#  D${\,}^{\\!2}$=H+$\frac{1}{4}$ with point interactions

Andrea Posilicano  and  Linda Reginato  DiSAT, Sezione di Matematica,
Università dell’Insubria, via Valleggio 11, I-22100 Como, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let ${\mathsf{D}}$ and ${\mathsf{H}}$ be the self-adjoint, one-dimensional
Dirac and Schrödinger operators in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ and
$L^{2}(\mathbb{R};\mathbb{C})$ respectively. It is well known that, in absence
of an external potential, the two operators are related through the equality
${\mathsf{D}}^{2}=({\mathsf{H}}+\frac{1}{4}){\mathbb{1}}$. We show that such a
kind of relation also holds in the case of $n$-point singular perturbations:
given any self-adjoint realization $\widehat{\mathsf{D}}$ of the formal sum
${\mathsf{D}}+\sum_{k=1}^{n}\gamma_{k}\delta_{y_{k}}$, we explicitly determine
the self-adjoint realization $\widehat{\mathsf{H}}$ of
${\mathsf{H}}{\mathbb{1}}+\sum_{k=1}^{n}(\alpha_{k}\delta_{y_{k}}+\beta_{k}\delta^{\prime}_{y_{k}})$
such that
${\widehat{\mathsf{D}}}^{2}=\widehat{\mathsf{H}}+\frac{{\mathbb{1}}}{4}$. The
found correspondence preserves the subclasses of self-adjoint realizations
corresponding to both the local and the separating boundary conditions. The
case on nonlocal boundary conditions allows the study of the relation
${\mathsf{D}}^{2}={\mathsf{H}}+\frac{1}{4}$ for quantum graphs with (at most)
two ends; in particular, the square of the extension corresponding to
Kirchhoff-type boundary conditions for the Dirac operator on the graph gives
the direct sum of two Schrödinger operators on the same graph, one with the
usual Kirchhoff boundary conditions and the other with a sort of reversed
Kirchhoff ones.

##  1\. Introduction

Let $L^{2}(\mathbb{R};\mathbb{C}^{d})$ be the Hilbert space of
$\mathbb{C}^{d}$-valued square integrable functions with scalar product
$\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$; likewise,
$H^{2}(\mathbb{R};\mathbb{C}^{d})\subset
H^{1}(\mathbb{R};\mathbb{C}^{d})\subset C_{b}(\mathbb{R};\mathbb{C}^{d})$
denote the Sobolev space on $\mathbb{R}$ of order 1 and 2 and the space of
bounded continuous functions with values in $\mathbb{C}^{d}$ respectively.
Whenever $d=1$, we simply write $L^{2}(\mathbb{R})$, $H^{k}(\mathbb{R})$ and
$C_{b}(\mathbb{R})$. In $L^{2}(\mathbb{R};\mathbb{C}^{2})$ we consider the
free self-adjoint Dirac operator $\mathsf{D}$ defined by

$\mathsf{D}:H^{1}(\mathbb{R};\mathbb{C}^{2})\subseteq
L^{2}(\mathbb{R};\mathbb{C}^{2})\to
L^{2}(\mathbb{R};\mathbb{C}^{2})\,,\quad\mathsf{D}:=-i\,\frac{d\,}{dx}\,\sigma_{1}+\frac{1}{2}\,\sigma_{3}$

where

$\sigma_{1}=\begin{bmatrix}0&1\\\ 1&0\\\
\end{bmatrix},\quad\sigma_{2}=\begin{bmatrix}0&-i\\\ i&0\\\
\end{bmatrix},\quad\sigma_{3}=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix}$

are the Pauli matrices. Furthermore, we consider the free self-adjoint
Schrödinger operator in $L^{2}(\mathbb{R})$

$\mathsf{H}:H^{2}(\mathbb{R})\subseteq L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),\quad\mathsf{H}:=-\frac{d^{2}}{dx^{2}}\,.$

It is well known and easy to check that in this free case there exists a
relation between the two operators:

(1.1)
$\displaystyle\mathsf{D}^{2}=\left(\mathsf{H}+\frac{1}{4}\right)\\!\mathbb{1}\,.$

Here and below, we use the isomorphism $L^{2}(\mathbb{R};\mathbb{C}^{2})\simeq
L^{2}(\mathbb{R})\oplus L^{2}(\mathbb{R})$ and the identification
${\mathsf{L}}\mathbb{1}\equiv{\mathsf{L}}\oplus{\mathsf{L}}$, ${\mathsf{L}}$ a
linear operator in $L^{2}(\mathbb{R})$. More generally, in the following we
use the shorthand notation
${\mathsf{L}}\mathbb{1}\equiv{\mathsf{L}}\oplus{\mathsf{L}}$ for a linear
operator $L:\text{\rm dom}(L)\subseteq H_{1}\to H_{2}$.

Notice that (1.1) entails a relation between the resolvent operators:

(1.2)
$(-\mathsf{D}+z)^{-1}=(\mathsf{D}+z)\left(-\mathsf{H}+z^{2}-\frac{1}{4}\right)^{\\!\\!-1}\\!\\!\\!\mathbb{1}\,,\qquad
z\in\mathbb{C}\backslash((-\infty,-{1}/{2}]\cup[{1}/{2},+\infty))\,.$

The aim of this paper is to extend this connection between Dirac’s and
Schrödinger’s operators to the case where $\mathsf{D}$ is perturbed by a sum
of $\delta$’s potential, equivalently, given any self-adjoint extension
$\mathsf{D}_{\Pi,\Theta}$ of the symmetric operator
$\mathsf{D}|C^{\infty}_{comp}(\mathbb{R}\backslash\\{y_{1},\dots,y_{n}\\};\mathbb{C}^{2})$,
we explicitly determine the couple $(\widehat{\Pi},\widehat{\Theta})$ such
that

(1.3)
$(\mathsf{D}_{\Pi,\Theta})^{2}=\left(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+\frac{\mathbb{1}}{4}\right)\,.$

Here, we parametrize the self-adjoint extensions of
$\mathsf{D}|C^{\infty}_{comp}(\mathbb{R}\backslash\\{y_{1},\dots,y_{n}\\};\mathbb{C}^{2})$
by couples $(\Pi,\Theta)$, $\Pi:\mathbb{C}^{2n}\to\mathbb{C}^{2n}$ an
orthogonal projector, $\Theta:\text{\rm ran}(\Pi)\to\text{\rm ran}(\Pi)$ a
symmetric operator, and likewise
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ denotes the self-
adjoint extension of
$\mathsf{H}\mathbb{1}|C^{\infty}_{comp}(\mathbb{R}\backslash\\{y_{1},\dots,y_{n}\\};\mathbb{C}^{2})$
corresponding to the couple $(\widehat{\Pi},\widehat{\Theta})$,
$\widehat{\Pi}:\mathbb{C}^{4n}\to\widehat{\mathbb{C}}^{4n}$ an orthogonal
projector, $\widehat{\Theta}:\text{\rm ran}(\widehat{\Pi})\to\text{\rm
ran}(\widehat{\Pi})$ a symmetric operator. Any operator of the kind
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ is a self-adjoint
realization of a singular perturbation of $\mathsf{H}\mathbb{1}$ by a sum of
$\delta$’s and $\delta^{\prime}$’s potentials. As in the free case, the
relation (1.3) entails another one for the resolvents:

$(-\mathsf{D}_{\Pi,\Theta}+z)^{-1}=(\mathsf{D}_{\Pi,\Theta}+z)\left(-\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+z^{2}-\frac{\mathbb{1}}{4}\right)^{\\!\\!-1}\,,$

where $\pm z\in\varrho(\mathsf{D}_{\Pi,\Theta})$ if and only if
$(z^{2}-\frac{1}{4})\in\varrho(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})$;
here, $\varrho(L)$ denotes the resolvent set of the closed operator $L$.

The specific case here considered is an example of solution of the problem
concerning the representation of the square of a singular perturbation of a
self-adjoint operator $A$ by a singular perturbation of $A^{2}$. This problem
has been studied in [2]; however, in such a paper only the case $A>0$ has been
considered and the explicit examples there presented are limited to rank-one
singular perturbations. The methods here used are different from the ones in
[2], we do not use the resolvent formulae directly but instead use the self-
adjointness domains.

