb{C}}^{n}}.$

When $n=1$ for instance, the Lagrangian planes are the one-dimensional
vectorial spaces $L={\rm Vect}_{\mathbb{C}}({\mathbf{z}})$ with
${\mathbf{z}}=(z,z^{\prime})$ satisfying the extra condition
$\overline{z}z^{\prime}\in{\mathbb{R}}$. Up to a phase, we may always assume
$z\in{\mathbb{R}}$, in which case $z^{\prime}\in{\mathbb{R}}$ as well. So the
Lagrangian planes are the one-dimensional subspaces of ${\mathbb{C}}^{2}$ of
the form ${\rm Vect}_{\mathbb{C}}({\mathbf{z}})$ with
${\mathbf{z}}\in{\mathbb{R}}^{2n}$.

######  Example 4 (In ${\mathbb{C}}^{N}$).

Another example is given by the symplectic form

$\forall{\mathbf{z}}_{1},{\mathbf{z}}_{2}\in{\mathbb{C}}^{N},\quad\widetilde{\omega}({\mathbf{z}}_{1},{\mathbf{z}}_{2})={\mathrm{i}}\langle{\mathbf{z}}_{1},{\mathbf{z}}_{2}\rangle_{{\mathbb{C}}^{N}}.$

With this symplectic form, a vector ${\mathbf{z}}\in{\mathbb{C}}^{N}$ is never
isotropic, since
$\widetilde{\omega}({\mathbf{z}},{\mathbf{z}})={\mathrm{i}}\|{\mathbf{z}}\|^{2}\neq
0$ for $z\neq 0$. In particular, $({\mathbb{C}}^{N},\widetilde{\omega})$ does
not have Lagrangian subspaces.

We record the following result.

######  Lemma 5.

If $\ell_{1}\subset\ell_{1}^{\circ}$ and $\ell_{2}\subset\ell_{2}^{\circ}$ are
two isotropic subspaces with $\ell_{1}+\ell_{2}=E$, then

$\ell_{1}=\ell_{1}^{\circ},\quad\ell_{2}=\ell_{2}^{\circ},\quad\text{are
Lagrangians, and}\quad\ell_{1}\oplus\ell_{2}=E.$

###### Proof.

Since $\ell_{1}+\ell_{2}=E$, we have
$\\{0\\}=\ell_{1}^{\circ}\cap\ell_{2}^{\circ}$. In particular,
$\ell_{1}\cap\ell_{2}\subset\ell_{1}^{\circ}\cap\ell_{2}^{\circ}=\\{0\\}$ as
well, so $\ell_{1}\oplus\ell_{2}=E$. Let $x\in\ell_{1}^{\circ}\subset E$, and
write $x=x_{1}+x_{2}$ with $x_{1}\in\ell_{1}$ and $x_{2}\in\ell_{2}$. Then, we
have $x_{2}\in\ell_{1}^{\circ}$, so
$x_{2}\in\ell_{1}^{\circ}\cap\ell_{2}\subset\ell_{1}^{\circ}\cap\ell_{2}^{\circ}=\\{0\\}$.
This proves that $x=x_{1}\in\ell_{1}$, hence $\ell_{1}^{\circ}=\ell_{1}$. The
proof for $\ell_{2}$ is similar. ∎

####  2.1.2. Lagrangian planes of Hilbert spaces and unitaries.

In the case where $E={\mathcal{H}}_{b}$ is an Hilbert space, with inner
product $\langle\cdot,\cdot\rangle_{{\mathcal{H}}_{b}}$, for all
$x\in{\mathcal{H}}_{b}$, the map

$T_{x}:y\mapsto\omega(x,y)$

is linear and bounded, so, by Riesz’ representation theorem, there exists
$v\in{\mathcal{H}}_{b}$ so that $T_{x}(y)=\langle
v,y\rangle_{{\mathcal{H}}_{b}}$. We denote by $Jx:=v$ this element. This
defines a operator $J:{\mathcal{H}}_{b}\to{\mathcal{H}}_{b}$, satisfying

$\forall x,y\in{\mathcal{H}}_{b},\quad\omega(x,y)=\langle
x,Jy\rangle_{{\mathcal{H}}_{b}}.$

In particular, since $\omega$ is bounded, we have

$\|Jy\|_{{\mathcal{H}}_{b}}^{2}=\langle
Jy,Jy\rangle_{{\mathcal{H}}_{b}}=\omega(Jy,y)\leq C_{\omega}\|Jy\|\cdot\|y\|,$

so $J$ is bounded. In addition, from the relation
$\omega(x,y)=-\overline{\omega(y,x)}$, we get that

$\langle x,Jy\rangle_{{\mathcal{H}}_{b}}=-\langle
Jx,y\rangle_{{\mathcal{H}}_{b}}.$

In other words, $J=-J^{*}$. Finally, since $\omega$ is not degenerate, we have
${\rm Ker}(J)=\\{0\\}$.

######  Example 6.

On ${\mathbb{C}}^{2n}$ with the canonical symplectic form $\omega(x,y)=\langle
x_{1},y_{2}\rangle_{{\mathbb{C}}_{n}}-\langle
x_{2},y_{1}\rangle_{{\mathbb{C}}^{n}}$, we have

$J=\begin{pmatrix}0_{n}&{\mathbb{I}}_{n}\\\
-{\mathbb{I}}_{n}&0_{n}\end{pmatrix}.$

Later in the article, we will make the following Assumption A:

$\textbf{Assumption A:}\quad J^{2}={\mathbb{I}}_{{\mathcal{H}}_{b}}.$ (2)

In this case, since $J$ is bounded skew self-adjoint with
$J^{2}=-{\mathbb{I}}$, we can write

${\mathcal{H}}_{b}={\rm Ker}\left(J-{\mathrm{i}}\right)\oplus{\rm
Ker}\left(J+{\mathrm{i}}\right).$ (3)

The hermitian form $-{\mathrm{i}}\omega$ is positive definite on ${\rm
Ker}\left(J-{\mathrm{i}}\right)$, negative definite on ${\rm
Ker}\left(J+{\mathrm{i}}\right)$. We have

$\forall x\in{\rm
Ker}(J-{\mathrm{i}}),\quad\omega(x,x)={\mathrm{i}}\|x\|_{{\mathcal{H}}_{b}}^{2},\quad\forall
y\in{\rm
Ker}(J+{\mathrm{i}}),\quad\omega(y,y)=-{\mathrm{i}}\|y\|_{{\mathcal{H}}_{b}}^{2},$

and finally

$\forall x\in{\rm Ker}\left(J-{\mathrm{i}}\right),\ \forall y\in{\rm
Ker}\left(J+{\mathrm{i}}\right),\quad\omega(x,y)=0.$ (4)

The following result goes back to Leray in its seminar [Ler78] (see also
[BBF98] and [BBZ13, Lemma 2 and 3]). We skip its proof for the sake of
brevity.

######  Lemma 7.

If Assumption A holds, then there is a one-to-one correspondence between the
Lagrangian planes $\ell$ of ${\mathcal{H}}_{b}$ and the unitaries $U$ from
${\rm Ker}\left(J-{\mathrm{i}}\right)$ to ${\rm
Ker}\left(J+{\mathrm{i}}\right)$, with

$\ell=\left\\{x+Ux,\quad x\in{\rm Ker}\left(J-{\mathrm{i}}\right)\right\\}.$

######  Corollary 8.

If $\dim{\rm Ker}\left(J-{\mathrm{i}}\right)\neq\dim{\rm
Ker}\left(J+{\mathrm{i}}\right)$, then there are no Lagrangian planes. This
happens for instance for the symplectic space
$({\mathbb{C}}^{n},\widetilde{\omega})$, with
$\widetilde{\omega}(z,z^{\prime})={\mathrm{i}}\langle
z,z^{\prime}\rangle_{{\mathbb{C}}^{n}}$ (see Example 4), for which we have
$Jz={\mathrm{i}}z$, so ${\rm Ker}(J-{\mathrm{i}})={\mathbb{C}}^{n}$ while
${\rm Ker}(J+{\mathrm{i}})=\\{0\\}$.

The next Lemma shows that the crossing of two Lagrangian planes can be read
from their respective unitaries. It can be found e.g. in [BBZ13, Lemma 2]. Our
definition slightly differs from the one in [BBZ13], since we put the adjoint
on the left operator (this seems more natural for what follows).

######  Lemma 9.

Let $\ell_{1}$ and $\ell_{2}$ be two Lagrangian planes of
$\Lambda({\mathcal{H}}_{b})$, with corresponding unitaries $U_{1}$ and $U_{2}$
from ${\rm Ker}(J-{\mathrm{i}})$ to ${\rm Ker}(J+{\mathrm{i}})$. Then there is
a natural isomorphism

${\rm Ker}\left(U_{2}^{*}U_{1}-{\mathbb{I}}_{{\rm
Ker}(J-{\mathrm{i}})}\right)\approx\ell_{1}\cap\ell_{2}.$

###### Proof.

If $x^{-}\in{\rm Ker}\left(J-{\mathrm{i}}\right)$ is such that
$U_{2}^{*}U_{1}x^{-}=x^{-}$, then we have $U_{1}x^{-}=U_{2}x^{+}$ in ${\rm
Ker}(J+{\mathrm{i}})$, so $x:=x^{-}+U_{1}x^{-}=x^{-}+U_{2}x^{-}$ is in
$\ell_{1}\cap\ell_{2}$. Conversely, if $x\in\ell_{1}\cap\ell_{2}$, then,
writing $x=x^{-}+x^{+}$ shows that $U_{1}x^{-}=U_{2}x^{-}$, so
$U_{2}^{*}U_{1}x^{-}=x^{-}$. ∎

####  2.1.3. Another unitary

In Section 3.1 below, we will consider periodic paths of Lagrangians
$\ell_{1}(t)$ and $\ell_{2}(t)$, and define the Maslov index of the pair
$(\ell_{1},\ell_{2})$. When ${\mathcal{H}}_{b}$ is finite dimensional, we will
prove that it equals the winding number of the determinant of
$U_{2}^{*}(t)U_{1}(t)$. Unfortunately, since $U_{1}$ and $U_{2}$ are not
endomorphism, we cannot split $\det(U_{2}^{*}U_{1})$ into
$\det(U_{1})/\det(U_{2})$. In this section, we present another one-to-one
correspondence between Lagrangian planes and some other unitaries (which will
be endomorphisms). The results of this section are new to the best of our
knowledge.

We now make the stronger assumption that ${\mathcal{H}}_{b}$ is of the form
${\mathcal{H}}_{b}={\mathcal{H}}_{1}\times{\mathcal{H}}_{2}$, where
${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$ are two complex Hilbert spaces,
and that, relative to this decomposition, $J$ is of the form

$\textbf{Assumption B:}\quad J=\begin{pmatrix}0&V^{*}\\\
-V&0\end{pmatrix},\quad\text{for some (fixed) unitary
$V:{\mathcal{H}}_{1}\to{\mathcal{H}}_{2}$}.$ (5)

It implies $J^{2}=-{\mathbb{I}}$ as before. Furthermore, we can identify

${\rm Ker}(J-{\mathrm{i}})=\begin{pmatrix}1\\\
{\mathrm{i}}V\end{pmatrix}{\mathcal{H}}_{1},\quad\text{and}\quad{\rm
Ker}(J+{\mathrm{i}})=\begin{pmatrix}1\\\
-{\mathrm{i}}V\end{pmatrix}{\mathcal{H}}_{1}.$

Defining the maps $Q^{\pm}:{\mathcal{H}}_{1}\to{\rm Ker}(K\pm{\mathrm{i}})$ by

$\forall x\in{\mathcal{H}}_{1},\quad
Q_{\pm}(x):=\frac{1}{\sqrt{2}}\begin{pmatrix}x\\\
\mp{\mathrm{i}}Vx\end{pmatrix},\quad\text{with dual}\quad
Q_{\pm}^{*}\begin{pmatrix}x\\\ \mp{\mathrm{i}}x\end{pmatrix}=\sqrt{2}x,$

we see that, for all $x\in{\mathcal{H}}_{1}$,

$\left\|Q^{\pm}x\right\|^{2}_{{\mathcal{H}}_{b}}=\frac{1}{2}\left\|\begin{pmatrix}x\\\
\pm{\mathrm{i}}Vx\end{pmatrix}\right\|_{{\mathcal{H}}_{b}}=\frac{1}{2}\left\|x\right\|_{{\mathcal{H}}_{1}}+\frac{1}{2}\left\|Vx\right\|_{{\mathcal{H}}_{2}}=\|x\|_{{\mathcal{H}}_{1}},$

so $Q^{\pm}$ are unitaries. In particular, if $U$ is a unitary from ${\rm
Ker}(J-{\mathrm{i}})$ to ${\rm Ker}(J+{\mathrm{i}})$, then

${\mathcal{U}}:=Q_{+}^{*}UQ_{-}$

is a unitary from ${\mathcal{H}}_{1}$ to itself, so an endomorphism. In what
follows, we use straight letters $U$ for unitaries from ${\rm
Ker}(J-{\mathrm{i}})\to{\rm Ker}(J+{\mathrm{i}})$, and curly letters
${\mathcal{U}}$ for unitaries of ${\mathcal{H}}_{1}$. We therefore proved the
following.

######  Lemma 10.

If Assumption B holds, then there is a one-to-one correspondence between the
Lagrangian planes $\ell$ of
$({\mathcal{H}}_{1}\times{\mathcal{H}}_{2},\omega)$ and the unitaries
${\mathcal{U}}$ of ${\mathcal{H}}_{1}$, with

$\ell=\left\\{\begin{pmatrix}1\\\
{\mathrm{i}}V\end{pmatrix}x+\begin{pmatrix}1\\\
-{\mathrm{i}}V\end{pmatrix}{\mathcal{U}}x,\quad
x\in{\mathcal{H}}_{1}\right\\}.$

In addition, if $\ell_{1}$ and $\ell_{2}$ are two Lagrangian planes with
corresponding unitaries ${\mathcal{U}}_{1}$ and ${\mathcal{U}}_{2}$, then
there is a natural isomorphism

${\rm
Ker}\left({\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1}-{\mathbb{I}}_{{\mathcal{H}}_{1}}\right)\approx\ell_{1}\cap\ell_{2}.$

###  2.2. Self-adjoint extensions of Hill’s operators

We now review some basic facts on self-adjoint operators (see e.g. [RS75,
Chapter X.1] for complete introduction). We first recall some general
definitions, and then focus on second order elliptic operators. We show the
connection with symplectic spaces using the second Green’s identity (see Lemma
11 below).

####  2.2.1. Self-adjoint operators

Let ${\mathcal{H}}$ be a separable Hilbert space, and let $A$ with dense
domain ${\mathcal{D}}_{A}$ be any operator on ${\mathcal{H}}$. In the sequel,
we sometime write $(A,{\mathcal{D}}_{A})$. The adjoint of
$(A,{\mathcal{D}}_{A})$ is denoted by $(A^{*},{\mathcal{D}}_{A^{*}})$.

For $A$ a symmetric, hence closable, operator on ${\mathcal{H}}$, we denote by
$(A_{\rm min},{\mathcal{D}}_{\rm min})$ its closure. The adjoint of $(A_{\rm
min},{\mathcal{D}}_{\rm min})$ is denoted by $(A_{\rm max},{\mathcal{D}}_{\rm
max})$. Since $A$ is symmetric, we have $A_{\rm min}\subset A_{\rm max}$
($A_{\rm max}$ is an extension of $A_{\rm min}$). The operator $A_{\rm min}$
is self-adjoint iff ${\mathcal{D}}_{\rm min}={\mathcal{D}}_{\rm max}$.
Otherwise, any self-adjoint extension of $A$, if exists, is of the form
$(\widetilde{A},\widetilde{{\mathcal{D}}})$ with

$A_{\rm min}\subset\widetilde{A}\subset A_{\rm max},\quad\text{in the
sense}\quad{\mathcal{D}}_{\rm
min}\subset\widetilde{{\mathcal{D}}}\subset{\mathcal{D}}_{\rm max}.$

In particular, once ${\mathcal{D}}_{\rm min}$ and ${\mathcal{D}}_{\rm max}$
have been identified, the self-adjoint extensions are simply given by domains
$\widetilde{{\mathcal{D}}}$ with ${\mathcal{D}}_{\rm
min}\subset\widetilde{{\mathcal{D}}}\subset{\mathcal{D}}_{\rm max}$, and the
operator $\widetilde{A}$ acts on this domain via

$\forall x\in\widetilde{{\mathcal{D}}},\quad\widetilde{A}x:=A_{\rm max}x.$

In what follows, we sometime write $(A_{\rm max},\widetilde{{\mathcal{D}}})$
instead of $(\widetilde{A},\widetilde{{\mathcal{D}}})$ to insist that only the
domain matters.

There are several ways to identify the self-adjoint extensions of $A$. The
original proof by von Neumann [vN30] uses the Cayley transform. As noticed in
[RS75, Chapter X.1] following [DS65], the connection with boundary values is
not so clear which this approach. Another approach can be found e.g. in
[BBF98, Section 3.1], where the authors give a correspondence between the
self-adjoint extensions of $A$ and the Lagrangian planes of the abstract space
${\mathcal{D}}_{\rm max}/{\mathcal{D}}_{\rm min}$, with the symplectic form

$\forall[x],[y]\in{\mathcal{D}}_{\rm max}/{\mathcal{D}}_{\rm
min},\quad\omega\left([x],[y]\right):=\langle x,A_{\rm
max}y\rangle_{{\mathcal{H}}}-\langle A_{\rm max}x,y\rangle_{\mathcal{H}}.$

Again, the connection with boundary conditions is not so clear in this
setting.