In more detail, the content of the paper is the following. In Section 2 we
build the whole families of the self-adjoint extensions of
$\mathsf{D}|C^{\infty}_{comp}(\mathbb{R}\backslash\\{y_{1},\dots,y_{n}\\};\mathbb{C}^{2})$
and
$\mathsf{H}\mathbb{1}|C^{\infty}_{comp}(\mathbb{R}\backslash\\{y_{1},\dots,y_{n}\\};\mathbb{C}^{2})$.
Instead of using the standard von Neumann theory (see, e.g., [1], [8], [16]),
which gives a parametrization in terms of unitary operators between the defect
spaces, we found more convenient to use the equivalent approach proposed in
[21] and [22], which gives a parametrization in terms of couples
$(\Pi,\Theta)$, where $\Pi$ is an orthogonal projection and $\Theta$ is a
self-adjoint operator in $\text{\rm ran}(\Pi)$; this allows for an easy
writing of the corresponding resolvents. Then, in Section 3, by a comparison
of the self-adjointness domains, we found the correspondence between the
couple $(\Pi,\Theta)$ and $(\widehat{\Pi},\widehat{\Theta})$ such that (1.3)
holds. In order to enhance the reader intuition, we start with simplest case,
where $n=1$ and $\Pi=\mathbb{1}$ and then proceede step-by-step towards the
most general case. Finally, in Section 4, we present various applications. In
Subsection 4.1 we consider the subclass of self-adjoint extensions for the
Dirac operator corresponding to local boundary conditions, i.e., to the ones
which do not couple different points $y_{k}$ and show that the corresponding
extensions for the Schrödinger operator provide local boundary conditions as
well. As a particular case of such a result, in Subsection 4.2 we consider the
Gesztesy-Šeba realizations; they are the self-adjoint realizations of the
Dirac operator with local point interactions corresponding, in the non
relativistic limit, to Schrödinger operators with local point interactions
either of $\delta$-type or of $\delta^{\prime}$-type (see [17], [1, Appendix
J], [14]). Then, in Subsection 4.3, we consider the subclass of self-adjoint
extensions for the Dirac operator corresponding to separating boundary
conditions, i.e., to the local ones for which, at any point, left limits are
independent from right limits. This entails that the corresponding Dirac
operator is the direct sum of self-adjoint Dirac operators $\mathsf{D}_{k}$ in
$L^{2}(I_{k})$, where the $I_{k}$’s are either the half-lines
$(-\infty,y_{1})$ and $(y_{n},+\infty)$ or the bounded intervals
$(y_{k},y_{k+1})$; the same is true for the corresponding corresponding
Schrödinger operator and
$(\mathsf{D}_{k})^{2}=\widehat{\mathsf{H}}_{k}+\frac{\mathbb{1}}{4}$. In
Subsection 4.4, some connections with supersymmetry are discussed and a simple
criterion of spontaneous supersymmetry breaking is provided (see [23], [3] and
references therein for somehow different aspects of supersymmetry in presence
of point interactions). In Subsection 4.5, we point out that our results, in
the case of non local boundary conditions, allow the study of the connection
between the square of the Dirac operator and the Schrödinger operator on
quantum graphs with (at most) two ends. In particular, as an explicit example,
we consider the Dirac operator on the eye graph with Kirchhoff-type boundary
conditions at the vertices and show that its square is the direct sum of two
Schrödinger operators on the same graph, one with Kirchhoff boundary
conditions and the other with a sort of inverse Kirchhoff ones. These latter
boundary conditions, like the Kirchhoff ones, reduce, in the case of the real
line, to the free boundary conditions; this is consistent with (1.1). The
procedure used for the eye graph can be extended, without substantial changes,
to any kind of graph, thus showing that the property of conservation of
Kirchhoff-like boundary conditions holds in general.

We presume that the results here presented can be extended to the more
involved cases corresponding to extensions of symmetric operators with
infinite deficiency indices as the 1-dimensional Dirac and Schrödinger
operators with singular perturbations on discrete sets (see [18] and [14]) and
the 3-dimensional Dirac and Schrödinger operators with singular perturbations
on 2-dimensional surfaces (see, e.g., [5], [6] and [7], [19]).

##  2\. $\mathsf{D}$ and $\mathsf{H}$ with point interactions

Given a finite set of points $Y=\\{y_{1},\cdots,y_{n}\\}$,
$y_{1}<y_{2},\dots<y_{n}$, we define

(2.1) $H^{1}(\mathbb{R}\backslash
Y;\mathbb{C}^{d}):=H^{1}(I_{0};\mathbb{C}^{d})\oplus\cdots\oplus
H^{1}(I_{n};\mathbb{C}^{d})\,,$

where,

(2.2) $I_{0}:=(-\infty,y_{1})\,,\quad
I_{1}:=(y_{1},y_{2})\,,\quad\dots\dots\quad I_{n-1}:=(y_{n-1},y_{n})\,,\quad
I_{n}:=(y_{n},+\infty)\,,$

and

$H^{1}(I_{j};\mathbb{C}^{d}):=\\{f\in
L^{2}(I_{j};\mathbb{C}^{d}):f^{\prime}\in
L^{2}(I_{j};\mathbb{C}^{d})\\}\,,\quad j=0,\dots,n\,.$

Here and below, $f^{\prime}$ denotes the (distributional) derivative of $f$.
Notice that the left and right limits $f(y_{k}^{\pm})$ exists and are finite
for any $f\in H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{d})$. We define

$H^{2}(\mathbb{R}\backslash
Y;\mathbb{C}^{d}):=H^{2}(I_{0};\mathbb{C}^{d})\oplus\cdots\oplus
H^{2}(I_{n};\mathbb{C}^{d})\,,$

where

$H^{2}(I_{j};\mathbb{C}^{d}):=\\{f\in
H^{1}(I_{j};\mathbb{C}^{d}):f^{\prime\prime}\in
L^{2}(I_{j};\mathbb{C}^{d})\\}\,,\quad j=0,\dots,n\,.$

Obviously, $H^{2}(\mathbb{R}\backslash Y;\mathbb{C}^{d})\subset
H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{d})$ and $f\in
H^{2}(\mathbb{R}\backslash Y;\mathbb{C}^{d})$ implies $f^{\prime}\in
H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{d})$. We simply write
$H^{k}(\mathbb{R}\backslash Y)$, $k=1,2$, whenever $d=1$. Next, we introduce
the two bounded operators

(2.3) $\displaystyle\tau:H^{1}(\mathbb{R}\backslash
Y;\mathbb{C}^{2})\to\mathbb{C}^{2n}\,,\quad\tau\Psi:=(\tau_{y_{1}}\Psi\,,\dots,\tau_{y_{n}}\Psi)\,,\qquad\tau_{y}\Psi:=\langle\Psi\rangle_{y}\,,$

and

(2.4) $\displaystyle\widehat{\tau}:H^{2}(\mathbb{R}\backslash
Y)\to\mathbb{C}^{2n}\,,\qquad\widehat{\tau}\psi:=(\widehat{\tau}_{y_{1}}\psi\,,\dots,\widehat{\tau}_{y_{n}}\psi)\,,\qquad\widehat{\tau}_{y}\psi:=\langle\psi\rangle_{y}\oplus\langle\psi^{\prime}\rangle_{y}\,,$

where

$\langle f\rangle_{y}:=\frac{1}{2}\,\big{(}f(y^{-})+f(y^{+})\big{)}\,.$

Obviously, $\langle f\rangle_{y_{k}}=f(y_{k})$ whenever $f\in
H^{1}(\mathbb{R};\mathbb{C}^{d})\subset C_{b}(\mathbb{R};\mathbb{C}^{d})$.

In this section, following the scheme proposed in [22] (for the equivalent
approaches which use either von Neuman’s theory or Boundary Triples theory,
see [16], [8] and [20, Sect. 4.1], [18], [14] respectively), we review the
construction of the self-adjoint extensions of the closed symmetric operators

$S:=\mathsf{D}{|\text{\rm
ker}(\tau|H^{1}(\mathbb{R};\mathbb{C}^{2}))}\,,\qquad\widehat{S}:=\mathsf{H}{|\text{\rm
ker}(\widehat{\tau}|H^{2}(\mathbb{R}))}\,.$

Both $S$ and $\widehat{S}$ have defect indices $(2n,2n)$; they are the
closures of the symmetric operators

$S^{\circ}:=\mathsf{D}{|C^{\infty}_{comp}(\mathbb{R}\backslash
Y;\mathbb{C}^{2})}\,,\qquad\widehat{S}^{\circ}:=\mathsf{H}{|C^{\infty}_{comp}(\mathbb{R}\backslash
Y)}\,.$

Let $\widehat{g}_{z}(x-y)$ be the kernel of the free Schrödinger resolvent
$(-\mathsf{H}+z)^{-1}=\left(\frac{d^{2}}{dx^{2}}+z\right)^{-1}$, with
$z\in\varrho(\mathsf{H})=\mathbb{C}\backslash[0,+\infty)$, i.e.,

(2.5)
$\displaystyle\widehat{g}_{z}(x)=\frac{e^{i\sqrt{z}\,|x|}}{2i\sqrt{z}}\,,\qquad\text{Im}(\sqrt{z})>0\,.$

By (1.2), setting $w_{z}:=z^{2}-\frac{1}{4}$, one then obtains the kernel
$g_{z}(x-y)$ of the free Dirac resolvent $(-\mathsf{D}+z)^{-1}$,
$z\in\varrho(\mathsf{D})=\mathbb{C}\backslash((-\infty,-{1}/{2}]\cup[{1}/{2},+\infty))$,

(2.6) $\displaystyle
g_{z}(x)=(\mathsf{D}+z)\widehat{g}_{w_{z}}\mathbb{1}=\frac{e^{i\sqrt{w_{z}}\,|x|}}{2i}\begin{bmatrix}\zeta_{z}&\text{sgn}(x)\\\
\text{sgn}(x)&\zeta_{z}^{-1}\\\ \end{bmatrix},$

where $\zeta_{z}:=({\frac{1}{2}-z})/{\sqrt{w_{z}}}$ and $\text{Im}(w_{z})>0$.
By such kernels, one gets that the bounded operators

$G_{z}:\mathbb{C}^{2n}\to L^{2}(\mathbb{R};\mathbb{C}^{2})\,,\quad
G_{z}:=(\tau(-\mathsf{D}+\bar{z})^{-1})^{*}\,,\quad
z\in\mathbb{C}\backslash((-\infty,-{1}/{2}]\cup[{1}/{2},+\infty))\,,$

and

$\widehat{G}_{z}:\mathbb{C}^{2n}\to
L^{2}(\mathbb{R})\,,\quad\widehat{G}_{z}:=(\widehat{\tau}(-\mathsf{H}+\bar{z})^{-1})^{*}\,,\quad
z\in\mathbb{C}\backslash(-\infty,0]\,,$

represents as

$[G_{z}\xi](x)=\sum_{k=1}^{n}g_{z}(y_{k}-x)\,\xi_{k}\,,\qquad\xi\equiv(\xi_{1},\dots,\xi_{n})\,,\quad\xi_{k}\in\mathbb{C}^{2}\,.$

and

$[\widehat{G}_{z}\xi](x)=\sum_{k=1}^{n}(\widehat{g}_{z}(y_{k}-x)\,\xi_{k,1}+\widehat{g}^{\,\prime}_{z}(y_{k}-x)\,\xi_{k,2})\,,\quad\xi\equiv((\xi_{1,1},\xi_{1,2}),\dots,(\xi_{n,1},\xi_{n,2}))\,.$

Their adjoints

$G^{*}_{\bar{z}}:L^{2}(\mathbb{R};\mathbb{C}^{2})\to\mathbb{C}^{2n}\,,\qquad\widehat{G}^{*}_{\bar{z}}:L^{2}(\mathbb{R})\to\mathbb{C}^{2n}$

are given by

$G^{*}_{\bar{z}}\Psi=\big{(}(G^{*}_{z}\Psi)_{1},\dots,(G^{*}_{z}\Psi)_{n}\big{)}\,,\qquad(G^{*}_{\bar{z}}\Psi)_{k}:=\int_{\mathbb{R}}g_{z}(y_{k}-x)\Psi(x)\,dx$

and

$\widehat{G}^{*}_{\bar{z}}\psi=\big{(}(\widehat{G}^{*}_{z}\Psi)_{1},\dots,(\widehat{G}^{*}_{z}\Psi)_{n}\big{)}\,,\qquad(\widehat{G}^{*}_{\bar{z}}\psi)_{k}:=\left(\int_{\mathbb{R}}\widehat{g}_{z}(y_{k}-x)\psi(x)\,dx,\int_{\mathbb{R}}\widehat{g}^{\,\prime}_{z}(y_{k}-x)\psi(x)\,dx\right).$

Since

$G_{z}\xi\in H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{2})\quad$ and
$\quad\widehat{G}_{z}\xi\in H^{2}(\mathbb{R}\backslash Y)$,

both

$\tau G_{z}:\mathbb{C}^{2n}\to\mathbb{C}^{2n}\quad$ and
$\quad\widehat{\tau}\widehat{G}_{z}:\mathbb{C}^{2n}\to\mathbb{C}^{2n}$

are well defined and are represented by the two $n\times n$ block matrices
with the $2\times 2$ blocks

(2.7) $[\tau
G_{z}]_{jk}=\frac{e^{i\sqrt{w_{z}}\,|y_{k}-y_{j}|}}{2i}\begin{bmatrix}\zeta_{z}&\text{sgn}(y_{k}-y_{j})\\\
\text{sgn}(y_{k}-y_{j})&\zeta_{z}^{-1}\\\ \end{bmatrix},$ (2.8)
$[\widehat{\tau}\widehat{G}_{z}]_{jk}=\frac{e^{i\sqrt{z}\,|y_{k}-y_{j}|}}{2}\begin{bmatrix}(i\sqrt{z})^{-1}&\text{sgn}(y_{k}-y_{j})\\\
-\text{sgn}(y_{k}-y_{j})&i\sqrt{z}\end{bmatrix},$

where

$\text{sgn}(x):=\begin{cases}-1&x<0\\\ 0&x=0\\\ +1&x>0\,.\end{cases}$

In the following, given an orthogonal projection
$P:\mathbb{C}^{d}\to\mathbb{C}^{d}$, by a slight abuse of notation, we use the
same symbol to denote both the surjection $P:\mathbb{C}^{d}\to\text{\rm
ran}(P)$ and the injection $P:\text{\rm ran}(P)\to\mathbb{C}^{d}$.

######  Theorem 2.1.

The sets of self-adjoint extensions of $S$ and $\widehat{S}$ are both
parametrized by couples $(\Pi,\Theta)$, where
$\Pi:\mathbb{C}^{2n}\to\mathbb{C}^{2n}$ is an othogonal projector and
$\Theta:\text{\rm ran}(\Pi)\to\text{\rm ran}(\Pi)$ is symmetric. The
extensions $\mathsf{D}_{\Pi,\Theta}$ and $\mathsf{H}_{\Pi,\Theta}$ have
resolvents

$(-\mathsf{D}_{\Pi,\Theta}+z)^{-1}=(-\mathsf{D}+z)^{-1}+G_{z}\Pi(\Theta-\Pi\,\tau
G_{z}\Pi)^{-1}\Pi G_{\bar{z}}^{*}\,,\qquad
z\in\varrho(\mathsf{D}_{\Pi,\Theta})\cap\varrho(\mathsf{D})$
$(-\mathsf{H}_{\Pi,\Theta}+z)^{-1}=(-\mathsf{H}+z)^{-1}+\widehat{G}_{z}\Pi(\Theta-\Pi\,\widehat{\tau}\widehat{G}_{z}\Pi)^{-1}\Pi\widehat{G}_{\bar{z}}^{*}\,,\qquad
z\in\varrho(\mathsf{H}_{\Pi,\Theta})\cap\varrho(\mathsf{H})\,.$

Moreover,

$\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})=\\{\Psi\in
L^{2}(\mathbb{R};\mathbb{C}^{2}):\Psi=\Psi_{z}+G_{z}\xi\,,\ \Psi_{z}\in
H^{1}(\mathbb{R};\mathbb{C}^{2})\,,\ \xi\in\text{\rm ran}(\Pi)\,,\
\Pi\tau\Psi=\Theta\xi\\}$
$(-\mathsf{D}_{\Pi,\Theta}+z)\Psi=(-\mathsf{D}+z)\Psi_{z}\,,$ $\text{\rm
dom}(\mathsf{H}_{\Pi,\Theta})=\\{\psi\in
L^{2}(\mathbb{R}):\psi=\psi_{z}+\widehat{G}_{z}\xi\,,\ \psi_{z}\in
H^{2}(\mathbb{R})\,,\ \xi\in\text{\rm ran}(\Pi)\,,\
\Pi\widehat{\tau}\psi=\Theta\xi\\},$
$(-\mathsf{H}_{\Pi,\Theta}+z)\psi=(-\mathsf{H}+z)\psi_{z}\,;$

such representations are $z$-independent and the decompositions of $\Psi$ in
$\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})$ and of $\psi$ in $\text{\rm
dom}(\mathsf{H}_{\Pi,\Theta})$ are unique.

###### Proof.

The statements regarding the resolvents and the actions of the extensions
follow from [22, Theorem 2.1] with $\Gamma_{\Pi,\Theta}(z)$ there defined
either as $\Gamma_{\Pi,\Theta}(z):=\Theta-\Pi\tau G_{z}\Pi$ or as
$\Gamma_{\Pi,\Theta}(z):=\widehat{\Theta}-\Pi\widehat{\tau}\widehat{G}_{z}\Pi$.

As regards the operators domains, we give the proof only for
$\mathsf{D}_{\Pi,\Theta}$, since the one for $\mathsf{H}_{\Pi,\Theta}$ is of
the same kind. By the resolvent formula, one has

$\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})=\\{\Psi\in
L^{2}(\mathbb{R};\mathbb{C}^{2}):\Psi=\Psi_{z}+G_{z}\Pi(\Theta-\Pi\tau
G_{z}\Pi)^{-1}\Pi\tau\Psi_{z}\,,\ \Psi_{z}\in
H^{1}(\mathbb{R};\mathbb{C}^{2})\\}\,.$

Let us define $\xi_{z}:=(\Theta-\Pi\,\tau
G_{z}\Pi)^{-1}\Pi\tau\Psi_{z}\in\text{\rm ran}(\Pi)$; it is not difficult to
check that $\xi_{z}$ does not depend on $z$ and so $\Psi=\Psi_{z}+G_{z}\xi$.
Then

$\Pi\tau\Psi-\Theta\xi=\Pi\tau\Psi_{z}+\Pi\tau
G_{z}\xi-\Theta\xi=\Pi\tau\Psi_{z}-(\Theta-\Pi\tau G_{z}\Pi)\xi=0\,.$

∎

######  Remark 2.2.

Notice that the choice $\Pi=\mathbb{0}$ gives the self-adjoint extensions
$\mathsf{D}$ and $\mathsf{H}$. Therefore, in the following we always suppose
$\Pi\not=\mathbb{0}$

Since we want to extend the relation (1.1) to the case with point
interactions, we also need to consider the self-adjoint extensions of
$\widehat{S}^{\circ}\mathbb{1}$. There are no essential changes with respect
to the case of $\mathbb{C}$-valued functions, the only relevant one being that
the defect indices increase to $(4n,4n)$. The result is of the same kind as in
Theorem 2.1.

######  Theorem 2.3.

The set of the self-adjoint extensions of $\widehat{S}\mathbb{1}$ is
parametrized by couples $(\widehat{\Pi},\widehat{\Theta})$, where
$\widehat{\Pi}:\mathbb{C}^{4n}\to\mathbb{C}^{4n}$ is an othogonal projector
and $\widehat{\Theta}:\text{\rm ran}(\widehat{\Pi})\to\text{\rm
ran}(\widehat{\Pi})$ is symmetric. The extension
$\widehat{\mathsf{H}}_{\Pi,\Theta}$ has resolvent

$\displaystyle(-\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+z)^{-1}=(-\mathsf{H}+z)^{-1}\mathbb{1}+(\widehat{G}_{z}\mathbb{1})\widehat{\Pi}(\widehat{\Theta}-\widehat{\Pi}(\widehat{\tau}\widehat{G}_{z}\mathbb{1})\widehat{\Pi})^{-1}\widehat{\Pi}(\widehat{G}_{\bar{z}}^{*}\mathbb{1}),\quad
z\in\varrho(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})\cap\varrho(\mathsf{H}).$

Moreover,

$\displaystyle\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})=\\{\Psi\in
L^{2}(\mathbb{R};\mathbb{C}^{2}):\Psi=\Psi_{z}+(\widehat{G}_{z}\mathbb{1})\widehat{\xi},\
\Psi_{z}\in H^{2}(\mathbb{R};\mathbb{C}^{2}),\ \widehat{\xi}\in\text{\rm
ran}(\widehat{\Pi}),\
\widehat{\Pi}(\widehat{\tau}\mathbb{1})\Psi=\widehat{\Theta}\widehat{\xi}\,\\},$
$(-\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+z)\Psi=(-\mathsf{H}+z)\mathbb{1}\Psi_{z}\,;$

such representation is $z$-independent and the decomposition of $\Psi$ in
$\text{\rm dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})$ is
unique.