In what follows, we present the approach in [LS18] (see also [CJM15]), which
is specific to second order elliptic operators. It uses the second Green’s
identity.

####  2.2.2. Self-adjoint extensions of Hill’s operators on the semi line

We first present the theory in the case where $A=h$ is a second order ODE
(Hill’s operators). We postpone the analysis for general second order elliptic
operator to Section 4 below.

Let $n\in{\mathbb{N}}$ and let $V:{\mathbb{R}}\to{\mathcal{S}}_{n}$ be a
bounded potential with values in ${\mathcal{S}}_{n}$, the set of $n\times n$
hermitian matrices. We consider the Hill’s operator

$h:=-\partial_{xx}^{2}+V(x)\quad\text{acting
on}\quad{\mathcal{H}}:=L^{2}({\mathbb{R}},{\mathbb{C}}^{n}).$

The bulk operator $h$ with core domain
$C^{\infty}_{0}({\mathbb{R}},{\mathbb{C}}^{n})$ is symmetric. Since the
potential $V$ is bounded, the operator $h$ is essentially self-adjoint, with
domain (see [Kat13, Chapter 4])

${\mathcal{D}}:={\mathcal{D}}_{\rm min}={\mathcal{D}}_{\rm
max}=H^{2}({\mathbb{R}},{\mathbb{C}}^{n}).$

When restricting this operator to the half line, we obtain the edge operator

$h^{\sharp}:=-\partial_{xx}^{2}+V(x)\quad\text{acting
on}\quad{\mathcal{H}}^{\sharp}:=L^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n}).$

On the core $C^{\infty}_{0}({\mathbb{R}}^{+},{\mathbb{C}}^{n})$, it is
symmetric, and its closure has domain

${\mathcal{D}}_{\rm
min}^{\sharp}:=H^{2}_{0}({\mathbb{R}}^{+},{\mathbb{C}}^{n}).$

The adjoint of $(h^{\sharp}_{\rm min},{\mathcal{D}}_{\rm min}^{\sharp})$ is
the operator $(h^{\sharp}_{\rm max},{\mathcal{D}}_{\rm max}^{\sharp})$ where
$h_{\rm max}^{\sharp}:=-\partial_{xx}^{2}+V(x)$ has domain

${\mathcal{D}}_{\rm max}^{\sharp}:=H^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n}).$

We have ${\mathcal{D}}_{\rm min}^{\sharp}\subsetneq{\mathcal{D}}_{\rm
max}^{\sharp}$, so $h^{\sharp}$ is not essentially self-adjoint. This reflects
the fact that some boundary conditions must be chosen at $x=0$. The
particularity of second order elliptic operator comes from the second Green’s
identity.

######  Lemma 11 (Green’s formula).

For all $\phi,\psi\in{\mathcal{D}}_{\rm max}^{\sharp}$,

$\langle\phi,h_{\rm max}^{\sharp}\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle
h_{\rm
max}^{\sharp}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=\langle\phi(0),\psi^{\prime}(0)\rangle_{{\mathbb{C}}^{n}}-\langle\phi^{\prime}(0),\psi(0)\rangle_{{\mathbb{C}}^{n}}.$

This suggests to introduce the boundary space

${\mathcal{H}}_{b}:={\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$

with its canonical symplectic form $\omega$ defined in Example 3. We also
introduce the map ${\rm Tr}:{\mathcal{D}}_{\rm
max}^{\sharp}\to{\mathcal{H}}_{b}$ defined by

$\forall\phi\in{\mathcal{D}}_{\rm max}^{\sharp},\quad{\rm
Tr}(\phi):=(\phi(0),\phi^{\prime}(0))\in{\mathcal{H}}_{b}.$ (6)

With these notations, the second Green’s identity reads

$\forall\phi,\psi\in{\mathcal{D}}^{\sharp}_{\rm max},\quad\langle\phi,h_{\rm
max}^{\sharp}\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle h_{\rm
max}^{\sharp}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=\omega\left({\rm
Tr}(\phi),{\rm Tr}(\psi)\right).$

We denote by $\|\cdot\|_{\sharp}$ the graph norm of $h^{\sharp}$, that is

$\forall\phi\in{\mathcal{D}}_{\rm
max}^{\sharp},\quad\|\phi\|_{\sharp}^{2}:=\|\phi\|_{{\mathcal{H}}^{\sharp}}^{2}+\|h^{\sharp}_{\rm
max}\phi\|_{{\mathcal{H}}^{\sharp}}^{2}.$

Recall that, by definition, a closed extension of $h^{\sharp}$ has a domain
which is closed for this norm.

######  Lemma 12.

The map ${\rm Tr}:({\mathcal{D}}_{\rm
max}^{\sharp},\|\cdot\|_{\sharp})\to{\mathcal{H}}_{b}$ is well-defined,
continuous and onto.

###### Proof.

Since $V$ is bounded, the graph norm $\|\cdot\|_{\sharp}$ is equivalent to the
usual $H^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n})$ norm on
${\mathcal{D}}^{\sharp}_{\rm max}=H^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n})$.
The Sobolev embedding shows that
$H^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n})\hookrightarrow
C^{1}([0,\infty),{\mathbb{C}}^{n})$ with continuous embedding. This implies
that ${\rm Tr}$ is a bounded linear operator. Let $C,S\in
C^{\infty}({\mathbb{R}}^{+},{\mathbb{R}})$ be two compactly supported smooth
functions with $C(0)=S^{\prime}(0)=1$ and $C^{\prime}(0)=S(0)=0$. Given an
element $(u,u^{\prime})\in{\mathcal{H}}_{b}$, we have $(u,u^{\prime})={\rm
Tr}(\psi)$ for $\psi(x):=uC(x)+u^{\prime}S(x)\in{\mathcal{D}}_{\rm
max}^{\sharp}$, so ${\rm Tr}$ is onto. ∎

The next result shows that self-adjoint extensions of $h^{\sharp}$ can be seen
as Lagrangian planes of ${\mathcal{H}}_{b}$.

######  Theorem 13.

Let ${\mathcal{D}}^{\sharp}$ be a domain satisfying ${\mathcal{D}}_{\rm
min}^{\sharp}\subset{\mathcal{D}}^{\sharp}\subset{\mathcal{D}}_{\rm
max}^{\sharp}$, and let $\ell:={\rm Tr}({\mathcal{D}}^{\sharp})$. The adjoint
of $(h^{\sharp}_{\rm max},{\mathcal{D}}^{\sharp})$ is the operator
$(h^{\sharp}_{\rm max},({\mathcal{D}}^{\sharp})^{*})$ with domain

$({\mathcal{D}}^{\sharp})^{*}:={\rm Tr}^{-1}\left(\ell^{\circ}\right).$

In particular, ${\mathcal{D}}^{\sharp}\subset{\mathcal{D}}_{\rm max}$ defines
a self-adjoint extension of $h^{\sharp}$ iff it is of the form

${\mathcal{D}}^{\sharp}:={\rm Tr}^{-1}(\ell),\quad\text{for some Lagrangian
plane}\quad\ell\in\Lambda({\mathcal{H}}_{b}).$

######  Remark 14.

Since ${\rm Tr}$ is a continuous map, and since any subspace of the form
$\ell^{\circ}$ is closed in ${\mathcal{H}}_{b}$, we deduce that
$({\mathcal{D}}^{\sharp})^{*}={\rm Tr}^{-1}(\ell^{\circ})$ is closed for the
graph norm of ${\mathcal{H}}^{\sharp}$. Note that ${\mathcal{D}}^{\sharp}$ is
not required to be closed.

###### Proof.

The following proof only uses that ${\rm Tr}$ is onto. It will therefore be
valid also for the Schrödinger PDE case (with the corresponding boundary space
and ${\rm Tr}$ operator, see Section 4 below).

Since ${\mathcal{D}}_{\rm
min}^{\sharp}\subset{\mathcal{D}}^{\sharp}\subset{\mathcal{D}}_{\rm
max}^{\sharp}$ and $({\mathcal{D}}_{\rm max}^{\sharp})^{*}={\mathcal{D}}_{\rm
min}^{\sharp}$, we have ${\mathcal{D}}_{\rm
min}^{\sharp}\subset\left({\mathcal{D}}^{\sharp}\right)^{*}\subset{\mathcal{D}}_{\rm
max}$ as well. Let
$\phi\in({\mathcal{D}}^{\sharp})^{*}\subset{\mathcal{D}}_{\rm max}^{\sharp}$ .
By definition of the adjoint, and the second Green’s identity, we have

$\forall\psi\in{\mathcal{D}}^{\sharp},\quad 0=\langle\phi,h^{\sharp}_{\rm
max}\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle h^{\sharp}_{\rm
max}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=\omega({\rm Tr}(\phi),{\rm
Tr}(\psi)).$

We deduce that ${\rm Tr}(\phi)\in\ell^{\circ}$.

Conversely, let $(u,u^{\prime})\in\ell^{\circ}\subset{\mathcal{H}}_{b}$ and
let $\psi_{0}\in{\mathcal{D}}_{\rm max}^{\sharp}$ be such that ${\rm
Tr}(\psi_{0})=(u,u^{\prime})$. By definition of $\ell^{\circ}$ and the second
Green’s identity, we get

$\forall\phi\in{\mathcal{D}}^{\sharp},\quad 0=\omega\left({\rm
Tr}(\psi_{0}),{\rm Tr}(\phi)\right)=\langle\psi_{0},h_{\rm
max}^{\sharp}\phi\rangle_{{\mathcal{H}}^{\sharp}}-\langle h_{\rm
max}^{\sharp}\psi_{0},\phi\rangle_{{\mathcal{H}}^{\sharp}}.$

In particular, the map $T_{\psi_{0}}:{\mathcal{D}}^{\sharp}\to{\mathbb{C}}$
defined by

$T_{\psi_{0}}:\phi\mapsto\langle\psi_{0},h_{\rm
max}^{\sharp}\phi\rangle_{{\mathcal{H}}^{\sharp}}=\langle h_{\rm
max}^{\sharp}\psi_{0},\phi\rangle_{{\mathcal{H}}^{\sharp}}$

is bounded on ${\mathcal{D}}^{\sharp}$ with
$\|T_{\psi_{0}}\phi\|_{{\mathcal{H}}^{\sharp}}\leq\|h^{\sharp}_{\rm
max}\psi_{0}\|_{{\mathcal{H}}^{\sharp}}\|\phi\|_{{\mathcal{H}}^{\sharp}}$. So
$\psi_{0}$ is in the adjoint domain $({\mathcal{D}}^{\sharp})^{*}$. This
proves that $(u,u^{\prime})={\rm Tr}(\psi_{0})\in{\rm
Tr}(({\mathcal{D}}^{\sharp})^{*})$. Altogether, we proved that

${\rm
Tr}\left(({\mathcal{D}}^{\sharp})^{*}\right)=\ell^{\circ},\quad\text{hence}\quad({\mathcal{D}}^{\sharp})^{*}={\rm
Tr}^{-1}(\ell^{\circ}),$

where we used in the last inequality that $({\mathcal{D}}^{\sharp})^{*}$ is
closed for the graph norm of ${\mathcal{H}}^{\sharp}$. ∎

In what follows, we denote by $\left(h^{\sharp},\ell^{\sharp}\right)$ the
self-adjoint extensions of $h^{\sharp}$ with domain ${\rm
Tr}^{-1}(\ell^{\sharp})$.

Before we go on, let us gives some examples of Lagrangian planes and their
corresponding unitaries ${\mathcal{U}}$ for some usual self-adjoint
extensions. Note that we have
${\mathcal{H}}_{b}={\mathcal{H}}_{1}\times{\mathcal{H}}_{2}$ with
${\mathcal{H}}_{1}={\mathcal{H}}_{2}={\mathbb{C}}^{n}$, so the unitary
${\mathcal{U}}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ is an element in ${\rm
U}(n)$.

######  Example 15 (Dirichlet and Neumann boundary conditions).

The Dirichlet extension of $h^{\sharp}$ corresponds to the Lagrangian plane
$\ell_{D}:=\\{0\\}\times{\mathbb{C}}^{n}$, and the Neumann one corresponds to
$\ell_{N}:={\mathbb{C}}^{n}\times\\{0\\}$. To identify the corresponding
unitary, we note that $(0,u^{\prime})\in\ell_{D}$ can be written as

$\begin{pmatrix}0\\\ u^{\prime}\end{pmatrix}=\begin{pmatrix}1\\\
{\mathrm{i}}\end{pmatrix}(-\tfrac{{\mathrm{i}}}{2}u^{\prime})+\begin{pmatrix}1\\\
-{\mathrm{i}}\end{pmatrix}(\tfrac{{\mathrm{i}}}{2}u^{\prime})$

Comparing with Lemma 10, this gives the unitary
${\mathcal{U}}_{D}:=-{\mathbb{I}}_{n}\in{\rm U}(n)$. The proof for Neumann
boundary conditions is similar, and we find
${\mathcal{U}}_{N}:={\mathbb{I}}_{n}\in{\rm U}(n)$.

######  Example 16 (Robin boundary conditions).

Consider $\Theta$ and $\Pi$ two hermitian $n\times n$ matrices so that

$\Theta^{*}=\Theta,\ \Pi^{*}=\Pi,\ \Theta\Pi=\Pi\Theta,\quad\Theta\ \text{or}\
\Pi\quad\text{is invertible}.$

Let $\ell_{\Theta,\Pi}$ be the subspace

$\ell_{\Theta,\Pi}:=\left\\{(\Theta x,\Pi x),\
x\in{\mathbb{C}}^{n}\right\\}\subset{\mathcal{H}}_{b}.$

We claim that $\ell_{\Theta,\Pi}$ is Lagrangian. First, we have

$\omega((\Theta x,\Pi x),(\Theta y,\Pi y))=\langle\Theta x,\Pi
y\rangle_{{\mathbb{C}}^{n}}-\langle\Pi x,\Theta
y\rangle_{{\mathbb{C}}^{n}}=0,$

so $\ell_{\Theta,\Pi}\subset\ell_{\Theta,\Pi}^{\circ}$. On the other hand, let
$(z,z^{\prime})\in\ell_{\Theta,\Pi}^{\circ}$. Assume that $\Theta$ is
invertible (the proof when $\Pi$ is invertible is similar). We have,

$\forall x\in{\mathbb{C}}^{n},\quad\langle z,\Pi
x\rangle_{{\mathbb{C}}^{n}}=\langle z^{\prime},\Theta
x\rangle_{{\mathbb{C}}^{n}},\quad\text{so}\quad\langle\Pi z-\Theta
z^{\prime},x\rangle_{{\mathbb{C}}^{n}}=0.$

We deduce that $\Pi z=\Theta z^{\prime}$. In particular, setting
$z_{0}=\Theta^{-1}z$, we have $z=\Theta z_{0}$ and $z^{\prime}=\Pi z_{0}$, so
$(z,z^{\prime})=(\Theta z_{0},\Pi z_{0})\in\ell_{\Theta,\Pi}$. This proves
that $\ell_{\Theta,\Pi}$ is Lagrangian. We say that the corresponding self-
adjoint extension has the $(\Theta,\Pi)$-Robin boundary condition. To identify
the corresponding unitary, we write

$\begin{pmatrix}\Theta x\\\ \Pi x\end{pmatrix}=\begin{pmatrix}1\\\
{\mathrm{i}}\end{pmatrix}\tfrac{1}{2}\left(\Theta-{\mathrm{i}}\Pi\right)x+\begin{pmatrix}1\\\
-{\mathrm{i}}\end{pmatrix}\tfrac{1}{2}\left(\Theta+{\mathrm{i}}\Pi\right)x.$

Identifying with Lemma 10, we recognise the unitary

${\mathcal{U}}_{\Theta,\Pi}:=(\Theta+{\mathrm{i}}\Pi)(\Theta-{\mathrm{i}}\Pi)^{-1}\quad\in{\rm
U}(n).$

Note that the Dirichlet boundary condition corresponds to the pair
$(0,{\mathbb{I}}_{n})$ and the Neumann boundary condition corresponds to the
pair $({\mathbb{I}}_{n},0)$.

######  Example 17 (Dirac operator on the half-line).

Let $\not{D}^{\sharp}$ be a Dirac operator of the form
$\not{D}^{\sharp}:=-{\mathrm{i}}\partial_{x}+V$ acting on
$L^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n})$. The operator $\not{D}^{\sharp}$ is
symmetric on the core $C^{\infty}_{0}({\mathbb{R}}^{+})$. The first Green’s
identity shows that

$\forall\phi,\psi\in
H^{1}({\mathbb{R}}^{+},{\mathbb{C}}^{n}),\quad\langle\phi,\not{D}^{\sharp}\psi\rangle-\langle\not{D}^{\sharp}\phi,\psi\rangle=-{\mathrm{i}}\langle\phi(0),\psi(0)\rangle_{{\mathbb{C}}^{n}}.$

In this case, the boundary space is ${\mathcal{H}}_{b}={\mathbb{C}}^{n}$, with
the symplectic form $\widetilde{\omega}(z_{1},z_{2})={\mathrm{i}}\langle
z_{1},z_{2}\rangle$. A theorem similar to Theorem 13 holds. However, as
noticed in Example 4, the symplectic space
$({\mathcal{H}}_{b},\widetilde{\omega})$ has no Lagrangian planes. We recover
the classical result that $\not{D}^{\sharp}$ has no self-adjoint extensions on
the half line.

###  2.3. The Lagrangian planes $\ell^{\pm}(E)$

In the previous section, we linked the boundary conditions at $x=0$ with the
Lagrangian planes of the boundary space ${\mathcal{H}}_{b}$. We now focus on
the Cauchy solutions of $H\psi=E\psi$. Since we are also interested in the
behaviour at $-\infty$, we introduce
${\mathcal{H}}^{\sharp,\pm}:=L^{2}({\mathbb{R}}^{\pm})$ and the maximal
domains

${\mathcal{D}}^{\sharp,\pm}_{\rm
max}:=H^{2}({\mathbb{R}}^{\pm},{\mathbb{C}}^{n}).$

The space ${\mathcal{D}}^{\sharp}_{\rm max}$ considered previously corresponds
to ${\mathcal{D}}^{\sharp,\pm}_{\rm max}$. We also denote by ${\rm
Tr}^{\pm}:{\mathcal{D}}^{\sharp,\pm}_{\rm max}\to{\mathcal{H}}_{b}$ the
corresponding boundary trace operator

$\forall\psi\in{\mathcal{D}}^{\sharp,\pm}_{\rm max},\quad{\rm
Tr}^{\pm}(\psi)=(\psi(0),\psi^{\prime}(0)).$

Note that, due to the orientation of the line ${\mathbb{R}}$, the Green’s
identity on the left-side reads

$\forall\phi,\psi\in{\mathcal{D}}^{\sharp,-}_{\rm max},\quad\langle\phi,h_{\rm
max}^{\sharp,-}\psi\rangle_{{\mathcal{H}}^{\sharp,-}}-\langle h_{\rm
max}^{\sharp,-}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp,-}}=-\omega\left({\rm
Tr}^{-}(\phi),{\rm Tr}^{-}(\phi)\right).$

The following result, which is straightforward in the finite dimensional
Hill’s case, is an essential tool for the proof of the next theorem.

######  Lemma 18.

Let ${\mathds{1}}_{{\mathbb{R}}^{\pm}}$ be the restriction operators from
${\mathcal{H}}$ to ${\mathcal{H}}^{\sharp,\pm}$. Then

${\mathcal{D}}^{\sharp,\pm}_{\rm
max}={\mathds{1}}_{{\mathbb{R}}^{\pm}}\left({\mathcal{D}}\right).$

In addition, for all $\psi\in{\mathcal{D}}$, we have ${\rm
Tr}^{-}\left({\mathds{1}}_{{\mathbb{R}}^{-}}\psi\right)={\rm
Tr}^{+}\left({\mathds{1}}_{{\mathbb{R}}^{+}}\psi\right)$.

This lemma states that all elements in ${\mathcal{D}}^{\sharp,\pm}_{\rm max}$
can be extended to the whole line ${\mathbb{R}}$ while keeping smoothness at
the cut. It is linked to the existence of an extension operator. Conversely,
we do not lose regularity when restricting a function to some domain. We leave
the proof, as it is similar to the one of Lemma 12.

We now set

${\mathcal{S}}^{\pm}(E):={\rm Ker}\left(h_{\rm
max}^{\sharp,\pm}-E\right)=\left\\{\psi\in{\mathcal{D}}^{\sharp,\pm}_{\rm
max},\quad-\psi^{\prime\prime}+V\psi=E\psi\right\\}.$

The solutions in ${\mathcal{S}}^{\pm}(E)$ can be seen as Cauchy solutions. In
particular, elements $\psi^{\pm}$ of ${\mathcal{S}}^{\pm}(E)$ can be
reconstructed from their boundary values ${\rm
Tr}^{\pm}\left(\psi^{\pm}\right)\in{\mathcal{H}}_{b}$. We set

$\ell^{\pm}(E):=\left\\{{\rm
Tr}^{\pm}(\psi),\quad\psi\in{\mathcal{S}}^{\pm}(E)\right\\}\quad\subset{\mathcal{H}}_{b}.$
(7)

It is unclear at this point that any Cauchy solutions of
$-\psi^{\prime\prime}+V\psi=E$ is either in ${\mathcal{S}}^{+}(E)$ or
${\mathcal{S}}^{-}(E)$. For instance, one can have solutions which are not
integrable at $\pm\infty$. This happens in particular when $V$ is periodic and
$E$ is in the essential spectrum of the corresponding Hill’s operator $h$. In
this case, we expect all Cauchy solutions to be quasi-periodic. These
solutions are not in ${\mathcal{S}}^{+}(E)$ nor in ${\mathcal{S}}^{-}(E)$.

On the other hand, these two spaces might cross. Actually, we have

######  Lemma 19.

For the bulk operator $h$, we have that for all $E\in{\mathbb{R}}$,

$\dim{\rm Ker}\left(h-E\right)=\dim\left(\ell^{+}(E)\cap\ell^{-}(E)\right).$

In particular, $E$ is eigenvalue of $h$ iff
$\ell^{+}(E)\cap\ell^{-}(E)\neq\\{0\\}$.

###### Proof.

A simple proof uses Cauchy’s theorem. Let us give an alternating proof, which
will also work in the next PDE setting.

If $\psi\in{\mathcal{D}}$ satisfies $(h-E)\psi=0$, then, by Lemma 18, its
restrictions $\psi^{\pm}:={\mathds{1}}_{{\mathbb{R}}^{\pm}}\psi$ are in
${\mathcal{D}}_{\rm max}^{\sharp,\pm}$. In addition, they satisfy $(h_{\rm
max}^{\sharp}-E)\psi^{\pm}=0$, so $\psi^{\pm}\in{\mathcal{S}}^{\pm}(E)$.
Taking traces and using again Lemma 18 shows that ${\rm Tr}^{+}(\psi^{+})={\rm
Tr}^{-}(\psi^{-})\in\ell^{+}(E)\cap\ell^{-}(E)$.

Conversely, let $\psi^{\pm}\in{\mathcal{S}}^{\pm}(E)$ be such that ${\rm
Tr}^{+}(\psi^{+})={\rm Tr}^{-}(\psi^{-})$, and consider the function
$\psi\in{\mathcal{H}}$ defined by

$\psi(x):=\begin{cases}\psi^{+}(x)\quad\text{for}\quad x>0,\\\
\psi^{-}(x)\quad\text{for}\quad x<0.\end{cases}$

It is unclear yet that $\psi$ is regular enough (i.e. belongs to
${\mathcal{D}}$). For $f\in{\mathcal{D}}$, we have

$\displaystyle\langle\psi,(h-E)f\rangle_{{\mathcal{H}}}$
$\displaystyle=\langle\psi^{+},{\mathds{1}}_{{\mathbb{R}}^{+}}(h-E)f\rangle_{{\mathcal{H}}^{+}}+\langle\psi^{-},{\mathds{1}}_{{\mathbb{R}}^{-}}(h-E)f\rangle_{{\mathcal{H}}^{-}}$
$\displaystyle=\langle\psi^{+},(h_{\rm
max}^{\sharp,+}-E)f^{+}\rangle_{{\mathcal{H}}^{+}}+\langle\psi^{-},(h_{\rm
max}^{\sharp,-}-E)f^{-}\rangle_{{\mathcal{H}}^{-}}$
$\displaystyle=\omega\left({\rm Tr}^{+}(\psi^{+}),{\rm
Tr}^{+}(f^{+})\right)-\omega\left({\rm Tr}^{-}(\psi^{-}),{\rm
Tr}^{-}(f^{-})\right)=0.$

So $T_{\psi}:f\mapsto\langle\psi,(h-E)f\rangle_{{\mathcal{H}}}$ is bounded on
${\mathcal{D}}$. We deduce first that $\psi$ is in the domain
${\mathcal{D}}^{*}={\mathcal{D}}$, and that $(h-E)\psi=0$. So $\psi$ is an
eigenvector for the eigenvalue $E$. ∎

The previous result shows that one can detect eigenvalues as the crossings of
$\ell^{+}(E)$ and $\ell^{-}(E)$. We now prove that, when $E$ is in the
resolvent set, we have instead
$\ell^{+}(E)\oplus\ell^{-}(E)={\mathcal{H}}_{b}$. Our proof only uses the fact
that the bulk operator $h$ is essentially self-adjoint.

######  Theorem 20.

For all $E\in{\mathbb{R}}\setminus\sigma(h)$, the sets $\ell^{\pm}(E)$ are
Lagrangian planes of ${\mathcal{H}}_{b}$, and

${\mathcal{H}}_{b}:=\ell^{+}(E)\oplus\ell^{-}(E).$

This shows for instance that there are as many Cauchy’s solutions which decay
to $+\infty$ as solutions which decay to $-\infty$ (here, they both form
subspaces of dimension $n$). This is somehow reminiscent of the Weyl’s
criterion [Wey10] (see also [RS75, Theorem X.7]).

###### Proof.

We first claim that for any $E\in{\mathbb{R}}$, $\ell^{\pm}(E)$ are isotropic
spaces. Let $\phi,\psi\in{\mathcal{S}}^{+}(E)$. By Green’s identity, we have

$\omega({\rm Tr}(\phi),{\rm Tr}(\psi))=\langle\phi,h_{\rm
max}^{\sharp}\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle h_{\rm
max}^{\sharp}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=\langle\phi,E\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle
E\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=0.$

In the last equality, we used that $E$ is real-valued. This proves that
$\ell^{+}(E)\subset\ell^{+}(E)^{\circ}$. The proof for $\ell^{-}(E)$ is
similar.

Let $E\in{\mathbb{R}}\setminus\sigma(h)$, so that the bulk operator $(h-E)$ is
invertible. We have (recall that
${\mathcal{H}}=L^{2}({\mathbb{R}},{\mathbb{C}}^{n})$)

${\mathcal{H}}={\mathcal{H}}^{+}\oplus{\mathcal{H}}^{-},\quad\text{where}\quad{\mathcal{H}}^{\pm}:=\left\\{\psi\in{\mathcal{H}},\
\psi=0\ \text{on}\ \overline{{\mathbb{R}}^{\mp}}\right\\}.$

Since $(h-E)$ is invertible with ${\mathcal{D}}=(h-E)^{-1}{\mathcal{H}}$, this
gives a decomposition

${\mathcal{D}}={\mathcal{D}}^{+}\oplus{\mathcal{D}}^{-},\quad{\mathcal{D}}^{\pm}:=(h-E)^{-1}{\mathcal{H}}^{\pm},$

and

${\mathcal{H}}_{b}={\rm Tr}\left({\mathcal{D}}^{+}\right)+{\rm
Tr}\left({\mathcal{D}}^{-}\right).$

The elements $\psi\in{\mathcal{D}}^{+}$ are such that
$(-\partial_{xx}^{2}+V-E)\psi=f$, for some $f\in{\mathcal{H}}$ with support
contained in ${\mathbb{R}}^{+}$. In particular, its restriction
$\psi^{-}:={\mathds{1}}_{{\mathbb{R}}^{-}}\psi$ is in ${\mathcal{D}}_{\rm
max}^{\sharp,-}$ and satisfies $(h_{\rm max}^{\sharp,-}-E)\psi^{-}=0$ on
${\mathbb{R}}^{-}$. So $\psi^{-}\in{\mathcal{S}}^{-}(E)$. Taking boundary
traces show that

${\rm Tr}\left({\mathcal{D}}^{+}\right)\subset\ell^{-}(E),\quad\text{and,
similarly, }\quad{\rm Tr}\left({\mathcal{D}}^{-}\right)\subset\ell^{+}(E).$

In particular, we have ${\mathcal{H}}_{b}=\ell^{+}(E)+\ell^{-}(E)$. We
conclude with Lemma 5.

∎

######  Remark 21.

In the proof, we use that $h=-\partial_{xx}^{2}+V$ is self-adjoint on the
whole line, and deduce that $\ell^{+}(E)$ and $\ell^{-}(E)$ are both
Lagrangian planes. Note however that $\ell^{+}(E)$ is independent of $V$ on
${\mathbb{R}}^{-}$. So $\ell^{+}(E)$ is a Lagrangian plane whenever there
exists an extension of $V$ on ${\mathbb{R}}^{-}$ for which the corresponding
bulk operator has $E$ in its resolvent set.

At this point, we defined two types of Lagrangian planes for a given operator
$h$. First, we defined the planes $\ell^{\sharp}$ representing the boundary
conditions of a self-adjoint extension of the edge Hamiltonian $h^{\sharp}$.
Then, we defined the planes $\ell^{+}(E)$ as the set of traces of ${\rm
Ker}(h_{\rm max}^{\sharp}-E)$. If ${\rm
Tr}(\psi)\in\ell^{+}(E)\cap\ell^{\sharp}$, then $\psi$ is in the domain of
$h^{\sharp}$, and satisfies $(h^{\sharp}-E)\psi=0$. So $\psi$ is an
eigenvector for the eigenvalue $E$. This proves the following result (compare
with Lemma 19).

######  Lemma 22.

Let $E\in{\mathbb{R}}\setminus\sigma(h)$, and consider a self-adjoint
extension $(h^{\sharp},\ell^{\sharp})$ of the edge operator. Then

$\dim{\rm
Ker}\left(h^{\sharp}-E\right)=\dim\left(\ell^{+}(E)\cap\ell^{\sharp}\right).$

If Assumption B holds, we can introduce ${\mathcal{U}}^{+}(E)$ and
${\mathcal{U}}^{\sharp}$ the unitaries corresponding to the Lagrangian planes
$\ell^{+}(E)$ and $\ell^{\sharp}$ respectively, and we have

$\dim{\rm
Ker}\left(h^{\sharp}-E\right)=\dim\left(\ell^{+}(E)\cap\ell^{\sharp}\right)=\dim\left(({\mathcal{U}}^{\sharp})^{*}{\mathcal{U}}^{+}(E)-1\right).$

For instance, the self-adjoint extension $(h^{\sharp},\ell^{+}(E))$ has the
energy $E$ as eigenvalue of multiplicity $n$.

##  3\. Families of Hill’s operators

In the previous section, we exhibit the relationships between self-adjoint
extensions, Lagrangian planes, and unitaries. We now consider periodic
families of these objects, namely $h^{\sharp}_{t}$, $\ell_{t}^{\sharp}$ and
${\mathcal{U}}_{t}^{\sharp}$. For each such family, we define an index, namely
a spectral flow across $E$ for the family $h^{\sharp}_{t}$, a Maslov index for
the bifamily $(\ell_{t}^{+}(E),\ell_{t}^{\sharp})$ and a spectral flow across
$1$ for the family $({\mathcal{U}}_{t}^{\sharp})^{*}{\mathcal{U}}_{t}^{+}(E)$.
All these objects are defined in the following sections, and we prove that
they all coincide.

All these indices can be defined for continuous families. However, since the
proofs are simpler in the continuously differentiable case, we restrict
ourselves to this case. All these indices depends only on the homotopy class
of the corresponding loops, so similar results hold in the continuous case.

###  3.1. Families of Lagrangians, and Maslov index

We first define the Maslov index of two families of Lagrangian spaces. This
index originates from the work of Maslov in [MBA72, Arn67]. In these works,
the index was defined for finite dimensional real symplectic spaces (namely
${\mathbb{R}}^{2n}$ in Example 2). A modern approach can be found in [Fur04],
where the infinite dimensional case is studied as well. Here, we present a
simple version of the theory, which is enough for our purpose.

Let $({\mathcal{H}}_{b},\omega)$ be a symplectic Hilbert space (not
necessarily finite dimensional). We define a topology on
$\Lambda({\mathcal{H}}_{b})$ by setting

$\forall\ell_{1},\ell_{2}\in\Lambda({\mathcal{H}}_{b}),\quad{\rm
dist}(\ell_{1},\ell_{2}):=\|P_{1}-P_{2}\|_{\rm op},$

where $P_{1}$ and $P_{2}$ are the orthogonal projectors on $\ell_{1}$ and
$\ell_{2}$ respectively. A family $\ell(t)$ in $\Lambda({\mathcal{H}}_{b})$ is
said to be continuous, continuously differentiable, etc. if the corresponding
family of projectors $P(t)$ is so in
${\mathcal{B}}\left({\mathcal{H}}_{b}\right)$.

####  3.1.1. Definition with quadratic crossing forms

Consider two continuously differentiable families
${\mathbb{T}}^{1}\mapsto\ell_{1}(t)$ and ${\mathbb{T}}^{1}\mapsto\ell_{2}(t)$.
Let $t^{*}\in{\mathbb{T}}^{1}$ be such that
$\ell_{1}(t^{*})\cap\ell_{2}(t^{*})\neq\\{{\mathbf{0}}\\}$. We define the
sequilinear form $b$ on $\ell_{1}(t^{*})\cap\ell_{2}(t^{*})$ by

$\forall x,y\in\ell_{1}(t^{*})\cap\ell_{2}(t^{*}),\quad
b(x,y):=\omega(x,P_{1}^{\prime}(t^{*})y)-\omega(x,P_{2}^{\prime}(t^{*})y).$
(8)

######  Lemma 23.

The sesquilinear form $b$ is hermitian, that is $b(x,y)=\overline{b(y,x)}$.