######  Remark 2.4.

By Theorems 2.1 and 2.3, if both $\widehat{\Pi}$ and $\widehat{\Theta}$ are
block diagonal, i.e., $\widehat{\Pi}=\Pi_{1}\oplus\Pi_{2}$ and
$\widehat{\Theta}=\Theta_{1}\oplus\Theta_{1}$, then

$(-\widehat{\mathsf{H}}_{\Pi_{1}\oplus\Pi_{2},\Theta_{1}\oplus\Pi_{2}}+z)^{-1}=(-\mathsf{H}_{\Pi_{1},\Theta_{1}}+z)^{-1}\oplus(-\mathsf{H}_{\Pi_{2},\Theta_{2}}+z)^{-1},$

equivalently,

$\widehat{\mathsf{H}}_{\Pi_{1}\oplus\Pi_{2},\Theta_{1}\oplus\Pi_{2}}=\mathsf{H}_{\Pi_{1},\Theta_{1}}\oplus\mathsf{H}_{\Pi_{2},\Theta_{2}}.$

In particular,

$\widehat{\mathsf{H}}_{\Pi\mathbb{1},\Theta\mathbb{1}}=\mathsf{H}_{\Pi,\Theta}\mathbb{1}\,.$

######  Remark 2.5.

Since $g_{z}$ is the fundamental solution of $-\mathsf{D}+z$, one has

$(-\mathsf{D}_{\Pi,\Theta}+z)\Psi=(-\mathsf{D}_{\Pi,\Theta}+z)(\Psi-
G_{z}\xi)=(-\mathsf{D}+z)\Psi-\sum_{k=1}^{n}\xi_{k}\delta_{y_{k}}\,,$

i.e.,

$\mathsf{D}_{\Pi,\Theta}\Psi=\mathsf{D}\Psi+\sum_{k=1}^{n}\xi_{k}\delta_{y_{k}}\,,\quad\xi\equiv(\xi_{1},\dots,\xi_{n})\,,$

where the action of $\mathsf{D}$ on $\Psi\in L^{2}(\mathbb{R};\mathbb{C}^{2})$
is to be understood in a distributional sense. Analogously,

$\mathsf{H}_{\Pi,\Theta}\psi=\mathsf{H}\psi+\sum_{k=1}^{n}(\xi_{k,1}\delta_{y_{k}}+\xi_{k,2}\delta^{\prime}_{y_{k}}),\quad\xi\equiv((\xi_{1,1},\xi_{1,2}),\dots,(\xi_{n,1},\xi_{n,2}))\,,$
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}\Psi=\mathsf{H}\mathbb{1}\Psi+\sum_{k=1}^{n}(\widehat{\xi}_{k,1}\delta_{y_{k}}+\widehat{\xi}_{k,2}\delta^{\prime}_{y_{k}}),\quad\widehat{\xi}\equiv((\widehat{\xi}_{1,1},\widehat{\xi}_{1,2}),\dots,(\widehat{\xi}_{n,1},\widehat{\xi}_{n,2}))\,.$

In the following, we use the abbreviated notations
$\mathsf{D}_{\Theta}\equiv\mathsf{D}_{\mathbb{1},\Theta}$,
$\mathsf{H}_{\Theta}\equiv\mathsf{H}_{\mathbb{1},\Theta}$,
$\widehat{\mathsf{H}}_{\widehat{\Theta}}\equiv\widehat{\mathsf{H}}_{\mathbb{1},\widehat{\Theta}}\,.$

##  3\. $\mathsf{D}^{2}=\mathsf{H}+\frac{1}{4}$ with point interations

We begin this section by providing an equivalent representation of the domains
and actions of the self-adjoint operators we built in Section 2.

######  Theorem 3.1.

Let $\mathsf{D}_{\Pi,\Theta}$, $\mathsf{H}_{\Pi,\Theta}$ and
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ as in Section 2. Then

$\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})=\\{\Psi\in H^{1}(\mathbb{R}\backslash
Y;\mathbb{C}^{2}):\rho\Psi\in\text{\rm ran}(\Pi),\
\Pi\tau\Psi=\Theta\rho\Psi\\},\quad\mathsf{D}_{\Pi,\Theta}\Psi=\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi\,,$ $\text{\rm dom}(\mathsf{H}_{\Pi,\Theta})=\\{\psi\in
H^{2}(\mathbb{R}\backslash Y):\widehat{\rho}\psi\in\text{\rm ran}(\Pi),\
\Pi\widehat{\tau}\psi=\Theta\widehat{\rho}\psi\\},\quad\mathsf{H}_{\Pi,\Theta}\psi=\mathsf{H}_{\mathbb{R}\backslash
Y}\psi\,,$ $\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})=\\{\Psi\in
H^{2}(\mathbb{R}\backslash
Y;\mathbb{C}^{2}):(\widehat{\rho}\mathbb{1})\Psi\in\text{\rm
ran}(\widehat{\Pi}),\
\widehat{\Pi}(\widehat{\tau}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}\mathbb{1})\Psi\\},\quad\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}\Psi=(\mathsf{H}_{\mathbb{R}\backslash
Y}\mathbb{1})\Psi\,,$

where $\mathsf{D}_{\mathbb{R}\backslash Y}$ and
$\mathsf{H}_{\mathbb{R}\backslash Y}$ denote the free Dirac and Schrödinger
operators acting on distributions supported in $\mathbb{R}\backslash Y$,

$\displaystyle\rho:H^{1}(\mathbb{R}\backslash
Y;\mathbb{C}^{2})\to\mathbb{C}^{2n}\,,\quad\rho\Psi:=\big{(}\rho_{y_{1}}\Psi\,,\dots,\rho_{y_{n}}\Psi\big{)}\,,\qquad\rho_{y}\Psi:=i\sigma_{1}[\Psi]_{y}\,,$
$\widehat{\rho}:H^{2}(\mathbb{R}\backslash
Y)\to\mathbb{C}^{2n}\,,\qquad\widehat{\rho}\psi:=\big{(}\widehat{\rho}_{y_{1}}\psi\,,\dots,\widehat{\rho}_{y_{n}}\psi\big{)}\,,\qquad\widehat{\rho}_{y}\psi:=[\psi^{\prime}\,]_{y}\oplus[-\psi]_{y}\,,$

$[f]_{y}:=f(y^{+})-f(y^{-})\,.$

###### Proof.

Let $\Psi=\Psi_{z}+G_{z}\xi\in\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})$. One
has $\Psi_{z}\in H^{1}(\mathbb{R};\mathbb{C}^{2})\subset
H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{2})$ and $G_{z}\xi\in
H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{2})$; therefore, $\Psi\in
H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{2})$. By
$[G_{z}\xi]_{y}=i\sigma_{1}\xi$, one gets $\rho G_{z}\xi=\xi$; furthermore, by
$H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{2})\subset
C_{b}(\mathbb{R};\mathbb{C}^{2})$, one gets $\rho\Psi_{z}=0$. Therefore,

$\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})\subseteq{\mathscr{D}}:=\\{\Psi\in
H^{1}(\mathbb{R}\backslash Y;\mathbb{C}^{2}):\rho\Psi\in\text{\rm ran}(\Pi),\
\Pi\tau\Psi=\Theta\rho\Psi\\}\,.$

By Remark 2.5, $\mathsf{D}_{\Pi,\Theta}\Psi=\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi$ for any $\Psi\in\text{\rm dom}(\mathsf{D}_{\Pi,\Theta})$, i.e.,
$\mathsf{D}_{\Pi,\Theta}\subset\mathsf{D}_{\mathbb{R}\backslash
Y}|{\mathscr{D}}$. Moreover, by integration by parts,
$\mathsf{D}_{\mathbb{R}\backslash Y}|{\mathscr{D}}$ is symmetric; hence, since
$\mathsf{D}_{\Pi,\Theta}$ is self-adjoint, one gets
$\mathsf{D}_{\Pi,\Theta}=\mathsf{D}_{\mathbb{R}\backslash Y}|{\mathscr{D}}$.

The proofs for $\mathsf{H}_{\Pi,\Theta}$ and
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ are of the same kind,
using the relation $\widehat{\rho}\widehat{G}_{z}\xi=\xi$. ∎

######  Remark 3.2.

Notice that $\psi\in H^{1}(\mathbb{R}\backslash Y)$ belongs to
$H^{1}(\mathbb{R})$ if and only if $[\psi]_{y_{k}}=0$ for any $k$ and
consequently $\psi\in H^{2}(f\mathbb{R}\backslash Y)$ belongs to
$H^{2}(\mathbb{R})$ if and only if $[\psi]_{y_{k}}=[\psi^{\prime}]_{y_{k}}=0$
for any $k$.