###### Proof.

Let $P_{t}:=P_{1}(t)$. First, since ${\rm Ran}\,P_{1}(t)=\ell_{1}(t)$ is
isotropic for all $t$, we have

$\forall x,y\in{\mathcal{H}}_{b},\quad\forall
t\in{\mathbb{T}}^{1},\quad\omega(P_{t}(x),P_{t}(y))=0.$

Differentiating gives

$\omega(P_{t}(x),P^{\prime}_{t}(y))=-\omega(P^{\prime}_{t}(x),P_{t}(y))=\overline{\omega(P_{t}(y),P^{\prime}_{t}(x))}.$

Taking $t=t^{*}$ and $x,y\in\ell_{1}(t^{*})\cap\ell_{2}(t^{*})$, so that
$P_{t^{*}}(x)=x$ and $P_{t^{*}}(y)=y$ gives

$\forall
x,y\in\ell_{1}(t^{*})\cap\ell_{2}(t^{*}),\quad\omega(x,P_{t}^{\prime}(y))=\overline{\omega(y,P_{t}^{\prime}(x))}.$

A similar equality holds for $P_{t}=P_{2}(t)$, which proves that $b$ is
hermitian. ∎

In particular, all eigenvalues of $b$ are real-valued. We say that $t^{*}$ is
a regular crossing if $\ell_{1}(t^{*})\cap\ell_{2}(t^{*})$ is finite
dimensional (say of dimension $k\in{\mathbb{N}}$), and if all its eigenvalues
$(\mu_{1},\cdots,\mu_{k})$ are non null (so the corresponding quadratic form
is non degenerate). For such crossings, we set

${\rm deg}(t^{*})=\sum_{j=1}^{k}{\rm sgn}\left(\mu_{j}\right).$

The pair $(\ell_{1},\ell_{2})$ is regular if all crossings are regular. For
such pair, the Maslov index is defined by

$\boxed{{\rm Mas}\left(\ell_{1},\ell_{2},{\mathbb{T}}^{1}\right):=\sum_{t^{*}\
\text{regular crossing}}{\rm deg}(t^{*})\quad\in{\mathbb{Z}}.}$

It is clear from the definition that ${\rm
Mas}(\ell_{1},\ell_{2},{\mathbb{T}}^{1})=-{\rm
Mas}(\ell_{2},\ell_{1},{\mathbb{T}}^{1})$. This definition does not require
Assumptions A (nor B).

####  3.1.2. Definition with the unitaries $U$

In the case where Assumption $A$ holds, we can relate the Maslov index to a
spectral flow. Consider two continuously differentiable loops of Lagrangian
$\ell_{1}(t)$ and $\ell_{2}(t)$ from $t\in{\mathbb{T}}^{1}$ to
$\Lambda({\mathcal{H}}_{b})$. Let $U_{1}(t)$ and $U_{2}(t)$ be the
corresponding unitaries from ${\rm Ker}(J-{\mathrm{i}})$ to ${\rm
Ker}(J+{\mathrm{i}})$. Then $U_{1}$ and $U_{2}$ are continuously
differentiable for the operator norm topology. From Lemma 9, we have that for
all $t\in{\mathbb{T}}^{1}$,

$\dim{\rm
Ker}\left(U_{2}^{*}U_{1}-{\mathbb{I}}_{\ker(J-{\mathrm{i}})}\right)=\dim\left(\ell_{1}(t)\cap\ell_{2}(t)\right).$

In particular, if all crossings are regular, then
$\dim(\ell_{1}\cap\ell_{2})={\rm Ker}(U_{2}^{*}U_{1}-1)$ is always finite
dimensional. Let $t^{*}\in{\mathbb{T}}^{1}$ be such that the kernel is non
empty, of dimension $k\in{\mathbb{N}}$. By usual perturbation theory for
operators [Kat13], we deduce that there are $k$ continuously differentiable
branches of eigenvalues of the unitary $U_{2}^{*}U_{1}$ crossing $1$ around
$t^{*}$. More specifically, we have the following.

######  Lemma 24.

Let $U(t)$ be a periodic continuously differentiable family of unitaries, and
let $t^{*}\in{\mathbb{T}}^{1}$ and $\varepsilon>0$ be such that, for all
$t\in(t^{*}-\varepsilon,t^{*}+\varepsilon)$,

$t\neq t^{*}\implies
1\notin\sigma\left(U(t)\right),\quad\text{while}\quad\dim{\rm
Ker}\left(U(t^{*})-1\right)=k\in{\mathbb{N}}.$

Then, there is $\delta>0$ and $k$ continuously differentiable functions
$\left(\theta_{j}(t)\right)$ from $t\in(t^{*}-\varepsilon,t^{*}+\varepsilon)$
to $\SS^{1}:=\\{z\in{\mathbb{C}},\ |z|=1\\}$, so that

$\sigma\left(U(t)\right)\cap
B(1,\eta)=\\{\theta_{1}(t),\cdots,\theta_{k}(t)\\}\cap B(1,\eta).$

The functions $\theta_{j}$ are the branches of eigenvalues of $U$. We say that
$t^{*}$ is a regular crossing if $k:=\dim{\rm Ker}(U(t)-1)<\infty$, and if
$\theta_{j}^{\prime}(t^{*})\neq 0$ for all $1\leq j\leq k$. Note that since
$\theta_{j}$ has values in $\SS^{1}$, we have
$\theta_{j}^{\prime}(t^{*})\in{\mathrm{i}}{\mathbb{R}}$. The degree of $t^{*}$
is

${\rm deg}(t^{*}):=\sum_{j=1}^{k}{\rm
sgn}\left(-{\mathrm{i}}\theta^{\prime}_{j}(t^{*})\right).$

This is the net number of eigenvalues crossing $1$ in $\SS^{1}$ in the
positive (counter-clockwise) direction. Finally, if all crossing are regular,
the spectral flow of $U$ across $1$ is

$\boxed{{\rm Sf}\left(U,1,{\mathbb{T}}^{1}\right):=\sum_{t^{*}\ \text{regular
crossing}}{\rm deg(t^{*})}\quad\in{\mathbb{Z}}.}$

######  Lemma 25.

Let $\ell_{1}(t)$ and $\ell_{2}(t)$ be two continuously differentiable
families of Lagrangians in $\Lambda({\mathcal{H}}_{b})$, and let $U_{1}(t)$
and $U_{2}(t)$ be the corresponding unitaries. Then $t^{*}\in{\mathbb{T}}^{1}$
is a regular crossing of $(\ell_{1},\ell_{2})$ iff it is a regular crossing of
$U_{2}^{*}U_{1}$. If all crossings are regular, then,

${\rm Mas}\left(\ell_{1},\ell_{2},{\mathbb{T}}_{1}\right)={\rm
Sf}\left(U_{2}^{*}U_{1},1,{\mathbb{T}}^{1}\right).$

###### Proof.

For the sake of simplicity, we assume that only $\ell_{1}$ depends on $t$. The
proof is similar in the general case.

Let $t^{*}$ be a regular crossing point, and let

$k:=\dim\left(\ell_{1}(t^{*})\cap\ell_{2}\right)=\dim{\rm
Ker}(U_{2}^{*}(t^{*})U_{1}(t^{*})-1).$

Let $\theta_{1},\theta_{2},\cdots\theta_{k}$ be the branches of eigenvalues
crossing $1$ at $t=t^{*}$, as in Lemma 24, and let
$x_{1}^{-}(t),\cdots,x_{k}^{-}(t)$ be a corresponding continuously
differentiable set of orthonormal eigenfunctions in ${\rm
Ker}(J-{\mathrm{i}})$. First, we have, for all $1\leq i,j\leq k$, and all
$t\in(t^{*}-\varepsilon,t^{*}+\varepsilon)$,

$\langle
x_{i}^{-},\left[U_{2}^{*}U_{1}-\theta_{j}\right]x_{j}^{-}\rangle_{{\mathcal{H}}_{b}}=0.$

Differentiating and evaluating at $t=t^{*}$ shows that

$\langle
x_{i}^{-},\partial_{t}\left[U_{2}^{*}U_{1}-\theta_{j}\right]x_{j}^{-}\rangle_{{\mathcal{H}}_{b}}+\langle\left[U_{2}^{*}(t^{*})U_{1}-1\right]^{*}x_{i},(\partial_{t}x_{j}^{-})\rangle_{{\mathcal{H}}_{b}}=0.$

At $t=t^{*}$, we have $U_{2}^{*}U_{1}x_{i}^{-}=x_{i}^{-}$, so
$U_{1}^{*}U_{2}x_{i}^{-}=x_{i}^{-}$ as well, and the last term vanishes. We
get the Hellman-Feynman equation

$\delta_{ij}\theta_{j}^{\prime}(t^{*})=\langle
U_{2}x_{i}^{-},(\partial_{t}U_{1})x_{j}^{-}\rangle_{{\mathcal{H}}_{b}}\big{|}_{t=t^{*}}=\langle
U_{1}x_{i}^{-},(\partial_{t}U_{1})x_{j}^{-}\rangle_{{\mathcal{H}}_{b}}\big{|}_{t=t^{*}}.$

On the other hand, we set $x_{j}(t):=x_{j}^{-}(t)+U_{1}(t)x_{j}^{-}(t)$ in
$\ell_{1}$. Since $x_{j}\in\ell_{1}$ for all $t$, we have $P_{1}x_{j}=x_{j}$
for all $t$. Differentiating gives

$\left(\partial_{t}P_{1}\right)x_{j}+P_{1}\left(\partial_{t}x_{j}\right)=(\partial_{t}x_{j}).$

Since $P_{1}\left(\partial_{t}x_{j}\right)\in\ell_{1}$, which is Lagrangian,
we obtain

$\omega\left(x_{i},\left(\partial_{t}P_{1}\right)x_{j}\right)=\omega(x_{i},\partial_{t}x_{j}).$
(9)

In addition, we have from $x_{j}(t):=x_{j}^{-}(t)+U_{1}(t)x_{j}^{-}(t)$ that

$\partial_{t}x_{j}=\left[1+U_{1}\right](\partial_{t}x_{j}^{-})+\left(\partial_{t}U_{1}\right)x_{j}^{-}.$

Since $x_{j}^{-}\in{\rm Ker}(J-{\mathrm{i}})$ for all $t$, we have
$(\partial_{t}x_{j}^{-})\in{\rm Ker}(J-{\mathrm{i}})$ as well, and the first
term is in $\ell_{1}$. On the other hand, $(\partial_{t}U_{1})x_{j}^{-}$ is in
${\rm Ker}(J+{\mathrm{i}})$. Together with (4), this gives

$\omega\left(x_{i},\left(\partial_{t}P_{1}\right)x_{j}\right)=\omega(x_{i},\partial_{t}x_{j})=\omega(x_{i},\left(\partial_{t}U_{1}\right)x_{j}^{-})=\omega(U_{1}x_{i}^{-},\left(\partial_{t}U_{1}\right)x_{j}^{-}).$

Using that $\omega(x,y)=\langle x,Jy\rangle_{{\mathcal{H}}_{b}}$ and that
$(\partial_{t}U_{1})x_{j}^{-}\in{\rm Ker}(J+{\mathrm{i}})$, we obtain, at
$t=t^{*}$, and recalling the definition of $b$ in (8),

$b\left(x_{i},x_{j}\right)=-{\mathrm{i}}\langle
U_{1}x_{i}^{-},(\partial_{t}U_{1})x_{j}^{-}\rangle_{{\mathcal{H}}_{b}}=\delta_{ij}(-{\mathrm{i}})\theta^{\prime}_{j}(t_{*}).$

The sesquilinear form $b$ is therefore diagonal in the $(x_{1},\cdots,x_{k})$
basis, with corresponding eigenvalues $(-{\mathrm{i}}\theta_{j}^{\prime})$.
The proof follows. ∎

####  3.1.3. Definition with the unitary ${\mathcal{U}}$.

In the case where the stronger Assumption B in (5) is also satisfies, one has
a similar result with the unitaries ${\mathcal{U}}$ instead of $U$. We state
it without proof, as it is similar to the previous one.

######  Lemma 26.

Assume $({\mathcal{H}}_{b}={\mathcal{H}}_{1}\times{\mathcal{H}}_{2},\omega)$
satisfies Assumption 5. Let $\ell_{1}(t)$ and $\ell_{2}(t)$ be two
continuously differentiable families of Lagrangians in
$\Lambda({\mathcal{H}}_{b})$, and let ${\mathcal{U}}_{1}(t)$ and
${\mathcal{U}}_{2}(t)$ be the corresponding unitaries of ${\mathcal{H}}_{1}$.
Then $t^{*}\in{\mathbb{T}}^{1}$ is a regular crossing of $(\ell_{1},\ell_{2})$
it is a regular crossing of ${\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1}$. If all
crossings are regular, then,

${\rm Mas}\left(\ell_{1},\ell_{2},{\mathbb{T}}_{1}\right)={\rm
Sf}\left({\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1},1,{\mathbb{T}}^{1}\right).$

The importance of this lemma comes from the fact that, in the finite
dimensional case (${\mathcal{H}}_{1}\approx{\mathbb{C}}^{n}$), the spectral
flow of a periodic family ${\mathcal{U}}(t)\in{\rm U}(n)$ across $1$ (or any
other point in $\SS^{1}$) equals the winding number of $\det{\mathcal{U}}(t)$:

${\rm Sf}\left({\mathcal{U}},z\in\SS^{1},{\mathbb{T}}^{1}\right)={\rm
Winding}\left(\det({\mathcal{U}}),{\mathbb{T}}^{1}\right).$

In our case where ${\mathcal{U}}={\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1}$, we
have
$\det({\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1})=\det({\mathcal{U}}_{1})/\det({\mathcal{U}}_{2})$,
hence

${\rm
Winding}\left(\det({\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1}),{\mathbb{T}}^{1}\right)={\rm
Winding}\left({\mathcal{U}}_{1},{\mathbb{T}}^{1}\right)-{\rm
Winding}\left({\mathcal{U}}_{2},{\mathbb{T}}^{1}\right).$

We obtain the following result.

######  Lemma 27.

In the case where ${\mathcal{H}}_{b}={\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$
with its canonical symplectic form, we have

${\rm
Mas}\left(\ell_{1},\ell_{2},{\mathbb{T}}^{1}\right)={\mathcal{I}}(\ell_{1},{\mathbb{T}}^{1})-{\mathcal{I}}(\ell_{2},{\mathbb{T}}^{1}),$

where, for a periodic continuous family $\ell(t)$ in
$\Lambda({\mathcal{H}}_{b})$, with corresponding unitary
${\mathcal{U}}(t)\in{\rm U}(n)$, we set

$\boxed{{\mathcal{I}}(\ell,{\mathbb{T}}^{1}):={\rm
Winding}\left(\det{\mathcal{U}},{\mathbb{T}}^{1}\right)\quad\in{\mathbb{Z}}.}$

In the infinite dimensional case, such a splitting is not valid. This gives
some complications (see Section 4). In order to have a unified definition with
the next Section, we recall that, in Example 15, we proved that Dirichlet
boundary conditions corresponds to the Lagrangian plane
$\ell_{D}=-{\mathbb{I}}_{d}$. So, setting
$\ell_{D}:=\\{0\\}\times{\mathbb{C}}^{n}$, we have the alternative definition

${\mathcal{I}}(\ell,{\mathbb{T}}^{1})={\rm
Mas}(\ell,\ell_{D},{\mathbb{T}}^{1})={\rm
Sf}\left({\mathcal{U}},-1,{\mathbb{T}}^{1}\right).$

###  3.2. Families of Hill’s operators, spectral flow

We now focus on a periodic family of Hill’s operator $h_{t}$. Let
${\mathbb{T}}^{1}\mapsto V_{t}$ be a periodic family of potentials satisfying
(1), and set

$h_{t}:=-\partial_{xx}^{2}+V_{t}(x).$

We assume that $t\mapsto V_{t}$ is continuously differentiable as a map from
${\mathbb{T}}^{1}$ to the Banach space
$L^{\infty}({\mathbb{R}},{\mathcal{S}}_{n})$. Since ${\mathbb{T}}^{1}$ is
compact, $V(t,x)$ is uniformly bounded. In particular, as in Section 2.2.2,
the operator $h_{t}$ is essentially self-adjoint with fixed domain
${\mathcal{D}}=H^{2}({\mathbb{R}}^{+},{\mathbb{C}}^{n})$ for all
$t\in{\mathbb{T}}^{1}$.

The spectrum of the family $(h_{t})_{t\in{\mathbb{T}}^{1}}$ is the set

$\sigma\left(h_{t},{\mathbb{T}}^{1}\right):=\bigcup_{t\in{\mathbb{T}}^{1}}\sigma(h_{t}).$

It is the compact union of all spectra of $(h_{t})$ for
$t\in{\mathbb{T}}^{1}$. Since $t\mapsto\sigma(h_{t})$ is continuous,
$\sigma\left(h_{t},{\mathbb{T}}^{1}\right)$ is a closed set in ${\mathbb{R}}$.
The complement of $\sigma\left(h_{t},{\mathbb{T}}^{1}\right)$ is the resolvent
set of the family $(h_{t})_{t\in{\mathbb{T}}^{1}}$.