By Theorem 3.1 and by

$(\mathsf{D}_{\mathbb{R}\backslash
Y})^{2}=\left(\mathsf{H}_{\mathbb{R}\backslash
Y}+\frac{1}{4}\right)\mathbb{1}\,,$

given the couple $(\Pi,\Theta)$, one gets that the couple
$(\widehat{\Pi},\widehat{\Theta})$ is such that

(3.1)
$(\mathsf{D}_{\Pi,\Theta})^{2}=\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+\frac{\mathbb{1}}{4}\,,$

if and only if

(3.2) $\text{\rm dom}((\mathsf{D}_{\Pi,\Theta})^{2})=\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})\,.$

Therefore, exploiting the definitions of the operator domains in Theorem 3.1,
there exists a couple $(\widehat{\Pi},\widehat{\Theta})$ for which (3.1) holds
if and only if, given $(\Pi,\Theta)$, there exists
$(\widehat{\Pi},\widehat{\Theta})$, $\widehat{\Pi}$ an orthogonal projector in
$\mathbb{C}^{4n}$ and $\widehat{\Theta}$ symmetric in $\text{\rm
ran}(\widehat{\Pi})$, such that

(3.3) $\begin{cases}\rho\Psi\oplus\rho\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi\in\text{\rm ran}(\Pi\oplus\Pi)\\\
(\Pi\oplus\Pi)\tau\Psi\oplus\tau\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi=(\Theta\oplus\Theta)\rho\Psi\oplus\rho\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi\end{cases}\quad\iff\qquad\begin{cases}(\widehat{\rho}\mathbb{1})\Psi\in\text{\rm
ran}(\widehat{\Pi})\\\
\widehat{\Pi}(\widehat{\tau}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}\mathbb{1})\Psi\,.\end{cases}$

###  3.1. Spectral correspondence

The relation (3.1) entails $\pm z\in\varrho(\mathsf{D}_{\Pi,\Theta})$ if and
only if
$z^{2}-\frac{1}{4}\in\varrho(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})$,
equivalently, $\pm\lambda\in\sigma(\mathsf{D}_{\Pi,\Theta})$ if and only if
$\lambda^{2}-\frac{1}{4}\in\sigma(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})$,
and

(3.4)
$(-\mathsf{D}_{\Pi,\Theta}+z)^{-1}=(\mathsf{D}_{\Pi,\Theta}+z)\left(-\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+\left(z^{2}-\frac{1}{4}\right)\mathbb{1}\right)^{\\!\\!-1}\,.$

Furthermore, since, by the invariance of the essential spectrum by finite-rank
perturbations,

$\sigma_{ess}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})=\sigma_{ess}(\mathsf{H}\mathbb{1})=[0,\infty)\,,\qquad\sigma_{ess}(\mathsf{D}_{\widehat{\Pi},\widehat{\Theta}})=\sigma_{ess}(\mathsf{D})=\left[-\infty,-\frac{1}{2}\right)\cup\left[\frac{1}{2},+\infty\right)\,,$

one gets

$\lambda\in\sigma_{disc}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})\cap\left[-\frac{1}{4},0\right)\quad\iff\quad\pm\left(\lambda+\frac{1}{4}\right)^{\frac{1}{2}}\in\sigma_{disc}(\mathsf{D}_{\Pi,\Theta})\,.$

By the resolvent formulae in Theorems 2.1 and 2.3,

(3.5)
$\lambda\in\sigma_{disc}(\mathsf{D}_{\Pi,\Theta})\quad\iff\quad\text{$\lambda\in(-1/2,1/2)\quad$
and $\quad\det(\Theta-\Pi\tau G_{\lambda}\Pi)=0$}\,,$ (3.6)
$\lambda\in\sigma_{disc}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})\quad\iff\quad\text{$\lambda\in(-\infty,0)\quad$
and
$\quad\det(\widehat{\Theta}-\widehat{\Pi}(\widehat{\tau}\widehat{G}_{\lambda}\mathbb{1})\widehat{\Pi})=0$}\,.$

Now, we solve (3.3) starting from the simplest case $n=1$, $\Pi=\mathbb{1}$
and then proceeding step-by-step towards the most general case.

###  3.2. The case $n=1$, $\Pi=\mathbb{1}$.

By (3.3), given the $2\times 2$ Hermitian matrix $\Theta$, we need to find the
$4\times 4$ Hermitian matrix $\widehat{\Theta}$ such that

(3.7) $\begin{bmatrix}\tau_{y}\Psi\\\
\tau_{y}\mathsf{D}_{\mathbb{R}\backslash\\{y\\}}\Psi\end{bmatrix}=\begin{bmatrix}\Theta&{\mathbb{0}}\\\
{\mathbb{0}}&\Theta\end{bmatrix}\begin{bmatrix}\rho_{y}\Psi\\\
\rho_{y}\mathsf{D}_{\mathbb{R}\backslash\\{y\\}}\Psi\end{bmatrix}\quad\iff\quad(\widehat{\tau}_{y}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}_{y}\mathbb{1})\Psi\,.$

To solve (3.7), at first we look for the two invertible matrices $M_{1}$ and
$M_{2}$ such that

(3.8)
$\displaystyle(\widehat{\tau}_{y}\mathbb{1})\Psi=M_{1}\begin{bmatrix}\tau_{y}\Psi\\\
\tau_{y}\mathsf{D}_{\mathbb{R}\backslash\\{y\\}}\Psi\end{bmatrix},\qquad(\widehat{\rho}_{y}\mathbb{1})\Psi=M_{2}\begin{bmatrix}\rho_{y}\Psi\\\
\rho_{y}\mathsf{D}_{\mathbb{R}\backslash\\{y\\}}\Psi\end{bmatrix}.$

By direct calculations, one gets

(3.9) $M_{1}=\begin{bmatrix}1&0&0&0\\\ 0&\frac{i}{2}&0&i\\\ 0&1&0&0\\\
-\frac{i}{2}&0&i&0\end{bmatrix},\qquad
M_{2}=\begin{bmatrix}\frac{1}{2}&0&1&0\\\ 0&i&0&0\\\ 0&-\frac{1}{2}&0&1\\\
i&0&0&0\end{bmatrix}.$

Therefore, (3.7) rewrites as

$M_{1}^{-1}(\widehat{\tau}_{y}\mathbb{1})\Psi=(\Theta\oplus\Theta)M_{2}^{-1}(\widehat{\rho}_{y}\mathbb{1})\Psi\quad\iff\quad(\widehat{\tau}_{y}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}_{y}\mathbb{1})\Psi$

and so the relation between $\widehat{\Theta}$ and $\Theta$ is given by

(3.10) $\displaystyle\widehat{\Theta}=$ $\displaystyle
M_{1}(\Theta\oplus\Theta)M_{2}^{-1}\,.$

By

$\widehat{\Theta}=\widehat{\Theta}^{*}\quad\iff\quad
M^{*}_{1}M_{2}(\Theta\oplus\Theta)=(\Theta\oplus\Theta)M_{2}^{*}M_{1},$

$\widehat{\Theta}$ is symmetric by the relations

(3.11) $M_{1}^{*}M_{2}=\begin{bmatrix}\mathbb{0}&\mathbb{1}\\\
\mathbb{1}&\mathbb{0}\end{bmatrix}=M_{2}^{*}M_{1}\,.$

More explicitly, if

$\Theta=\begin{bmatrix}a&b\\\ \bar{b}&d\end{bmatrix},\qquad
a,d\in\mathbb{R},\,b\in\mathbb{C}\,,$

then $\widehat{\Theta}$ is represented by the Hermitian matrix

$\widehat{\Theta}=\begin{bmatrix}0&-ib&0&-ia\\\ i\bar{b}&d&id&0\\\
0&-id&0&-i\bar{b}\\\ ia&0&ib&-a\end{bmatrix}\,.$

If $a=d=0$ and $b\in\mathbb{R}$, i.e., if $\Theta=b\sigma_{1}$, then
$\widehat{\Theta}=b(\sigma_{2}\oplus\sigma_{2})\equiv b\sigma_{2}\mathbb{1}$
and, by Remark 2.4, the corresponding Schrödinger operator in
$L^{2}(\mathbb{R};\mathbb{C}^{2})$ is block diagonal:

(3.12)
$(\mathsf{D}_{b\sigma_{1}})^{2}=\left(\mathsf{H}_{b\sigma_{2}}+\frac{1}{4}\right)\\!\mathbb{1}\,.$

###  3.3. The case $n=1$, $\Pi\not=\mathbb{1}$.

Here we take $\Pi:\mathbb{C}^{2}\to\mathbb{C}^{2}$ a not trivial orthogonal
projection, i.e., $\text{dim}(\text{\rm ran}(\Pi))=1$, and
$\Theta\in\mathbb{R}$. By (3.8), (3.3) rewrites as

(3.13) $\begin{cases}(\widehat{\rho}_{y}\mathbb{1})\Psi\in\text{\rm
ran}(M_{2}(\Pi\oplus\Pi))\\\
M_{2}(\Pi\oplus\Pi)M_{1}^{-1}(\widehat{\tau}_{y}\mathbb{1})\Psi=\Theta(\widehat{\rho}_{y}\mathbb{1})\Psi\end{cases}\quad\iff\qquad\begin{cases}(\widehat{\rho}_{y}\mathbb{1})\Psi\in\text{\rm
ran}(\widehat{\Pi})\\\
\widehat{\Pi}(\widehat{\tau}_{y}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}_{y}\mathbb{1})\Psi\,.\end{cases}$

Therefore, $\widehat{\Pi}:\mathbb{C}^{4}\to\mathbb{C}^{4}$ is the orthogonal
projection onto the $2$-dimensional subspace

$\text{\rm ran}(\widehat{\Pi})=\text{\rm ran}(M_{2}(\Pi\oplus\Pi))=\text{\rm
ran}(M_{2}(\Pi\oplus\Pi)M_{1}^{-1})\,,$

i.e.,

$\displaystyle\widehat{\Pi}=$ $\displaystyle
M_{2}(\Pi\oplus\Pi)((\Pi\oplus\Pi)M_{2}^{*}M_{2}(\Pi\oplus\Pi))^{-1}(\Pi\oplus\Pi)M_{2}^{*}$
$\displaystyle=$ $\displaystyle
M_{2}(\Pi\oplus\Pi)(M_{2}^{*}M_{2})^{-1}(\Pi\oplus\Pi)M_{2}^{*}$
$\displaystyle=$
$\displaystyle(M_{2}(\Pi\oplus\Pi)M_{2}^{-1})(M_{2}(\Pi\oplus\Pi)M_{2}^{-1})^{*}\,.$

By (3.11), $M_{2}(\Pi\oplus\Pi)M_{1}^{-1}$ is symmetric. Hence, $\text{\rm
ran}(\widehat{\Pi})=\text{\rm ker}(M_{2}(\Pi\oplus\Pi)M_{1}^{-1})^{\perp}$ and
the symmetric operator