We now consider a corresponding family of edge self-adjoint operators, of the
form $(h_{t}^{\sharp},\ell_{t}^{\sharp})$. We say that this family is
continuous, continuously differentiable, etc. if the corresponding family of
Lagrangian planes $\ell_{t}^{\sharp}$ is (in $\Lambda({\mathcal{H}}_{b})$).

Fix $E\in{\mathbb{R}}$ in the resolvent set of
$(h_{t})_{t\in{\mathbb{T}}^{1}}$. As $t$ varies in ${\mathbb{T}}^{1}$, the
spectrum of the bulk operator $h_{t}$ stays away from $E$. However, for the
edge operators $(h^{\sharp}_{t},\ell_{t}^{\sharp})$, some eigenvalues may
touch the energy $E$. If $t^{*}\in{\mathbb{T}}^{1}$ is such that $\dim{\rm
Ker}\left(h^{\sharp}_{t^{*}}-E\right)=k\in{\mathbb{N}}$, then, as in Lemma 24,
we can find $\varepsilon>0$, $\eta>0$ and $k$ continuously differentiable
branches of eigenvalues $\lambda_{j}(t)\in{\mathbb{R}}$ so that, for
$t\in(t^{*}-\varepsilon,t^{*}+\varepsilon)$,

$\sigma\left(h_{t}^{\sharp}\right)\cap
B(E,\eta)=\\{\lambda_{1}(t),\cdots,\lambda_{k}(t)\\}\cap B(E,\eta).$

At $t=t^{*}$, we have $\lambda_{1}(t^{*})=\cdots=\lambda_{k}(t^{*})=E$. The
crossing $t^{*}$ is regular if $\lambda_{j}^{\prime}(t^{*})\neq 0$ for all
$1\leq j\leq k$. For such a crossing, we set

${\rm deg}(t^{*})=\sum_{j=1}^{k}{\rm
sgn}\left(\lambda_{j}^{\prime}(t^{*})\right).$

We say that the energy $E$ is a regular energy if all crossings at $E$ are
regular. For such an energy, we define the spectral flow of $(h_{t}^{\sharp})$
across $E$ as the net number of eigenvalues crossing $E$ downwards:

$\boxed{{\rm Sf}(h^{\sharp}_{t},E,{\mathbb{T}}^{1}):=-\sum_{t^{*}\text{regular
crossing}}{\rm deg}(t^{*})\quad\in{\mathbb{Z}}.}$

The main result of this section is the following.

######  Theorem 28.

Let $(a,b)\subset{\mathbb{R}}$ be any interval in
${\mathbb{R}}\setminus\sigma\left(h_{t},{\mathbb{T}}_{1}\right)$. Then

  * •

almost all $E$ in $(a,b)$ is a regular energy for
$(h_{t}^{\sharp},\ell_{t}^{\sharp})$;

  * •

for any regular energy $E$ in $(a,b)$, we have

${\rm Sf}(h^{\sharp}_{t},E,{\mathbb{T}}^{1})={\rm
Mas}\left(\ell_{t}^{+}(E),\ell_{t}^{\sharp},{\mathbb{T}}_{1}\right)={\mathcal{I}}(\ell_{t}^{+}(E),{\mathbb{T}}^{1})-{\mathcal{I}}(\ell_{t}^{\sharp},{\mathbb{T}}^{1}).$

###### Proof.

The first part comes from Sard’s lemma, and can be proved as in [Gon20, Lemma
III.18].

Fix $E$ is regular energy, let $t^{*}$ be a crossing point so that $\dim{\rm
Ker}\left(h_{t^{*}}^{\sharp}-E\right)=k\in{\mathbb{N}}$, and let
$\lambda_{1},\cdots,\lambda_{k}$ be the corresponding branches of eigenvalues.
The idea of the proof is to follow the two families of branches
$(t,\lambda_{j}(t))$ and $(t,E)$, describing respectively $\ell^{\sharp}_{t}$
and $\ell^{+}_{t}(E)$.

For the first branch, let $\psi_{1}(t),\cdots,\psi_{k}(t)$ be a continuously
differentiable family or ${\mathcal{H}}^{\sharp}$-orthonormal eigenvectors in
${\mathcal{D}}^{\sharp}_{t}:={\rm Tr}^{-1}\left(\ell_{t}^{\sharp}\right)$, so
that $h_{t}^{\sharp}\psi_{j}(t)=\lambda_{k}(t)\psi_{j}(t)$, and let

$u_{j}:={\rm Tr}\left(\psi_{j}\right),\quad\text{so
that}\quad\ell^{+}_{t^{*}}(E)\cap\ell^{\sharp}_{t^{*}}={\rm
Vect}\left(u_{1}(t^{*}),\cdots,u_{k}(t^{*})\right).$

We have, for all $t\in(t^{*}-\varepsilon,t^{*}+\varepsilon)$, that (recall
that $h_{t}^{*}\subset h_{t,\rm max}^{*}$)

$\langle\psi_{i}(t),(h^{\sharp}_{t,{\rm
max}}-\lambda_{j}(t))\psi_{j}(t)\rangle_{{\mathcal{H}}^{\sharp}}=0.$

Differentiating the first equation and evaluating at $t=t^{*}$ gives

$\langle\psi_{i},\partial_{t}(h^{\sharp}_{t,{\rm
max}}-\lambda_{j})\psi_{j}\rangle_{{\mathcal{H}}^{\sharp}}\big{|}_{t=t^{*}}+\langle\psi_{i},(h^{\sharp}_{t^{*},{\rm
max}}-E)\left(\partial_{t}\psi_{j}\right)\rangle_{{\mathcal{H}}^{\sharp}}\big{|}_{t=t^{*}}=0$

For the last term, we put the operator $(h_{t^{*},{\rm max}}^{\sharp}-E)$ on
the other side using the second Green’s identity. Since
$(h^{\sharp}_{t^{*},{\rm max}}-E)\psi_{i}=0$. we get

$\langle\psi_{i},(h^{\sharp}_{t^{*},{\rm
max}}-E)\left(\partial_{t}\psi_{j}\right)\rangle_{{\mathcal{H}}^{\sharp}}\big{|}_{t=t^{*}}=\omega\left(u_{i},\partial_{t}u_{j}\right)\big{|}_{t=t^{*}}=\omega\left(u_{i},\left(\partial_{t}P^{\sharp}_{t^{*}}\right)u_{j}\right)\big{|}_{t=t^{*}}.$

For the last equality, we introduced $P^{\sharp}_{t}$ the projection on
$\ell_{t}^{\sharp}$, and used an equality similar to (9). This gives our first
identity

$\delta_{ij}\lambda_{j}^{\prime}(t^{*})=\langle\psi_{i},\partial_{t}(h^{\sharp}_{t,{\rm
max}})\psi_{j}\rangle_{{\mathcal{H}}^{\sharp}}\big{|}_{t=t^{*}}+\omega\left(u_{i},\partial_{t}\left(P^{\sharp}_{t}\right)u_{j}\right)\big{|}_{t=t^{*}}.$

######  Remark 29.

In the case where the domain $\ell^{\sharp}_{t}=\ell^{\sharp}$ is independent
of $t$, we recover the Hellman-Feynman identity
$\langle\psi_{i},\partial_{t}(h^{\sharp}_{t}-\lambda_{j})\psi_{j}\rangle_{{\mathcal{H}}^{\sharp}}\big{|}_{t=t^{*}}=0$.

For the second branch, let $(\phi_{1}(t),\cdots,\phi_{k}(t))$ be a smooth
family of linearly independent functions in ${\mathcal{S}}^{+}(t,E)$, and so
that, at $t=t^{*}$,
$\phi_{j}(t^{*})={\mathds{1}}_{{\mathbb{R}}^{+}}\psi_{j}(t^{*})$. We set

$v_{j}={\rm Tr}(\phi_{j}),\quad\text{so, at $t=t^{*}$},\quad
v_{j}(t^{*})=u_{j}(t^{*}).$

This time, we have, for all $t\in(t^{*}-\varepsilon,t^{*}+\varepsilon)$,

$\left\langle\phi_{i},\left(h^{\sharp}_{t,\rm
max}-E\right)\phi_{j}\right\rangle=0.$

Differentiating and evaluating at $t=t^{*}$ gives as before

$\left\langle\phi_{i},\partial_{t}\left(h^{\sharp}_{t^{*},\rm
max}\right)\phi_{j}\right\rangle=-\omega(v_{i},\partial_{t}v_{j})=-\omega\left(v_{i},\left(\partial_{t}P_{t}^{+}\right)v_{j}\right)=-\omega\left(u_{i},\left(\partial_{t}P_{t}^{+}\right)u_{j}\right)$

Gathering the two identities shows that

$\delta_{ij}\lambda_{j}^{\prime}(t^{*})=-\omega\left(u_{i},\left(\partial_{t}P_{t}^{+}\right)u_{j}\right)\big{|}_{t=t^{*}}+\omega\left(u_{i},\partial_{t}\left(P^{\sharp}_{t}\right)u_{j}\right)\big{|}_{t=t^{*}}.$

We recognise the sesquilinear form $b$ defined in (8). This form is therefore
diagonal in the $(u_{1},\cdots,u_{j})$ basis, and its eigenvalues are the
$-\lambda_{j}^{\prime}(t^{*})$. The result follows. ∎

While our proofs are valid only for continuously differentiable families, they
can be extended to the continuous case. Indeed, it is well-known that the
spectral flow, the Maslov index, and the winding number only depends only on
the homotopy class of the corresponding families.

For instance, the spectral flow is independent of the energy $E$ in the gap
$(a,b)$. So all indices are independent of $E$ in the gap, and Theorem 28 also
holds for non-regular $E\in(a,b)$ (with the corresponding continuous versions
for the definitions of the indices).

###  3.3. Applications

Let us give two applications of the previous theory. The first one shows that
a spectral flow must appear when modifying Robin boundary conditions. The
second one concerns the important case of junctions between two materials.

####  3.3.1. Robin boundary conditions

In the case $n=1$, consider a fixed (independent of $t$) bounded potential
$V_{t}(x)=V(x)$. We consider the self-adjoint Robin operator
$h^{\sharp}_{t}=-\partial_{xx}^{2}+V$ on $L^{2}({\mathbb{R}}^{+})$, with the
$t$-dependent domain

${\mathcal{D}}_{t}:=\left\\{\psi\in H^{2}({\mathbb{R}}^{+}),\quad\cos(\pi
t)\psi(0)-\sin(\pi t)\psi^{\prime}(0)=0\right\\}.$

Note that ${\mathcal{D}}_{t+1}={\mathcal{D}}_{t}$, so $H_{t}^{\sharp}$ is
indeed periodic in $t$. For $t=0$, we recover Dirichlet boundary condition,
and for $t=\frac{1}{2}$, we recover Neumann boundary conditions, so Robin
boundary conditions somehow interpolates between these two cases. The
Lagrangian plane of ${\mathcal{H}}_{b}={\mathbb{C}}\times{\mathbb{C}}$
corresponding to the extension ${\mathcal{D}}_{t}$ is

$\ell_{t}^{\sharp}={\rm Vect}_{\mathbb{C}}\begin{pmatrix}\sin(\pi t)\\\
\cos(\pi t)\end{pmatrix}\subset{\mathbb{C}}\times{\mathbb{C}}.$

It is of the form $\ell_{t}^{\sharp}\\{(\Theta x,\Pi x),\ x\in{\mathbb{C}}\\}$
for $\Theta=\sin(\pi t)$ and $\Pi=\cos(\pi t)$. So, by the results of Example
16, the corresponding unitary ${\mathcal{U}}(t)\in{\rm U}(1)\approx\SS^{1}$,
is

${\mathcal{U}}(t)=\dfrac{\sin(\pi t)+{\mathrm{i}}\cos(\pi t)}{\sin(\pi
t)-{\mathrm{i}}\cos(\pi t)}={\mathrm{e}}^{2{\mathrm{i}}\pi t}.$

We see that ${\mathcal{U}}(t)$ winds once positively around $\SS^{1}$ as $t$
runs through ${\mathbb{T}}^{1}$, that is

${\mathcal{I}}(\ell_{t}^{\sharp},{\mathbb{T}}_{1})=1.$

Using Theorem 28, and the fact that $\ell^{+}_{t}(E)$ is independent of $t$,
we obtain

${\rm Sf}\left(h_{t}^{\sharp},E,{\mathbb{T}}^{1}\right)=-1.$

We deduce that there is a spectral flow of exactly $1$ eigenvalue going
upwards in all spectral gaps of $h$. This includes the lower gap
$(-\infty,\inf\sigma(h))$.

####  3.3.2. Junction between two materials.

We now consider a junction between a left and a right potentials $V_{L,t}$ and
$V_{R,t}$, where $t\mapsto V_{L,t}$ and $t\mapsto V_{R,t}$ are two periodic
continuously differentiable families of potentials in
$L^{\infty}({\mathbb{R}},{\mathcal{S}}_{n})$. Take $\chi$ a bounded switch
function, satisfying, for some $X>0$ large enough,

$\forall x\leq-X,\ \chi(x)=1,\quad\text{and}\quad\forall x\geq
X,\quad\chi(x)=0.$

We consider the domain wall Hill’s operator

$h^{\chi}_{t}:=-\partial_{xx}^{2}+V_{L,t}(x)\chi(x)+V_{R,t}(x)(1-\chi(x)).$

Let $E\in{\mathbb{R}}$ be in the resolvent set of the bulk operators $h_{R,t}$
and $h_{L,t}$ for all $t\in{\mathbb{T}}^{1}$. Again, some eigenvalues of
$h^{\chi}_{t}$ might cross $E$ as $t$ goes from $0$ to $1$, and we can define
a corresponding spectral flow ${\rm Sf}(h_{t}^{\chi},E,{\mathbb{T}}^{1})$.

######  Theorem 30 (Bulk-edge correspondence for junctions).

With the previous notation, let
$(a,b)\subset{\mathbb{R}}\setminus\\{\sigma\left(h_{R,t},{\mathbb{T}}^{1}\right)\cup\sigma\left(h_{L,t},{\mathbb{T}}^{1}\right)\\}$.
Then,

  * •

almost all $E\in(a,b)$ is a regular energy for $h_{t}^{\chi}$;

  * •

for any such regular energy, we have

${\rm Sf}(h_{t}^{\chi},E,{\mathbb{T}}^{1})={\rm
Mas}\left(\ell_{R,t}^{+}(E),\ell_{L,t}^{-}(E),{\mathbb{T}}^{1}\right)={\mathcal{I}}(\ell_{R,t}^{+}(E),{\mathbb{T}}^{1})-{\mathcal{I}}(\ell_{L,t}^{-}(E),{\mathbb{T}}^{1}).$

The right-hand side is independent of the switch $\chi$.

######  Remark 31.

Applying this theorem with $V_{L}=V_{R}$ shows that, for $E$ in a gap of all
operators $h_{t}$ (so that there is no spectral flow), we have

$0={\rm
Mas}\left(\ell_{t}^{+}(E),\ell_{t}^{-}(E),{\mathbb{T}}^{1}\right)={\mathcal{I}}(\ell_{t}^{+}(E),{\mathbb{T}}^{1})-{\mathcal{I}}(\ell_{t}^{-}(E),{\mathbb{T}}^{1}).$

So
${\mathcal{I}}(\ell_{t}^{+}(E),{\mathbb{T}}^{1})={\mathcal{I}}(\ell_{t}^{-}(E),{\mathbb{T}}^{1})$.
This means that we can drop the upper-script $\pm$ in Theorem 30. The common
integer ${\mathcal{I}}(\ell_{t}^{\pm}(E),{\mathbb{T}}^{1})$ is the usual
bulk/edge index.

###### Proof.

The following proof uses the theory developed in the previous sections. We
give a different proof in the Schrödinger case (see Theorem 40 below).
Let us denote by $\ell^{\pm}_{\chi,t}(x_{0},E)$ the Lagrangian planes obtained
with the potential

$V^{\chi}_{t}(x):=V_{L,t}(x)\chi(x)+V_{R,t}(x)(1-\chi(x)),$

and when the real line ${\mathbb{R}}$ is cut at the location
$x_{0}\in{\mathbb{R}}$. By Lemma 19, we have

$\forall x_{0}\in{\mathbb{R}},\quad\dim{\rm
Ker}\left(h_{t}^{\chi}-E\right)=\dim\left(\ell_{\chi,t}^{+}(x_{0},E)\cap\ell_{\chi,t}^{-}(x_{0},E)\right).$

Adapting the proof of Theorem 28 shows that

$\displaystyle{\rm Sf}\left(h_{t}^{\chi},E,{\mathbb{T}}^{1}\right)$
$\displaystyle={\rm
Mas}\left(\ell^{+}_{\chi,t}(x_{0},E),\ell^{-}_{\chi,t}(x_{0},E),{\mathbb{T}}^{1}\right)$
$\displaystyle={\mathcal{I}}\left(\ell^{+}_{\chi,t}(x_{0},E),{\mathbb{T}}^{1}\right)-{\mathcal{I}}\left(\ell^{-}_{\chi,t}(x_{0},E),{\mathbb{T}}^{1}\right).$

Since $V$ is uniformly (hence locally) bounded, all Cauchy solutions to
$-\psi^{\prime\prime}+V\psi=E\psi$ are well defined and continuously
differentiable on the whole line ${\mathbb{R}}$. This implies that the maps
$x_{0}\mapsto\ell^{\pm}_{\chi,t}(x_{0},E)$ are also continuous. In particular,
since the index $M$ depends only on the homotopy class of the loops, we have

$M\left(\ell^{+}_{\chi,t}(x_{0},E),{\mathbb{T}}^{1}\right)=M\left(\ell^{+}_{\chi,t}(x_{0}=X,E),{\mathbb{T}}^{1}\right).$

The last quantity only involves the half space $\\{x\geq X\\}$. On this half
space, we have $V^{\chi}_{t}(x)=V_{R,t}(x)$, so

$M\left(\ell^{+}_{\chi,t}(x_{0}=X,E),{\mathbb{T}}^{1}\right)=M\left(\ell^{+}_{R,t}(X,E),{\mathbb{T}}^{1}\right)=M\left(\ell^{+}_{R,t}(E),{\mathbb{T}}^{1}\right).$

The proof that
$M\left(\ell^{-}_{\chi,t}(x_{0},E),{\mathbb{T}}^{1}\right)=M\left(\ell^{-}_{L,t}(E),{\mathbb{T}}^{1}\right)$
is similar, and the result follows. ∎

##  4\. The infinite dimensional case

In this section, we explain how to modify the previous proofs to handle the
infinite dimensional setting, by which we mean the PDE Schrödinger case. We
chose to put this section separately, since it introduces some technical
details, and since the results are slightly different.

###  4.1. Schrödinger operators on a tube

We now consider systems defined on a $d$-dimensional cylinder of the form

$\Omega:={\mathbb{R}}\times\Gamma\subset{\mathbb{R}}^{d}.$

Here and in what follows, $\Gamma=(0,1)^{d-1}$ is the $(d-1)$-dimensional unit
open square. A point in $\Omega$ is denoted by ${\mathbf{x}}=(x,{\mathbf{y}})$
with $x\in{\mathbb{R}}$ and ${\mathbf{y}}\in\Gamma$.

Let $V:\Omega\to{\mathbb{R}}$ be a real-valued potential, which we assume to
be bounded on $\Omega$. We consider bulk Schrödinger operator $H$ of the form

$H:=-\Delta+V,\quad\text{acting
on}\quad{\mathcal{H}}:=L^{2}(\Omega,{\mathbb{C}}).$

The case of general second order elliptic operators, and with values in
${\mathbb{C}}^{n}$, can be treated similarly. Again, we do not assume here
that $V$ is periodic, but only that $V$ is bounded.

The operator $H$ with core domain $C^{\infty}_{0}(\Omega)$ is symmetric, and
we have

${\mathcal{D}}_{\rm
min}=H^{2}_{0}(\Omega,{\mathbb{C}}),\quad\text{and}\quad{\mathcal{D}}_{\rm
max}=H^{2}(\Omega,{\mathbb{C}}).$

This time, the bulk operator is not self-adjoint, and indeed, boundary
conditions must be chosen on the boundary
$\partial\Omega={\mathbb{R}}\times\partial\Gamma$.

####  4.1.1. The bulk Schrödinger operators

Let us explain in detail how we choose the boundary conditions on
$\partial\Omega={\mathbb{R}}\times\partial\Gamma$ for the bulk operator.
First, we see $V$ as a perturbation of $-\Delta$, so, once the domain of
$-\Delta$ has been chosen, the Schrödinger operator $-\Delta+V$ will have the
same domain (see for instance [Kat13, Chapter 5.4.3]).

We have $\Omega={\mathbb{R}}\times\Gamma$, so

${\mathcal{H}}:=L^{2}(\Omega)=L^{2}({\mathbb{R}})\otimes L^{2}(\Gamma),$

and, with respect to this decomposition,

$-\Delta=\left(-\partial_{xx}^{2}\right)\otimes\left(-\Delta_{\Gamma}\right).$

The operator $-\partial_{xx}^{2}$ is essentially self-adjoint on
$L^{2}({\mathbb{R}})$, so $-\Delta$ is self-adjoint on $L^{2}(\Omega)$
whenever $-\Delta_{\Gamma}$ is so on $L^{2}(\Gamma)$. In what follows, we fix
one such extension. We consider here the periodic realisation of
$-\Delta_{\Gamma}$, but the proofs are similar for other choices, such as the
Dirichlet extension, the Neumann one, or the quasi-periodic one. So we see
$\Gamma=(0,1)^{d-1}\approx{\mathbb{T}}^{d-1}$ as the $(d-1)$-torus, and we
consider the domain

${\mathcal{D}}_{\Gamma,{\rm per}}:=H^{2}({\mathbb{T}}^{d-1}).$

The operator $\left(-\Delta_{\Gamma,{\rm per}},{\mathcal{D}}_{\Gamma,{\rm
per}}\right)$ is self-adjoint. It is compact resolvent, diagonal in the
Fourier basis, and admits a spectral decomposition of the form

$-\Delta_{\Gamma,{\rm
per}}:=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}4\pi|{\mathbf{k}}|^{2}|e_{\mathbf{k}}\rangle\langle
e_{\mathbf{k}}|,\quad\text{with}\quad
e_{\mathbf{k}}({\mathbf{y}}):={\mathrm{e}}^{{\mathrm{i}}2\pi{\mathbf{k}}\cdot{\mathbf{y}}}.$

The corresponding operator $-\Delta_{\rm
per}:=(-\partial_{xx}^{2})\otimes(-\Delta_{\Gamma,{\rm per}})$ is self-adjoint
on $L^{2}(\Omega)$, with domain

${\mathcal{D}}_{\rm
per}:=\overline{H^{2}({\mathbb{R}}){\otimes}{\mathcal{D}}_{\Gamma,{\rm
per}}}^{{\mathcal{D}}_{\rm max}}.$

The subscript ”${\rm per}$” only refers to periodicity in the last $(d-1)$
dimensions. The elements in ${\mathcal{H}}$ can be written in the partial
Fourier form

$f(x,{\mathbf{y}})=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}f_{\mathbf{k}}(x)e_{\mathbf{k}}({\mathbf{y}}),\quad\text{with}\quad\|f\|_{{\mathcal{H}}}^{2}:=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\|f_{\mathbf{k}}\|_{L^{2}({\mathbb{R}})}^{2}<\infty.$
(10)

We have $f\in{\mathcal{D}}_{\rm per}$ if $\|(-\Delta_{{\rm
per}})f\|_{{\mathcal{H}}}<\infty$ as well, where

$\|(-\Delta_{\rm
per})f\|_{{\mathcal{H}}}^{2}=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\left(\|f_{\mathbf{k}}\|_{L^{2}({\mathbb{R}})}^{2}+(4\pi|{\mathbf{k}}|^{2})^{2}\|f_{\mathbf{k}}\|_{L^{2}({\mathbb{R}})}^{2}\right)<\infty.$

####  4.1.2. Edge Schrödinger operators on a tube

We now define the edge Schrödinger operator

$H^{\sharp}:=-\Delta+V\quad\text{acting on}\quad
L^{2}(\Omega^{+},{\mathbb{C}}),\quad\text{where}\quad\Omega^{+}:={\mathbb{R}}^{+}\times\Gamma.$

This operator acts on the right half tube. We sometime write $H^{\sharp,+}$
for $H^{\sharp}$ and define $H^{\sharp,-}$ for the corresponding operator on
the left half tube $\Omega^{-}:={\mathbb{R}}^{-}\times\Gamma$. The operator
$H^{\sharp}$ with core domain $C^{\infty}_{0}(\Omega^{+})$ is symmetric, and
we have,

${\mathcal{D}}_{\rm
min}^{\sharp}=H^{2}_{0}(\Omega^{+},{\mathbb{R}}),\quad\text{and}\quad{\mathcal{D}}_{\rm
max}^{\sharp}=H^{2}(\Omega^{+},{\mathbb{R}}).$

Again, we need to specify the boundary conditions. On the boundary
${\mathbb{R}}^{+}\times\partial\Gamma$, we keep the same boundary conditions
as for the bulk one. In other words, writing

${\mathcal{H}}^{\sharp}:=L^{2}(\Omega^{+})=L^{2}({\mathbb{R}}^{+})\otimes
L^{2}(\Gamma),$

and setting

$-\Delta^{\sharp}_{{\rm per},\rm min}:=(-\partial_{xx}^{2})^{\sharp}_{\rm
min}\otimes\left(-\Delta_{\Gamma,{\rm
per}}\right)\quad\text{and}\quad-\Delta^{\sharp}_{{\rm per},\rm
max}:=(-\partial_{xx}^{2})^{\sharp}_{\rm max}\otimes\left(-\Delta_{\Gamma,{\rm
per}}\right),$

we only look for self-adjoint extensions $(-\Delta^{\sharp})$ of the Laplacian
satisfying

$-\Delta^{\sharp}_{{\rm per},\rm
min}\subset-\Delta^{\sharp}\subset-\Delta^{\sharp}_{{\rm per},\rm max}.$

We are only looking at the self-adjoint domains contained in
${\mathcal{D}}_{{\rm per},{\rm max}}^{\sharp}$. We identify this last domain
in the next Lemma (compare with Lemma 18).

######  Lemma 32.

A function is in ${\mathcal{D}}_{{\rm per},{\rm max}}^{\sharp}$ iff it is the
restriction to $\Omega^{+}$ of an element in the bulk domain
${\mathcal{D}}_{\rm per}$.

###### Proof.

This follows from the fact that there is an extension operator
$H^{2}(\Omega^{+})\to H^{2}(\Omega)$ which can be constructed with reflection
operators, see e.g. [GT15, Theorem 7.25] or [LM12, Theorem 8.1]. These
reflection operators keep the periodic properties at the boundary
${\mathbb{R}}\times\partial\Gamma$. ∎

In our case of periodic boundary conditions, we obtain

${\mathcal{D}}^{\sharp}_{{\rm per},{\rm
min}}=H^{2}_{0}({\mathbb{R}}^{+}\times{\mathbb{T}}^{d-1},{\mathbb{C}}),\quad\text{and}\quad{\mathcal{D}}^{\sharp}_{{\rm
per},{\rm max}}=H^{2}({\mathbb{R}}^{+}\times{\mathbb{T}}^{d-1},{\mathbb{C}}).$

###  4.2. Trace maps, and the boundary space ${\mathcal{H}}_{b}^{\prime}$

In order to express the second Green’s identity in this setting, we recall
some basic facts on the Dirichlet and Neumann trace operators.

####  4.2.1. Boundary Sobolev-like spaces

We will need the boundary Hilbert spaces $H^{s}_{\rm per}(\Gamma)$. To define
them, we note that $(-\Delta_{\Gamma,{\rm per}})$ is positive. For $0\leq
s\leq 2$, and for functions $f,g\in{\mathcal{D}}_{\Gamma,{\rm per}}$, we set

$\langle f,g\rangle_{H^{s}_{\rm per}(\Gamma)}^{2}=\frac{1}{2}\langle
f,g\rangle_{L^{2}(\Gamma)}^{2}+\frac{1}{2}\left\langle(-\Delta_{\Gamma,{\rm
per}})^{s/2}f,(-\Delta_{\Gamma,{\rm
per}})^{s/2}g\right\rangle_{L^{2}(\Gamma)}.$

The Hilbert space $H^{s}_{\rm per}(\Gamma)$ is the completion of
${\mathcal{D}}_{\Gamma,{\rm per}}$ for the corresponding norm. The
$\frac{1}{2}$ factor is here so that $H^{s}_{\rm per}(\Gamma)=L^{2}(\Gamma)$.
In terms of Fourier coefficients, we have

$\langle
f,g\rangle_{H^{s}(\Gamma)}=\frac{1}{2}\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\overline{f_{\mathbf{k}}}g_{\mathbf{k}}\left(1+(4\pi|{\mathbf{k}}|^{2})^{s}\right).$

Finally, for $0\leq s\leq 2$, we define $H^{-s}_{\rm per}(\Gamma)$ as the dual
of $H^{s}_{\rm per}(\Gamma)$ for the $L^{2}(\Gamma)$-inner product. It is also
an Hilbert space with corresponding inner product

$\langle f,g\rangle_{H^{-s}_{\rm
per}(\Gamma)}=\frac{1}{2}\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\overline{f_{\mathbf{k}}}g_{\mathbf{k}}\left(1+(4\pi|{\mathbf{k}}|^{2})^{s}\right)^{-1}.$

It is classical that, for $-2\leq s^{\prime}\leq s\leq 2$, we have
$H^{s}_{{\rm per}}(\Gamma)\hookrightarrow H^{s^{\prime}}_{\rm per}(\Gamma)$
with compact embedding, and that $H^{s}_{{\rm per}}(\Gamma)$ is dense in
$H^{s^{\prime}}_{\rm per}(\Gamma)$. For $s<0$, the subscript ”${\rm per}$” can
be dropped, since it equals the usual $H^{s}(\Gamma)$ Sobolev space.

######  Remark 33.

Should we choose the Dirichlet Laplacian $-\Delta_{\Gamma,D}$, instead of the
periodic ones, we would obtain the usual $H^{s}_{0}(\Gamma)$ Sobolev spaces
for $s>0$, and the subscript $0$ can be dropped for $s\leq 0$.

####  4.2.2. Dirichlet and Neumann trace operators

For $\psi\in C^{\infty}(\Omega^{+})$, we introduce the functions
$\gamma^{D}\psi$ and $\gamma^{N}\psi$ defined on $\Gamma$ by

$\forall{\mathbf{y}}\in\Gamma,\quad(\gamma^{D}\psi)({\mathbf{y}}):=\psi(x=0,{\mathbf{y}}),\quad\left(\gamma^{N}\psi\right)({\mathbf{y}})=\partial_{x}\psi(x=0,{\mathbf{y}}).$

Our definition differs from the usual one
$\gamma^{N}\psi=-\partial_{x}\psi(0,\cdot)$, where the minus sign comes the
outward normal direction of $\Gamma$ from the $\Omega^{+}$ perspective. Our
definition without the minus sign matches the one of the previous section.

It is classical that $\gamma^{D}$ can be extended from $H^{2}(\Omega^{+})$ to
$H^{3/2}(\Gamma)$ and that $\gamma^{N}$ can be extended from
$H^{2}(\Omega^{+})$ to $H^{1/2}(\Gamma)$, and that these extensions are onto
(see for instance [LM12, Theorem 8.3]). This suggests to introduce the
boundary space

${\mathcal{H}}_{b}:=H^{3/2}_{\rm per}(\Gamma)\times H^{1/2}_{\rm
per}(\Gamma),$

and the boundary trace operator

${\rm Tr}:{\mathcal{D}}_{{\rm per},{\rm
max}}^{\sharp}\to{\mathcal{H}}_{b},\quad{\rm
Tr}(\psi):=\left(\gamma^{D}\psi,\gamma^{N}\psi\right).$

The counterpart of Lemma 12 is the following.

######  Lemma 34.

The map ${\rm Tr}:\left({\mathcal{D}}_{{\rm per},{\rm
max}}^{\sharp},\|\cdot\|_{\sharp}\right)$ to ${\mathcal{H}}_{b}$ is well-
defined, continuous and onto.

In other terms, we do not loose the surjectivity of ${\rm Tr}$ by restricting
all domains to be periodic in the last $(d-1)$-direction.

###### Proof.

The fact that ${\rm Tr}$ is well-defined and continuous follows from the fact
that $\|\cdot\|_{\sharp}$ is equivalent to the usual $H^{2}(\Omega^{+})$ norm,
together with the continuity of the trace maps. Tos prove that ${\rm Tr}$ is
onto, one can adapt the proof of [LM12, Theorem 8.3]. We provide here an
alternative proof.

An element $f\in H^{s}_{\rm per}(\Gamma)$ is of the form

$f({\mathbf{y}})=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}f_{\mathbf{k}}e_{\mathbf{k}}({\mathbf{y}}),\quad\text{with}\quad\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\left(1+(4\pi|{\mathbf{k}}|^{2})^{s}\right)|f_{\mathbf{k}}|^{2}<\infty.$

Let $f\in H^{3/2}_{\rm per}(\Gamma)$ and $f^{\prime}\in H^{1/2}_{\rm
per}(\Gamma)$ with respective coefficients $(f_{\mathbf{k}})$ and
$(f_{\mathbf{k}}^{\prime})$. Consider also a smooth cut-off function $\chi(x)$
with $\chi(x)=1$ for $0\leq x<1/2$, $\chi(x)=0$ for $x>2$ and
$\int_{{\mathbb{R}}^{+}}\chi^{2}=1$. We set $\chi_{\mathbf{0}}=\chi$, and for
${\mathbf{k}}\in{\mathbb{Z}}^{d-1}\setminus\\{0\\}$

$\chi_{\mathbf{k}}(x):=\chi\left(x|{\mathbf{k}}|^{1/2}\right).$

For all ${\mathbf{k}}$, the function $\chi_{\mathbf{k}}$ is smooth, compactly
supported, with $\chi_{\mathbf{k}}(x)=1$ for all $x<1/2|{\mathbf{k}}|$. In
addition, we have the scalings

$\int_{R^{+}}|\chi_{\mathbf{k}}|^{2}=\dfrac{1}{|{\mathbf{k}}|},\quad\int_{R^{+}}|\chi_{\mathbf{k}}^{\prime}|^{2}=|{\mathbf{k}}|,\quad\int_{R^{+}}|\chi_{\mathbf{k}}^{\prime\prime}|^{2}=|{\mathbf{k}}|^{3}.$

Finally, we consider the function defined on $\Omega^{+}$ by

$\Psi(x,{\mathbf{y}}):=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\left(f_{\mathbf{k}}+xf_{\mathbf{k}}^{\prime}\right)\chi_{\mathbf{k}}(x)e_{\mathbf{k}}({\mathbf{y}}).$

The function $\Psi$ is compactly supported and, we check that ${\rm
Tr}(\Psi)=(f,f^{\prime})$. It remains to check that $\Psi$ is in
$H^{2}(\Omega^{+})$. We have

$\int_{\Omega^{+}}\|-\Delta\Psi\|_{L^{2}}^{2}\lesssim\int_{{\mathbb{R}}^{+}}\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}\left(|f_{\mathbf{k}}|^{2}|{\mathbf{k}}|^{4}|\chi_{\mathbf{k}}|^{2}+|f_{\mathbf{k}}|^{2}|\chi_{\mathbf{k}}^{\prime\prime}|^{2}+|f_{\mathbf{k}}^{\prime}|^{2}|\chi_{\mathbf{k}}^{\prime}|^{2}\right).$

Thanks to our scaling, we deduce that

$\int_{\Omega^{+}}\|-\Delta\Psi\|_{L^{2}}^{2}\lesssim\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}|f_{\mathbf{k}}|^{2}\cdot|{\mathbf{k}}|^{3}+\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}|f_{\mathbf{k}}^{\prime}|^{2}|{\mathbf{k}}|\lesssim\|f\|_{H^{3/2}_{\rm
per}}^{2}+\|f^{\prime}\|_{H^{1/2}_{\rm per}}^{2}.$

The result follows. ∎

We can now state the second Green’s identity in the PDE case.