$M_{2}(\Pi\oplus\Pi)M_{1}^{-1}:\text{\rm ran}(\widehat{\Pi})\to\text{\rm
ran}(\widehat{\Pi})$

is a bijection. Then, (3.13) gives

$\widehat{\Theta}:\text{\rm ran}(\widehat{\Pi})\to\text{\rm
ran}(\widehat{\Pi}),\qquad\widehat{\Theta}:=\Theta(M_{2}(\Pi\oplus\Pi)M_{1}^{-1})^{-1}.$

###  3.4. The case $n>1$, $\Pi=\mathbb{1}$.

In order to exploit the results from the $n=1$ case, we introduce the unitary
operator

(3.14) $U:\mathbb{C}^{4n}\to\mathbb{C}^{4n}\,,\qquad
U(\xi_{1},\xi_{2},\dots,\xi_{2n}):=(\xi_{1},\xi_{n+1},\xi_{2},\xi_{n+2},\dots,\xi_{n},\xi_{2n})\,,\quad\xi_{k}\in\mathbb{C}^{2}\,.$

By such a definition,

$U(\tau\Psi\oplus\tau\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi)=\left(\,\begin{bmatrix}\tau_{y_{1}}\Psi\\\
\tau_{y_{1}}\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi\end{bmatrix},\dots,\begin{bmatrix}\tau_{y_{n}}\Psi\\\
\tau_{y_{n}}\mathsf{D}_{\mathbb{R}\backslash Y}\Psi\end{bmatrix}\,\right),$
$U(\rho\Psi\oplus\rho\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi)=\left(\,\begin{bmatrix}\rho_{y_{1}}\Psi\\\
\rho_{y_{1}}\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi\end{bmatrix},\dots,\begin{bmatrix}\rho_{y_{n}}\Psi\\\
\rho_{y_{n}}\mathsf{D}_{\mathbb{R}\backslash Y}\Psi\end{bmatrix}\,\right)\,.$

Therefore, setting

$M_{1}^{\oplus}:\mathbb{C}^{4n}\to\mathbb{C}^{4n}\,,\qquad
M_{1}^{\oplus}:=M_{1}\oplus\dots\oplus M_{1}\,,$
$M_{2}^{\oplus}:\mathbb{C}^{4n}\to\mathbb{C}^{4n}\,,\qquad
M_{2}^{\oplus}:=M_{2}\oplus\dots\oplus M_{2}\,,$

by (3.8), one gets

$M_{1}^{\oplus}U(\tau\Psi\oplus\tau\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi)=\big{(}(\widehat{\tau}_{y_{1}}\mathbb{1})\Psi,\dots,(\widehat{\tau}_{y_{n}}\mathbb{1})\Psi\big{)}=U(\widehat{\tau}\mathbb{1})\Psi\,,$
$M_{2}^{\oplus}U(\rho\Psi\oplus\rho\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi)=\big{(}(\widehat{\rho}_{y_{1}}\mathbb{1})\Psi,\dots,(\widehat{\rho}_{y_{n}}\mathbb{1})\Psi\big{)}=U(\widehat{\rho}\mathbb{1})\Psi$

and so (3.3) rewrites as

$U^{*}(M_{1}^{\oplus})^{-1}U(\widehat{\tau}\mathbb{1})\Psi=(\Theta\oplus\Theta)U^{*}(M_{2}^{\oplus})^{-1}U(\widehat{\rho}\mathbb{1})\Psi\quad\iff\quad(\widehat{\tau}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}\mathbb{1})\Psi\,.$

This gives

(3.15)
$\widehat{\Theta}=U^{*}M_{1}^{\oplus}U(\Theta\oplus\Theta)U^{*}(M_{2}^{\oplus})^{-1}U\,.$

Such a operator $\widehat{\Theta}$ is symmetric by

(3.16)
$U^{*}(M_{1}^{\oplus})^{*}M_{2}^{\oplus}U=\begin{bmatrix}\mathbb{0}&\mathbb{1}\\\
\mathbb{1}&\mathbb{0}\end{bmatrix}=U^{*}(M_{2}^{\oplus})^{*}M^{\oplus}_{1}U\,.$

The relations (3.16) generalize (3.11), since $U=\mathbb{1}$ whenever $n=1$,
and are a consequence of $\eqref{M1M2}$ itself and the definition (3.14).

###  3.5. The case $n>1$, $\Pi\not=\mathbb{1}$.

Finally, we consider the most general case. Using the unitary
$U:\mathbb{C}^{4n}\to\mathbb{C}^{4n}$ as in the previous section, (3.3)
rewrites as

(3.17)
$\begin{cases}U^{*}(M^{\oplus}_{2})^{-1}U(\widehat{\rho}\mathbb{1})\Psi\in\text{\rm
ran}(\Pi\oplus\Pi)\\\
(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U(\widehat{\tau}\mathbb{1})\Psi=(\Theta\oplus\Theta)U^{*}(M_{2}^{\oplus})^{-1}U(\widehat{\rho}\mathbb{1})\Psi\end{cases}\iff\quad\begin{cases}(\widehat{\rho}\mathbb{1})\Psi\in\text{\rm
ran}(\widehat{\Pi})\\\
\widehat{\Pi}(\widehat{\tau}\mathbb{1})\Psi=\widehat{\Theta}(\widehat{\rho}\mathbb{1})\Psi\,.\end{cases}$

This gives the orthogonal projector
$\widehat{\Pi}:\mathbb{C}^{4n}\to\mathbb{C}^{4n}$, with $\text{dim}(\text{\rm
ran}(\widehat{\Pi}))=2\,\text{dim}(\text{\rm ran}(\Pi))$, such that

(3.18) $\text{\rm ran}(\widehat{\Pi})=\text{\rm
ran}(U^{*}M_{2}^{\oplus}U(\Pi\oplus\Pi))=\text{\rm
ran}(U^{*}M_{2}^{\oplus}U(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U)\,,$

i.e.,

$\displaystyle\widehat{\Pi}=$
$\displaystyle(U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi))\big{(}(U^{*}M_{2}^{\oplus}U(\Pi\oplus\Pi))^{*}(U^{*}M_{2}^{\oplus}U(\Pi\oplus\Pi))\big{)}^{-1}(U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi))^{*}$
$\displaystyle=$ $\displaystyle
U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)((U^{*}M^{\oplus}_{2}U)^{*}U^{*}M^{\oplus}_{2}U)^{-1}U(\Pi\oplus\Pi)U^{*}(M^{\oplus}_{2})^{*}U$
$\displaystyle=$
$\displaystyle\big{(}U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)U^{*}(M^{\oplus}_{2})^{-1}\big{)}\big{(}U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)U^{*}(M^{\oplus}_{2})^{-1}\big{)}^{*}\,,$

and $\widehat{\Theta}:\text{\rm ran}(\widehat{\Pi})\to\text{\rm
ran}(\widehat{\Pi})$,

(3.19)
$\widehat{\Theta}:=\big{(}U^{*}M_{2}^{\oplus}U(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U\big{)}^{-1}U^{*}M_{2}^{\oplus}U(\Theta\oplus\Theta)U^{*}(M_{2}^{\oplus})^{-1}U.$

By (3.16), $U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U$ is
symmetric. Therefore, by

$\text{\rm ran}(\widehat{\Pi})=\text{\rm
ran}(U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U)=\text{\rm
ker}(U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U)^{\perp}\,,$

the operator

$U^{*}M^{\oplus}_{2}U(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}U:\text{\rm
ran}(\widehat{\Pi})\to\text{\rm ran}(\widehat{\Pi})$

is a bijection and $\widehat{\Theta}$ is well defined. To conclude, we have to
show that $\widehat{\Theta}$ is symmetric. By (3.19) and by
$U^{*}U=\mathbb{1}$,

$\text{$\widehat{\Theta}$ is
symmetric}\iff\text{$M_{2}^{\oplus}U(\Theta\oplus\Theta)(\Pi\oplus\Pi)U^{*}(M_{1}^{\oplus})^{-1}$
is symmetric}$

and so $\widehat{\Theta}$ is symmetric by (3.16).

######  Remark 3.3.

Let us point out that it is not necessary to determine $\widehat{\Pi}$ and
$\widehat{\Theta}$ explicitly in order to write down the domain of
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$. Indeed, by (3.3),

$\displaystyle\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})=\big{\\{}\Psi\in
H^{2}(\mathbb{R}\backslash
Y;\mathbb{C}^{2}):\rho\Psi\oplus\rho\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi\in\text{\rm ran}(\Pi\oplus\Pi)$
$\displaystyle(\Pi\tau\Psi)\oplus(\Pi\tau\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi)=(\Theta\rho\Psi)\oplus(\Theta\rho\mathsf{D}_{\mathbb{R}\backslash
Y}\Psi)$ $\displaystyle\big{\\}}\,.$

However, one needs to know $\widehat{\Pi}$ and $\widehat{\Theta}$ in order to
write down the resolvent of
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$, according to Theorem
2.3.

The above representation of $\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})$ suggests an
alternative way to build the self-adjoint extensions of
$\widehat{S}^{\circ}\mathbb{1}=\mathsf{H}\mathbb{1}|C^{\infty}_{comp}(\mathbb{R};\mathbb{C}^{2})$:
one can apply the results in [21] and [22] to $\mathsf{H}\mathbb{1}|\text{\rm
ker}(\widetilde{\tau})$, where

$\widetilde{\tau}:H^{2}(\mathbb{R};\mathbb{C}^{2})\to\mathbb{C}^{2n}\oplus\mathbb{C}^{2n}\,,\qquad\widetilde{\tau}\Psi:=\tau\Psi\oplus\tau\mathsf{D}\Psi\,.$

In that case, the family of self-adjoint extensions of
$\widehat{S}^{\circ}\mathbb{1}$ is represented by operators of the kind
$\widetilde{H}_{\widetilde{\Pi},\widetilde{\Theta}}$, where $\widetilde{\Pi}$
is an orthogonal projector in $\mathbb{C}^{2n}\oplus\mathbb{C}^{2n}$ and
$\widetilde{\Theta}$ is a symmetric operator in $\text{\rm
ran}(\widetilde{\Theta})$. With respect to this parametrization, one has that
$\mathsf{D}_{\Pi,\Theta}^{2}=\widetilde{\mathsf{H}}_{\widetilde{\Pi},\widetilde{\Theta}}+\frac{\mathbb{1}}{4}$
if and only if $\widetilde{\Pi}=\Pi\oplus\Pi$ and
$\widetilde{\Theta}=\Theta\oplus\Theta$. Even if such a correspondence is more
explicit than the one which uses the couple
$(\widehat{\Pi},\widehat{\Theta})$, it has the drawback that it works with a
representation of the family of self-adjoint extensions of
$\widehat{S}^{\circ}\mathbb{1}$ which is different from the usual one and
which lacks of the analogous of the property
$\widehat{\mathsf{H}}_{\Pi\mathbb{1},\Theta\mathbb{1}}=\mathsf{H}_{\Pi,\Theta}\mathbb{1}$.
Therefore, in this paper we prefer to work with the family
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$.