######  Lemma 35 (Second Green’s formula).

For all $\phi,\psi\in{\mathcal{D}}^{\sharp}_{{\rm per},{\rm max}}$,

$\langle\phi,H^{\sharp}_{{\rm per},{\rm
max}}\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle H^{\sharp}_{{\rm per},{\rm
max}}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=\langle\gamma^{D}\phi,\gamma^{N}\psi\rangle_{L^{2}(\Gamma)}-\langle\gamma^{N}\phi,\gamma^{D}\psi\rangle_{L^{2}(\Gamma)}.$

Introducing the symplectic form $\omega$ on ${\mathcal{H}}_{b}$, defined by

$\forall(f,f^{\prime}),(g,g^{\prime})\in{\mathcal{H}}_{b},\quad\omega^{\prime}((f,f^{\prime}),(g,g^{\prime})):=\langle
f,g^{\prime}\rangle_{L^{2}(\Gamma)}-\langle
f^{\prime},g\rangle_{L^{2}(\Gamma)},$

the second Green’s identity simply reads

$\langle\phi,H^{\sharp}_{{\rm per},{\rm
max}}\psi\rangle_{{\mathcal{H}}^{\sharp}}-\langle H^{\sharp}_{{\rm per},{\rm
max}}\phi,\psi\rangle_{{\mathcal{H}}^{\sharp}}=\omega\left({\rm Tr}(\phi),{\rm
Tr}(\psi)\right).$

######  Remark 36.

The symplectic Hilbert space $({\mathcal{H}}_{b},\omega)$ does not satisfy
Assumption A in (2). Introducing the map $A:H^{1/2}_{\rm per}(\Gamma)\to
H^{3/2}_{\rm per}(\Gamma)$ so that

$\forall f\in H^{3/2}_{\rm per}(\Gamma),\ \forall g\in H^{1/2}_{\rm
per}(\Gamma),\quad\langle f,g\rangle_{L^{2}(\Gamma)}=\langle
f,Ag\rangle_{H^{3/2}_{\rm per}}=\langle A^{*}f,g\rangle_{H^{1/2}_{\rm per}},$

we have $J=\begin{pmatrix}0&A^{*}\\\ -A&0\end{pmatrix}$, but the operators $A$
and $A^{*}$ are compact (hence $J$ as well, and
$J^{2}\neq-{\mathbb{I}}_{{\mathcal{H}}_{b}}$). In particular, we cannot
consider the unitaries $U$ nor ${\mathcal{U}}$. Such situation was considered
in [BBZ13]. We present below in Appendix A a new approach which allows the use
of unitaries as in the previous section. The ”more natural” symplectic space
$H^{1/2}_{\rm per}(\Gamma)\times H^{-1/2}(\Gamma)$ appears at this point.

Now that our functional spaces and operators have been identified, all our
previous results for Hill’s operators (Section 2) hold in the Schrödinger
case. Indeed, these ones never used finite dimensional arguments. More
specifically, we have the following.

######  Theorem 37.

  * •

For ${\mathcal{D}}^{\sharp}\subset{\mathcal{D}}^{\sharp}_{{\rm per},{\rm
max}}$, the operator $(H^{\sharp},{\mathcal{D}}^{\sharp})$ is self-adjoint iff

${\mathcal{D}}^{\sharp}:={\rm Tr}^{-1}\left(\ell\right),\quad\text{for a
Lagrangian plane}\ \ell\in\Lambda\left({\mathcal{H}}_{b}\right).$

  * •

If $E\in{\mathbb{R}}\setminus\sigma(H_{\rm per})$, then the sets

$\ell^{\pm}(E):={\rm Tr}\left[{\rm Ker}\left(H^{\sharp,\pm}_{{\rm per},{\rm
max}}-E\right)\right]$

are Lagrangian planes of $\Lambda({\mathcal{H}}_{b})$, and
$\ell^{+}(E)\oplus\ell^{-}(E)={\mathcal{H}}_{b}$.

  * •

For all $E\in{\mathbb{R}}$ (not necessarily in the resolvent set), we have

$\dim{\rm Ker}\left(H_{\rm
per}-E\right)=\dim\left(\ell^{+}(E)\cap\ell^{-}(E)\right).$

  * •

If $E\in{\mathbb{R}}\setminus\sigma(H_{\rm per})$, then,

$\dim{\rm
Ker}\left(H^{\sharp}-E\right)=\dim\left(\ell^{+}(E)\cap\ell^{\sharp}\right).$

Again, for $\ell^{\sharp}\in\Lambda({\mathcal{H}}_{b})$, we denote by
$(H^{\sharp},\ell^{\sharp})$ the self-adjoint extensions of $H^{\sharp}$
corresponding to this Lagrangian plane.

In the finite dimensional Hill’s case, for any such extension
$(h^{\sharp,+},\ell^{\sharp})$ and $(h^{\sharp,-},\ell^{\sharp})$ with the
same Lagrangian plane $\ell^{\sharp}$, we have

$\sigma_{\rm ess}(h)=\sigma_{\rm ess}(h^{\sharp,+})\cup\sigma_{\rm
ess}(h^{\sharp,-}).$

In some sense, boundary conditions are finite dimensional (hence compact)
perturbations. In the Schrödinger case, we always have

$\sigma_{\rm ess}(H)\subset\sigma_{\rm ess}(H^{\sharp,+})\cup\sigma_{\rm
ess}(H^{\sharp,-}),$

which comes from the fact that Weyl sequences for $H$ can only escape to
$\pm\infty$. However, the inclusion may be strict. For instance, for
$E\in{\mathbb{R}}\setminus\sigma(H)$, the extension
$(H^{\sharp,+},\ell^{+}(E))$ has the energy $E$ as an eigenvalue of infinite
multiplicity. So $E\in\sigma_{\rm ess}(H^{\sharp,+})$. The corresponding Weyl
sequences localise near the cut.

This makes bulk-boundary correspondence more subtle in the Schrödinger case:
different self-adjoint extensions may give different results. For the usual
extensions however, we prove that the result are independent of the choice
(see the proof of Theorem 40 below).

####  4.2.3. Families of Schrödinger operators

We now consider a family of Schrödinger operators of the form

$H_{t}:=-\Delta_{\rm per}+V_{t},\quad\text{acting on}\quad{\mathcal{H}}.$

Again, we assume that $t\mapsto V_{t}$ is continuously differentiable from
${\mathbb{T}}^{1}$ to $L^{\infty}(\Omega,{\mathbb{R}})$. We also consider a
family of (self-adjoint extensions of) edge operators
$\left(H_{t}^{\sharp},\ell_{t}^{\sharp}\right)$.

Let $E\in{\mathbb{R}}\setminus\sigma\left(H_{t}\right)$. We say that $E$ is a
regular energy if, for all $t\in{\mathbb{T}}^{1}$, the energy $E$ is not in
the essential spectrum of $H^{\sharp}_{t}$. In particular, this implies
$\dim{\rm
Ker}\left(H^{\sharp}_{t}-E\right)=\dim\left(\ell_{t}^{+}(E)\cap\ell^{\sharp}_{t}\right)<\infty$.
In addition, we require all corresponding crossings to be regular.

Noticing that the definition of the Maslov index in Section 3.1 does not
require Assumption A, we can apply the first part of the proof of Theorem 28
to the Schrödinger case, and we obtain the following.

######  Theorem 38.

Let $(a,b)\subset{\mathbb{R}}$ be such that, for all $t\in{\mathbb{T}}^{1}$,

$(a,b)\cap\sigma\left(H_{t}\right)=\emptyset\quad\text{and}\quad(a,b)\cap\sigma_{\rm
ess}\left(H_{t}^{\sharp}\right)=\emptyset.$

Then,

  * •

almost any $E\in(a,b)$ is a regular energy;

  * •

for such a regular energy, we have

$\displaystyle{\rm Sf}\left(H_{t}^{\sharp},E,{\mathbb{T}}^{1}\right)$
$\displaystyle={\rm
Mas}\left(\ell^{+}(t),\ell^{\sharp}_{t},{\mathbb{T}}^{1}\right)$

Unfortunately, since the symplectic space $({\mathcal{H}}_{b},\omega)$ does
not satisfy Assumption A, we can not yet relate the last index as a spectral
flow of unitaries. We prove in Appendix A the following.

######  Lemma 39.

The Lagrangian planes of ${\mathcal{H}}_{b}$ are in one-to-one correspondence
with the unitaries of $L^{2}(\Gamma)$. In addition, if
$(\ell_{1}(t),\ell_{2}(t))$ is regular with corresponding unitaries
${\mathcal{U}}_{1}$ and ${\mathcal{U}}_{2}$, then

${\rm Mas}\left(\ell_{1},\ell_{2},{\mathbb{T}}^{1}\right)={\rm
Sf}\left({\mathcal{U}}_{2}^{*}{\mathcal{U}}_{1},1,{\mathbb{T}}^{1}\right).$

However, since ${\mathcal{U}}_{1}$ and ${\mathcal{U}}_{2}$ are now unitaries
in the infinite dimensional space $L^{2}(\Gamma)$, we can no longer use the
fact that the spectral flow is the winding number of the determinant. It is
unclear that we can split this index has a contribution of $\ell_{1}$ minus a
contribution of $\ell_{2}$. This will however be the case for junctions, as we
prove in the next section.

###  4.3. Junctions for Schrödinger operators

Let us consider the two families of potentials $V_{L,t}$ and $V_{R,t}$,
continuously differentiable from ${\mathbb{T}}^{1}$ to $L^{\infty}(\Omega)$.
For $\chi:\Omega\to[0,1]$ a bounded switch function with
$\chi(x,{\mathbf{y}})=1$ for $x<-X$ and $\chi(x,{\mathbf{y}})=0$ for $x>X$, we
set

$H_{t}^{\chi}:=-\Delta_{\rm
per}+V^{\chi}_{t}({\mathbf{x}}),\quad\text{with}\quad
V^{\chi}_{t}:=V_{L,t}\chi+V_{R,t}(1-\chi).$

As in Section 3.3.2, $H_{t}^{\chi}$ models a junction between a left and right
potential.

We denote by $H_{L,t}$ and $H_{R,t}$ the corresponding Hamiltonians. Let
$E\in{\mathbb{R}}$ be in the resolvent set of both $H_{L,t}$ and $H_{R,t}$ for
all $t\in{\mathbb{T}}^{1}$, so that the Lagrangian planes
$\ell^{\pm}_{L,t}(E)$ and $\ell^{\pm}_{R,t}(E)$ are all well-defined. We also
set $\ell_{D}^{\sharp}:=\\{0\\}\times H^{1/2}_{\rm
per}\in\Lambda({\mathcal{H}}_{b})$ the Lagrangian plane corresponding to
Dirichlet boundary conditions, and we set

${\mathcal{I}}\left(\ell_{L/R,t}^{\pm}(E),{\mathbb{T}}^{1}\right):={\rm
Mas}\left(\ell_{L/R,t}^{\pm}(E),\ell^{\sharp}_{D},{\mathbb{T}}^{1}\right)={\rm
Sf}\left(H^{\sharp,\pm}_{L/R,t},E,{\mathbb{T}}^{1}\right).$

This defines an integer-valued index for the Lagrangian planes of the form
$\ell_{t}^{\pm}(E)$ (which is our bulk/edge index).

######  Theorem 40 (Junctions in the Schrödinger case).

Let $(a,b)\subset{\mathbb{R}}$ be such that, for all $t\in{\mathbb{T}}^{1}$,

$(a,b)\cap\left(\sigma(H_{t,R})\cup\sigma(H_{t,L})\right)=\emptyset.$

Then, for all $E\in(a,b)$, we have

${\rm Sf}\left(H_{t}^{\chi},E,{\mathbb{T}}^{1}\right)={\rm
Mas}\left(\ell_{R}^{+}(E),\ell_{L}^{-}(E),{\mathbb{T}}^{1}\right)={\mathcal{I}}\left(\ell_{R}^{+}(E),{\mathbb{T}}^{1}\right)-{\mathcal{I}}\left(\ell_{L}^{-}(E),{\mathbb{T}}^{1}\right).$

This number is independent of $\chi$ and of $E$ in the gap.

Reasoning as in Remark 31, we see that the upper-scripts $\pm$ may be dropped
in the last line.

###### Proof.

Let us prove first the result for $\chi_{0}(x)={\mathds{1}}(x<0)$. We set for
simplicity $H_{t}:=-\Delta_{\rm per}+V_{t}^{\chi_{0}}$ acting on
$L^{2}(\Omega)$, and $H_{D,t}$ the corresponding operator acting on
$L^{2}(\Omega)\approx L^{2}(\Omega^{-}\cup\Omega^{+})$, and with Dirichlet
boundary conditions at $\\{0\\}\times\Gamma$. Since $V_{t}$ is uniformly
bounded, the operators $H_{t}$ and $H_{D,t}$ are uniformly bounded from below.
Consider $\Sigma\in{\mathbb{R}}$ such that

$\Sigma<\inf_{t\in{\mathbb{T}}^{1}}\inf\sigma(H_{t})\quad\text{and}\quad\Sigma<\inf_{t\in{\mathbb{T}}^{1}}\inf\sigma(H_{D,t}).$

We set $R_{t}:=(H_{t}-\Sigma)^{-1}$ and $R_{D,t}:=(H_{D,t}-\Sigma)^{-1}$,
which are both bounded operators. It is a standard result (see for instance
[RS80, Theorem XI.79]) that, for some $m\in{\mathbb{N}}$,
$R_{t}^{m}-R_{D,t}^{m}$ is a compact (even trace-class) operator. In
particular, for all $t\in{\mathbb{T}}^{1}$, we have

$\sigma_{\rm ess}(H_{t})=\sigma_{\rm ess}(H_{D,t}).$

Let $(a,b)$ denote an essential gap of these operators, and let $E\in(a,b)$ be
a regular energy for both operators. Using spectral theory, we have

${\rm Sf}\left(H_{t},E,{\mathbb{T}}^{1}\right)=-{\rm
Sf}\left(R_{t}^{m},(\Sigma-E)^{-m},{\mathbb{T}}^{1}\right)$

and similarly for $H_{D,t}$. Introducing

$R^{m}_{t}(s):=R_{t}+s(R_{D,t}^{m}-R_{t}^{m}),$

we see that $s\mapsto R^{m}_{t}(s)$ is a continuous family of operators
connecting $R_{t}$ and $R_{D,t}$. Since for all $s\in[0,1]$, $R^{m}_{t}(s)$ is
a compact perturbation of $R_{t}$, the essential gap does not close as $s$
varies. We deduce that these two spectral flows are equal (see for instance
[Wat16, Theorem 4.2.3]). So

${\rm Sf}\left(R_{t}^{m},(\Sigma-E)^{-m},{\mathbb{T}}^{1}\right)={\rm
Sf}\left(R_{D,t}^{m},(\Sigma-E)^{-m},{\mathbb{T}}^{1}\right),$

which gives

${\rm Sf}\left(H_{t},E,{\mathbb{T}}^{1}\right)={\rm
Sf}\left(H_{D,t},E,{\mathbb{T}}^{1}\right).$

Now, since $H_{D,t}=H^{\sharp,+}_{t}\oplus H^{\sharp,-}_{t}$, we see that $E$
is an eigenvalue of $H_{D,t}$ iff it is an eigenvalue of either
$H^{\sharp,+}_{t}$ or $H^{\sharp,-}_{t}$. Actually, we have

${\rm Sf}\left(H_{D,t},E,{\mathbb{T}}^{1}\right)={\rm
Sf}\left(H_{t}^{\sharp,+},E,{\mathbb{T}}^{1}\right)+{\rm
Sf}\left(H_{t}^{\sharp,-},E,{\mathbb{T}}^{1}\right).$

so, recalling that $\chi_{0}(x)={\mathds{1}}(x<0)$, so that
$\ell^{\pm}_{L/R,\chi_{0},t}=\ell^{\pm}_{L/R,t}$ we get as wanted

${\rm
Sf}\left(H_{t}^{\chi_{0}},E,{\mathbb{T}}^{1}\right)=M\left(\ell^{+}_{t},{\mathbb{T}}^{1}\right)-M\left(\ell^{-}_{t},{\mathbb{T}}^{1}\right).$

The minus sign comes from the Green’s identity on the $\Omega^{-}$ region.