######  Remark 3.4.

Suppose that for any $\Psi\equiv(\psi_{1},\psi_{2})\in\text{\rm
dom}(\mathsf{D}_{\Pi,\Theta})$ one has

$\begin{cases}\rho\Psi\in\text{\rm ran}(\Pi)\\\
\Pi\tau\Psi=\Theta\rho\Psi\end{cases}\iff\quad\begin{cases}B_{1}(\psi_{1})=0\\\
B_{2}(\psi_{2})=0\,,\end{cases}$

with some linear operators $B_{1}:H^{1}(\mathbb{R}\backslash
Y)\to\mathbb{C}^{d_{1}}$ and $B_{2}:H^{1}(\mathbb{R}\backslash
Y)\to\mathbb{C}^{d_{2}}$. Then, by the representation of $\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})$ in Remark3.3,
there follows that the boundary conditions for
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ rewrites as

$\begin{cases}B_{1}(\psi_{1})=0\\\
B_{1}\big{(}-i\psi^{\prime}_{2}+\frac{1}{2}\,\psi_{1}\big{)}=0\\\
B_{2}(\psi_{2})=0\\\
B_{2}\big{(}-i\psi^{\prime}_{1}-\frac{1}{2}\,\psi_{1}\big{)}=0\end{cases}\equiv\quad\begin{cases}B_{1}(\psi_{1})=0\\\
B_{2}(\psi_{1}^{\prime})=0\\\ B_{2}(\psi_{2})=0\\\
B_{1}(\psi^{\prime}_{2})=0\,.\end{cases}$

This entails

$\text{\rm
dom}(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}})=\text{\rm
dom}(\mathsf{H}_{1,2})\oplus\text{\rm
dom}(\mathsf{H}_{2,1})\,,\qquad(\mathsf{D}_{\Pi,\Theta})^{2}=\left(\mathsf{H}_{1,2}+\frac{1}{4}\right)\oplus\left(\mathsf{H}_{2,1}+\frac{1}{4}\right)\,,$

where the self-adjoint operators $\mathsf{H}_{j,k}:\text{\rm
dom}(\mathsf{H}_{j,k})\subseteq L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ are
defined by

$\text{\rm dom}(\mathsf{H}_{1,2}):=\\{\psi\in H^{2}(\mathbb{R}\backslash
Y):B_{1}(\psi)=0,\
B_{2}(\psi^{\prime})=0\\}\,,\quad\mathsf{H}_{1,2}\psi:=\mathsf{H}_{\mathbb{R}\backslash
Y}\psi\,.$ $\text{\rm dom}(\mathsf{H}_{2,1}):=\\{\psi\in
H^{2}(\mathbb{R}\backslash Y):B_{2}(\psi)=0,\
B_{1}(\psi^{\prime})=0\\}\,,\quad\mathsf{H}_{2,1}\psi:=\mathsf{H}_{\mathbb{R}\backslash
Y}\psi\,.$

##  4\. Applications

###  4.1. Local boundary conditions.

Here we consider the case corresponding to local boundary conditions for the
Dirac operator, i.e., boundary conditions which do not couple the values of
$\Psi$ at different point. That means

$\Pi=\Pi_{1}\oplus\dots\oplus\Pi_{n}\,,\qquad\Pi_{k}:\mathbb{C}^{2}\to\mathbb{C}^{2}\,,\qquad
1\leq k\leq n\,,$
$\Theta=\Theta_{1}\oplus\dots\oplus\Theta_{n}\,,\qquad\Theta_{k}:\text{\rm
ran}(\Pi_{k})\to\text{\rm ran}(\Pi_{k})\,,\qquad 1\leq k\leq n\,.$

In this case, by

$U((\Pi_{1}\oplus\dots\oplus\Pi_{n})\oplus(\Pi_{1}\oplus\dots\oplus\Pi_{n}))U^{*}=(\Pi_{1}\oplus\Pi_{1})\oplus\dots\oplus(\Pi_{n}\oplus\Pi_{n})\,,$

one gets, by (3.18),

$\displaystyle\text{\rm ran}(\widehat{\Pi})=$ $\displaystyle\text{\rm
ran}(U^{*}(M_{2}(\Pi_{1}\oplus\Pi_{1})M_{1}^{-1})\oplus\dots\oplus(M_{2}(\Pi_{n}\oplus\Pi_{n})M_{1}^{-1})U)$
$\displaystyle=$ $\displaystyle\text{\rm
ran}(U^{*}(\widehat{\Pi}_{1}\oplus\dots\oplus\widehat{\Pi}_{n})U)\,,$

where, in the case $\Pi_{k}\not=\mathbb{1}$,
$\widehat{\Pi}_{k}:\mathbb{C}^{4}\to\mathbb{C}^{4}$ is the orthogonal
projector onto the $2$-dimensional subspace

(4.1) $\text{\rm ran}(\widehat{\Pi}_{k})=\text{\rm
ran}(M_{2}(\Pi_{k}\oplus\Pi_{k})M_{1}^{-1})\,,$

otherwise $\widehat{\Pi}_{k}=\mathbb{1}$. Then, by (3.19) and by

$\displaystyle
U((\Pi_{1}\Theta_{1}\Pi_{1}\oplus\dots\oplus\Pi_{n}\Theta_{n}\Pi_{n})\oplus(\Pi_{1}\Theta_{1}\Pi_{1}\oplus\dots\oplus\Pi_{n}\Theta_{n}\Pi_{n}))U^{*}$
$\displaystyle=$
$\displaystyle(\Pi_{1}\Theta_{1}\Pi_{1}\oplus\Pi_{1}\Theta_{1}\Pi_{1})\oplus\dots\oplus(\Pi_{n}\Theta_{n}\Pi_{n}\oplus\Pi_{n}\Theta_{n}\Pi_{n})\,,$

one obtains

$\displaystyle\widehat{\Theta}=U^{*}(\widehat{\Theta}_{1}\oplus\dots\oplus\widehat{\Theta}_{n})U\,,$

where, in the case $\Pi_{k}\not=\mathbb{1}$, $\Theta_{k}\in\mathbb{R}$,

$\widehat{\Theta}_{k}:\text{\rm ran}(\widehat{\Pi}_{k})\to\text{\rm
ran}(\widehat{\Pi}_{k})\,,\qquad\widehat{\Theta}_{k}=\Theta_{k}M_{1}(\Pi_{k}\oplus\Pi_{k})M_{2}^{-1}\,,$

otherwise,

$\widehat{\Theta}_{k}:\mathbb{C}^{4}\to\mathbb{C}^{4}\,,\qquad\widehat{\Theta}_{k}=M_{1}(\Theta_{k}\oplus\Theta_{k})M_{2}^{-1}\,.$

Therefore, the corresponding boundary conditions for
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ are

$\widehat{\rho}_{y_{k}}\Psi\in\text{\rm
ran}(\widehat{\Pi}_{k})\,,\qquad\widehat{\Pi}_{k}\widehat{\tau}_{y_{k}}\Psi=\widehat{\Theta}_{k}\widehat{\rho}_{y_{k}}\Psi\,,\qquad
1\leq k\leq n\,,$

and so they are local as well.

###  4.2. Gesztesy-Šeba realizations

These two families of self-adjoint realizations of the Dirac operator with
local point interactions correspond, in the non relativistic limit, to
Schrödinger operators with local point interactions either of $\delta$-type or
of $\delta^{\prime}$-type (see [17], [1, Appendix J], [14]). The operators in
the $\alpha$-family have self-adjointness domains

(4.2) $\text{\rm dom}(\mathsf{D}_{\alpha})=\\{\Psi\equiv(\psi_{1},\psi_{2})\in
H^{1}(\mathbb{R})\oplus H^{1}(\mathbb{R}\backslash
Y):[\psi_{2}]_{y_{k}}=-i\alpha_{k}\psi_{1}(y_{k}),\ 1\leq k\leq
n\\},\quad\alpha_{k}\in\mathbb{R}\,,$

and the ones in the $\beta$-family have self-adjointness domains

(4.3) $\text{\rm dom}(\mathsf{D}_{\beta})=\\{\Psi\equiv(\psi_{1},\psi_{2})\in
H^{1}(\mathbb{R}\backslash Y)\oplus
H^{1}(\mathbb{R}):[\psi_{1}]_{y_{k}}=-i\beta_{k}\psi_{2}(y_{k}),\ 1\leq k\leq
n\\},\quad\beta_{k}\in\mathbb{R}\,.$

Since the cases where all the $\alpha_{k}$’s or all the $\beta_{k}$’s are
equal to zero correspond to $\mathsf{D}$, and the cases where there are
$0<m<n$ $\alpha_{k}$’s or $\beta_{k}$’s which are zero reduce to the cases
with $(n-m)$ point interactions, without loss of generality we can suppose
that all the $\alpha_{k}$’s and $\beta_{k}$’s are different from zero. By
Theorem 3.1 and Remark 3.2, one has