For a general switch function $\chi$, the function $\chi-\chi_{0}$ is
compactly supported. In particular,

$H_{t}^{\chi}-H_{t}^{\chi_{0}}=(V_{L,t}-V_{R,t})(\chi-\chi_{0}),$

is a compact perturbation of $H_{t}^{\chi_{0}}$ for all
$t\in{\mathbb{T}}_{1}$. Again, by robustness of the spectral flow with respect
to compact perturbation, we obtain that ${\rm
Sf}(H_{t}^{\chi},E,{\mathbb{T}}^{1})={\rm
Sf}(H_{t}^{\chi_{0}},E,{\mathbb{T}}^{1})$. ∎

As we see from our proof, the main ingredient is the fact that the difference
of resolvent $(\Sigma-H)^{-1}-(\Sigma-H_{D})^{-1}$ is compact. Since this
holds also for Neumann boundary conditions instead of Dirichlet, the result is
also valid with Neumann boundary conditions. In particular, we must have

${\rm Mas}(\ell^{+}(E),\ell^{\sharp}_{N},{\mathbb{T}}^{1})={\rm
Mas}(\ell^{+}(E),\ell^{\sharp}_{D},{\mathbb{T}}^{1})+k,$

for some fixed constant $k\in{\mathbb{Z}}$, and for all families of potentials
$(V_{t})$. Taking $V_{t}$ independent of $t$ shows that $k=0$. So we have:

######  Corollary 41.

Let $H_{t}$ be a family of Schrödinger operators, and let $H^{\sharp}_{D,t}$
and $H^{\sharp}_{N,t}$ be the corresponding Dirichlet and Neumann extensions
on the semi-tube $\Omega^{+}$. Then, for all $E$ in the gap of $(H_{t})$, we
have

${\rm Sf}\left(H_{D,t}^{\sharp},E,{\mathbb{T}}^{1}\right)={\rm
Sf}\left(H_{N,t}^{\sharp},E,{\mathbb{T}}^{1}\right).$

As noticed in the discussion following 37, such an equality cannot hold for
all self-adjoint extensions of $H^{\sharp}_{t}$, since the essential spectrum
may not be conserved.

##  Appendix A Embedding of symplectic spaces, and unitaries.

As noticed in Remark 36, the symplectic form $({\mathcal{H}}_{b},\omega)$ does
not satisfy Assumption A. In order to use the results of Section 2.1.3, we
embed this space is a larger one. In this appendix, we denote by
${\mathcal{H}}_{b}^{\prime}:=H^{3/2}_{\rm per}(\Gamma)\times H^{1/2}_{\rm
per}(\Gamma)$ the previous space, and reserve the notation ${\mathcal{H}}_{b}$
for the larger space.

###  A.1. The boundary space ${\mathcal{H}}_{b}$.

There are several possibilities, one can choose

${\mathcal{H}}_{b}:=H^{1/2}_{\rm per}(\Gamma)\times
H^{-1/2}(\Gamma),\quad\text{or}\quad{\mathcal{H}}_{b}=L^{2}(\Gamma)\times
L^{2}(\Gamma).$

While the first choice seems more natural in the context of boundary
conditions (in particular, Robin boundary conditions are best described in
this space), we emphasise that the two choices are equivalent. For the ease of
notation, we consider the second choice ${\mathcal{H}}_{b}=L^{2}(\Gamma)\times
L^{2}(\Gamma)$. We explain at the end of the section what happens in the
second case, and why the two choices are equivalent.

We therefore set ${\mathcal{H}}_{b}:=L^{2}(\Gamma)\times L^{2}(\Gamma)$,
together with the symplectic form $\omega$ defined by

$\forall(f,f^{\prime}),(g,g^{\prime})\in{\mathcal{H}}_{b},\quad\omega((f,f^{\prime}),(g,g^{\prime}))=\langle
f,g^{\prime}\rangle_{L^{2}(\Gamma)}-\langle
f^{\prime},g\rangle_{L^{2}(\Gamma)}.$

We have ${\mathcal{H}}_{b}^{\prime}\hookrightarrow{\mathcal{H}}_{b}$ with
continuous (compact) embedding. In addition, ${\mathcal{H}}_{b}^{\prime}$ is
dense in ${\mathcal{H}}_{b}$, and

$\forall
x,y\in{\mathcal{H}}_{b}^{\prime}\subset{\mathcal{H}}_{b},\quad\omega^{\prime}(x,y)=\omega(x,y).$

######  Theorem 42.

Let $({\mathcal{H}}^{\prime},\omega^{\prime})$ and $({\mathcal{H}},\omega)$ be
two symplectic Hilbert spaces, with
${\mathcal{H}}^{\prime}\subset{\mathcal{H}}$ with continuous embedding,
${\mathcal{H}}^{\prime}$ dense in ${\mathcal{H}}$ and $\omega^{\prime}=\omega$
on ${\mathcal{H}}^{\prime}\times{\mathcal{H}}^{\prime}$. Then there is a one-
to-one correspondence between the Lagrangian planes of
${\mathcal{H}}^{\prime}$ and the ones of ${\mathcal{H}}$. More specifically,

$\displaystyle\ell^{\prime}\in\Lambda({\mathcal{H}}^{\prime})\quad\text{iff}\quad\ell^{\prime}=\ell\cap{\mathcal{H}}^{\prime}\
\text{for some}\ \ell\in\Lambda({\mathcal{H}}),$
$\displaystyle\ell\in\Lambda({\mathcal{H}}),\quad\text{iff}\quad\ell=\overline{\ell^{\prime}}^{{\mathcal{H}}}\
\text{for some}\ \ell^{\prime}\in\Lambda({\mathcal{H}}^{\prime}).$

In addition, if $(\ell_{1}^{\prime}(t),\ell_{2}^{\prime}(t))$ is a periodic
path of Lagrangian in ${\mathcal{H}}^{\prime}$, with corresponding path
$(\ell_{1}(t),\ell_{2}(t))$ in ${\mathcal{H}}$, we have

${\rm
Mas}^{\prime}\left(\ell_{1}^{\prime},\ell_{2}^{\prime},{\mathbb{T}}_{1}\right)={\rm
Mas}\left(\ell_{1},\ell_{2},{\mathbb{T}}_{1}\right).$

###### Proof.

Let $\ell^{\prime}$ be a any sub-vectorial space of ${\mathcal{H}}^{\prime}$,
and let $\ell:=\overline{\ell^{\prime}}^{{\mathcal{H}}}$, so that
$\ell^{\prime}=\ell\cap{\mathcal{H}}^{\prime}$. First, for all
$x^{\prime}\in(\ell^{\prime})^{\circ}$, we have

$\forall y^{\prime}\in\ell^{\prime},\
\omega(x^{\prime},y^{\prime})=\omega^{\prime}(x^{\prime},y^{\prime})=0,\quad\text{hence,
by density},\quad\forall y\in\ell,\ \omega(x^{\prime},y)=0.$

We deduce that $(\ell^{\prime})^{\circ}\subset\ell^{\circ}$, hence
$\overline{(\ell^{\prime})^{\circ}}_{{\mathcal{H}}}\subset\ell^{\circ}$ and
$(\ell^{\prime})^{\circ}\subset\ell^{\circ}\cap{\mathcal{H}}^{\prime}$. On the
other hand, if $x\in\ell^{\circ}\cap{\mathcal{H}}^{\prime}$, then we have

$\forall y^{\prime}\in\ell^{\prime}\subset\ell,\quad
0=\omega(x,y^{\prime})=\omega^{\prime}(x,y^{\prime}),$

so $x\in\ell^{\circ}$. We therefore proved

$\ell=\overline{\ell^{\prime}}^{\mathcal{H}}\Longleftrightarrow\ell^{\circ}=\overline{(\ell^{\prime})^{\circ}}^{\mathcal{H}}.$

In particular, $\ell$ is a Lagrangian plane of ${\mathcal{H}}$ iff
$\ell^{\prime}$ is a Lagrangian plane of ${\mathcal{H}}^{\prime}$.

If $\ell_{1}^{\prime}$ and $\ell_{2}^{\prime}$ are Lagrangian planes of
${\mathcal{H}}^{\prime}$, with corresponding planes $\ell_{1}$ and $\ell_{2}$
in ${\mathcal{H}}$, then by density of ${\mathcal{H}}^{\prime}$ in
${\mathcal{H}}$, we have

$\dim\left(\ell_{1}^{\prime}\cap\ell_{2}^{\prime}\right)=\dim\left(\ell_{1}\cap\ell_{2}\right).$

In the case where this dimension is finite, we even have
$\ell_{1}^{\prime}\cap\ell_{2}^{\prime}=\ell_{1}\cap\ell_{2}$, since finite
dimensional subspaces are always closed. As the Maslov index only involves
these finite dimensional spaces, the equality of the Maslov index follows. ∎

The advantage of ${\mathcal{H}}_{b}$ over ${\mathcal{H}}_{b}^{\prime}$ is
that, in the space ${\mathcal{H}}_{b}$, we have

$J=\begin{pmatrix}0&{\mathbb{I}}_{L^{2}(\Gamma)}\\\
-{\mathbb{I}}_{L^{2}(\Gamma)}&0\end{pmatrix},$

which satisfies Assumptions A and B. We obtain for instance the following.

######  Theorem 43.

The self-adjoint extensions of $H^{\sharp}$ are in one-to-one correspondence
with the Lagrangian planes of $({\mathcal{H}}_{b},\omega)$ which are in one-
to-one correspondence with the unitaries ${\mathcal{U}}$ of $L^{2}(\Gamma)$.
The correspondence is given by

${\mathcal{D}}^{\sharp}:=\left\\{\psi\in{\mathcal{D}}_{{\rm per},{\rm
max}}^{\sharp},\quad{\rm Tr}(\psi)=\begin{pmatrix}1\\\
{\mathrm{i}}\end{pmatrix}f+\begin{pmatrix}1\\\
-{\mathrm{i}}\end{pmatrix}{\mathcal{U}}f,\quad\text{for some $f\in
L^{2}(\Gamma)$}\right\\}$

We emphasise that the trace map ${\rm Tr}$ is not onto on ${\mathcal{H}}_{b}$,
so some $f\in L^{2}(\Gamma)$ may not be attained.

###  A.2. The other choice of boundary space.

Let us explain what happens in the case where we choose
$\widetilde{{\mathcal{H}}_{b}}:=H^{1/2}_{\rm per}(\Gamma)\times
H^{-1/2}(\Gamma)$, with symplectic form

$\forall(f,f^{\prime}),(g,g^{\prime})\in{\mathcal{H}}_{b},\quad\widetilde{\omega}\left((f,f^{\prime}),(g,g^{\prime})\right):=\langle
f,g^{\prime}\rangle_{H^{1/2}_{\rm per},H^{-1/2}}-\langle
f^{\prime},g\rangle_{H^{-1/2},H^{1/2}_{\rm per}}.$

One can again apply Theorem 42, and deduce that there is a one-to-one
correspondence between $\Lambda({\mathcal{H}}_{b}^{\prime})$ and
$\Lambda(\widetilde{{\mathcal{H}}_{b}})$. Let us prove that this last space
also satisfies Assumption B.

Let $V$ be the map $V:H^{1/2}_{\rm per}\to H^{-1/2}$ be such that for all
$f\in H^{1/2}_{\rm per}(\Gamma)$ and all $g\in H^{-1/2}(\Gamma)$, we have

$\langle f,g\rangle_{H^{1/2}_{\rm per},H^{-1/2}}=\langle
f,V^{*}g\rangle_{H^{1/2}_{\rm per}}+\langle Vf,g\rangle_{H^{-1/2}}.$ (11)

We can check that $V$ is well-defined and unitary. This time, we have
$J=\begin{pmatrix}0&V^{*}\\\ -V&0\end{pmatrix}$, so
$(\widetilde{{\mathcal{H}}_{b}},\widetilde{\omega})$ also satisfies Assumption
B. The previous theorem in this framework states that the self-adjoint
extensions of $H^{\sharp}$ are in one-to-one correspondence the unitaries
${\mathcal{U}}$ of $H^{1/2}_{\rm per}(\Gamma)$ with

${\mathcal{D}}^{\sharp}:=\left\\{\psi\in{\mathcal{D}}_{{\rm per},{\rm
max}}^{\sharp},\quad{\rm Tr}(\psi)=\begin{pmatrix}1\\\
{\mathrm{i}}V\end{pmatrix}f+\begin{pmatrix}1\\\
-{\mathrm{i}}V\end{pmatrix}\widetilde{{\mathcal{U}}}f,\quad\text{for some
$f\in H^{1/2}_{\rm per}(\Gamma)$}\right\\}.$

In order to see the connection between the two approaches using
$({\mathcal{H}}_{b},\omega)$ and
$(\widetilde{{\mathcal{H}}_{b}},\widetilde{\omega})$, we introduce the maps
$V_{1}:H^{1/2}_{\rm per}(\Gamma)\to L^{2}(\Gamma)$, and
$V_{2}:L^{2}(\Gamma)\to H^{-1/2}(\Gamma)$, defined by

$\text{for}\quad
f({\mathbf{y}})=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}f_{\mathbf{k}}e_{\mathbf{k}}({\mathbf{y}}),\quad\left(V_{1,2}f\right)({\mathbf{y}}):=\sum_{{\mathbf{k}}\in{\mathbb{Z}}^{d-1}}f_{\mathbf{k}}(1+2\pi|{\mathbf{k}}|)^{1/2}e_{\mathbf{k}}({\mathbf{y}}).$

Then $R_{1}$ is a unitary from $H^{1/2}_{\rm per}(\Gamma)$ and
$L^{2}(\Gamma)$, and $R_{2}$ is a unitary from $L^{2}(\Gamma)$ to
$H^{-1/2}(\Gamma)$. Note that the map $V$ defined in (11) is $V=R_{2}R_{1}$.
So the map $R:{\mathcal{H}}_{b}\to\widetilde{{\mathcal{H}}_{b}}$ defined by

$\forall(f,f^{\prime})\in{\mathcal{H}}_{b},\quad
R(f,f^{\prime}):=\left(R_{1}^{*}f,R_{2}f\right)\quad\in\widetilde{{\mathcal{H}}_{b}}$

is a unitary. In addition, we have

$\forall(f,f^{\prime}),(g,g^{\prime})\in{\mathcal{H}}_{b},\quad\omega((f,f^{\prime}),(g,g^{\prime}))=\widetilde{\omega}\left(R(f,f^{\prime}),R(g,g^{\prime})\right).$

We deduce that $\ell$ is a Lagrangian plane of ${\mathcal{H}}_{b}$ iff
$\widetilde{\ell}:=R\ell$ is a Lagrangian plane of
$\widetilde{{\mathcal{H}}_{b}}$. In addition, ${\mathcal{U}}$ is a unitary of
${\mathcal{H}}_{1}=L^{2}(\Gamma)$ iff
$\widetilde{{\mathcal{U}}}:=R_{1}^{*}UR_{1}$ is a unitary of
$\widetilde{{\mathcal{H}}_{1}}=H^{1/2}_{\rm per}(\Gamma)$. This proves that
the two approaches are equivalent.

######  Example 44 (Robin boundary conditions).

Let $\Theta:H^{1/2}_{\rm per}\to H^{-1/2}$ be an invertible operator,
symmetric in the sense

$\forall f\in H^{1/2}_{\rm per}(\Gamma),\ \forall g\in
H^{-1/2}(\Gamma),\quad\langle
f,\Theta^{*}g\rangle_{H^{1/2}}(\Gamma)=\langle\Theta f,g\rangle_{H^{-1/2}}.$

Then we can check that

$\ell_{\Theta}:=\left\\{\begin{pmatrix}f\\\ \Theta f\end{pmatrix},\quad f\in
H^{1/2}_{\rm per}\right\\}$

is a Lagrangian plane. The corresponding self-adjoint extension has the
$\Theta$-Robin conditions. To identity the corresponding unitary, we write
that

$\begin{pmatrix}f\\\ \Theta f\end{pmatrix}=\begin{pmatrix}1\\\
{\mathrm{i}}V\end{pmatrix}\frac{1}{2}\left(f-{\mathrm{i}}V^{*}\Theta
f\right)+\begin{pmatrix}1\\\
-{\mathrm{i}}V\end{pmatrix}\frac{1}{2}\left(f+{\mathrm{i}}V^{*}\Theta
f\right).$

We recognise the unitary $\widetilde{{\mathcal{U}}}$ of $H^{1/2}_{\rm
per}(\Gamma)$ defined by

$\widetilde{{\mathcal{U}}}:=\left({\mathbb{I}}-{\mathrm{i}}V^{*}\Theta\right)^{-1}({\mathbb{I}}+{\mathrm{i}}V^{*}\Theta).\quad\text{(Cayley
transform of $V^{*}\Theta$)}.$

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