$\mathsf{D}_{\alpha}=\mathsf{D}_{\Pi^{(\alpha)},\Theta^{(\alpha)}}\,,\quad\Pi^{(\alpha)}=\Pi^{(\alpha)}_{1}\oplus\dots\oplus\Pi^{(\alpha)}_{n}\,,\quad\Theta^{(\alpha)}=\Theta^{(\alpha)}_{1}\oplus\dots\oplus\Theta^{(\alpha)}_{n}\,,$

where

$\Pi^{(\alpha)}_{k}(\xi_{1},\xi_{2})=(\xi_{1},0)\,,\qquad\Theta_{k}^{(\alpha)}:\mathbb{C}\to\mathbb{C}\,,\quad\Theta_{k}^{(\alpha)}=\alpha_{k}^{-1}$

and

$\mathsf{D}_{\beta}=\mathsf{D}_{\Pi^{(\beta)},\Theta^{(\beta)}}\,,\quad\Pi^{(\beta)}=\Pi^{(\beta)}_{1}\oplus\dots\oplus\Pi^{(\beta)}_{n}\,,\quad\Theta^{(\beta)}=\Theta^{(\beta)}_{1}\oplus\dots\oplus\Theta^{(\beta)}_{n}\,,$

where

$\Pi^{(\beta)}_{k}(\xi_{1},\xi_{2})=(0,\xi_{2})\,,\qquad\Theta_{k}^{(\beta)}:\mathbb{C}\to\mathbb{C}\,,\quad\Theta_{k}^{(\beta)}=\beta_{k}^{-1}\,.$

Therefore,

$(\mathsf{D}_{\alpha})^{2}=\widehat{\mathsf{H}}_{\alpha}+\frac{\mathbb{1}}{4}\,,\qquad$

where

$\widehat{\mathsf{H}}_{\alpha}=\widehat{\mathsf{H}}_{\widehat{\Pi}^{(\alpha)},\widehat{\Theta}^{(\alpha)}}\,,$
$\text{\rm ran}\big{(}\widehat{\Pi}^{(\alpha)}\big{)}=\text{\rm
ran}\big{(}U^{*}\big{(}\widehat{\Pi}^{(\alpha)}_{1}\oplus\dots\oplus\widehat{\Pi}^{(\alpha)}_{n}\big{)}U\big{)}\,,\quad\widehat{\Theta}^{(\alpha)}=U^{*}\big{(}\widehat{\Theta}^{(\alpha)}_{1}\oplus\dots\oplus\widehat{\Theta}^{(\alpha)}_{n}\big{)}U\,,$
$\text{\rm ran}(\widehat{\Pi}^{(\alpha)}_{k})=\text{\rm
ran}(M_{2}(\Pi^{(\alpha)}_{k}\oplus\Pi^{(\alpha)}_{k}))=\mathbb{C}\oplus\\{0\\}\oplus\\{0\\}\oplus\mathbb{C}\equiv\mathbb{C}^{2}\,,$
$\widehat{\Theta}_{k}^{(\alpha)}=M_{1}(\Theta^{(\alpha)}_{k}\oplus\Theta^{(\alpha)}_{k})M_{2}^{-1}:\mathbb{C}^{2}\to\mathbb{C}^{2}\,,\quad\widehat{\Theta}_{k}^{(\alpha)}=\frac{1}{\alpha_{k}}\begin{bmatrix}0&-i\\\
i&-1\end{bmatrix}$

and

$(\mathsf{D}_{\beta})^{2}=\widehat{\mathsf{H}}_{\beta}+\frac{\mathbb{1}}{4}\,,$

where

$\widehat{\mathsf{H}}_{\beta}=\widehat{\mathsf{H}}_{\widehat{\Pi}^{(\beta)},\widehat{\Theta}^{(\beta)}}\,,$
$\text{\rm ran}\big{(}\widehat{\Pi}^{(\beta)}\big{)}=\text{\rm
ran}\big{(}U^{*}\big{(}\widehat{\Pi}^{(\beta)}_{1}\oplus\dots\oplus\widehat{\Pi}^{(\beta)}_{n}\big{)}U\big{)}\,,\quad\widehat{\Theta}^{(\beta)}=U^{*}\big{(}\widehat{\Theta}^{(\beta)}_{1}\oplus\dots\oplus\widehat{\Theta}^{(\beta)}_{n}\big{)}U\,,$
$\text{\rm ran}(\widehat{\Pi}^{(\beta)}_{k})=\text{\rm
ran}(M_{2}(\Pi^{(\beta)}_{k}\oplus\Pi^{(\beta)}_{k}))=\\{0\\}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\\{0\\}\equiv\mathbb{C}^{2}\,,$
$\widehat{\Theta}_{k}^{(\beta)}=M_{1}(\Theta^{(\beta)}_{k}\oplus\Theta^{(\beta)}_{k})M_{2}^{-1}:\mathbb{C}^{2}\to\mathbb{C}^{2}\,,\quad\widehat{\Theta}_{k}^{(\beta)}=\frac{1}{\beta_{k}}\begin{bmatrix}1&i\\\
-i&0\end{bmatrix}.$

Hence,

$\displaystyle\text{\rm
dom}(\widehat{\mathsf{H}}_{\alpha})=\\{\Psi\equiv(\psi_{1},\psi_{2})\in
H^{2}(\mathbb{R}\backslash
Y):[\psi_{1}]_{y_{k}}=[\psi^{\prime}_{2}]_{y_{k}}=0,\ $
$\displaystyle[\psi^{\prime}_{1}]_{y_{k}}=\alpha_{k}(\psi_{1}(y_{k})-i\psi^{\prime}_{2}(y_{k})),\
[\psi_{2}]_{y_{k}}=-i\alpha_{k}\psi_{1}(y_{k}),\ 1\leq k\leq n\\}\,,$

and

$\displaystyle\text{\rm
dom}(\widehat{\mathsf{H}}_{\beta})=\\{\Psi\equiv(\psi_{1},\psi_{2})\in
H^{2}(\mathbb{R}\backslash
Y):[\psi^{\prime}_{1}]_{y_{k}}=[\psi_{2}]_{y_{k}}=0,\ $
$\displaystyle[\psi_{1}]_{y_{k}}=-i\beta_{k}\psi_{2}(y_{k}),\
[\psi^{\prime}_{2}]_{y_{k}}=-\beta_{k}(\psi_{2}(y_{k})+i\psi^{\prime}_{1}(y_{k})),\
1\leq k\leq n\\}\,.$

###  4.3. Separating boundary conditions

Let $n=1$, and $\Pi=\mathbb{1}$. By $2\tau\Psi=\Psi(y^{-})+\Psi(y^{+})$ and
$\rho\Psi=i\sigma_{1}(\Psi(y_{+})-\Psi(y^{+}))$, the boundary condition
$\tau\Psi=\Theta\rho\Psi$ rewrites as

$(2i\Theta\sigma_{1}+\mathbb{1})\Psi(y^{-})=(2i\Theta\sigma_{1}-\mathbb{1})\Psi(y^{+})\,.$

If $\Theta$ is such that

(4.4) $\text{\rm ran}(2i\Theta\sigma_{1}+\mathbb{1})\cap\text{\rm
ran}(2i\Theta\sigma_{1}-\mathbb{1})=\\{0\\}\,,$

then $\tau\Psi=\Theta\rho\Psi$ is equivalent to the separating boundary
conditions

(4.5) $\displaystyle(2i\Theta\sigma_{1}+\mathbb{1})\Psi(y^{-})=0$ (4.6)
$\displaystyle(2i\Theta\sigma_{1}-\mathbb{1})\Psi(y^{+})=0\,.$

By the equivalence of (4.4) with

(4.7)
$\det(2i\sigma_{1}\Theta-\mathbb{1})=0=\det(2i\sigma_{1}\Theta+\mathbb{1})\,,$

one gets that (4.4) holds if and only if

(4.8)
$\Theta=\Theta_{\omega,\alpha,\beta}:=\frac{1}{2}\begin{bmatrix}\alpha&i\omega\sqrt{1+\alpha\beta}\
\\\ -i\omega\sqrt{1+\alpha\beta}&\beta\end{bmatrix},\qquad\omega=\pm 1\,,\
\alpha,\beta\in\mathbb{R}\,,\ \alpha\beta\geq-1\,,$

and (4.5), (4.6) rewrite, whenever $\Psi\equiv(\psi_{1},\psi_{2})$, as

(4.9)
$\begin{cases}(\omega\sqrt{1+\alpha\beta}-1)\psi_{1}(y^{-})-i\alpha\psi_{2}(y^{-})=0\\\
i\beta\psi_{1}(y^{-})+(\omega\sqrt{1+\alpha\beta}+1)\psi_{2}(y^{-})=0\,,\end{cases}$
(4.10)
$\begin{cases}(\omega\sqrt{1+\alpha\beta}+1)\psi_{1}(y^{+})-i\alpha\psi_{2}(y^{+})=0\\\
i\beta\psi_{1}(y^{+})+(\omega\sqrt{1+\alpha\beta}-1)\psi_{2}(y^{+})=0\,.\end{cases}$

Then

$\mathsf{D}_{\omega,\alpha,\beta}=\mathsf{D}^{-}_{\omega,\alpha,\beta}\oplus\mathsf{D}_{\omega,\alpha,\beta}^{+}\,,$

where
$\mathsf{D}_{\omega,\alpha,\beta}:=\mathsf{D}_{\Theta_{\omega,\alpha,\beta}}$
and the self-adjoint operators $\mathsf{D}_{\omega,\alpha,\beta}^{-}$ and
$\mathsf{D}^{+}_{\omega,\alpha,\beta}$ denote the Dirac operators in
$L^{2}((-\infty,y);\mathbb{C}^{2})$ and $L^{2}((y,+\infty);\mathbb{C}^{2})$,
with boundary conditions (4.9) and (4.10) respectively.

Rewriting the boundary condition
$\widehat{\tau}\Psi=\widehat{\Theta}_{\omega,\alpha,\beta}(\widehat{\rho}\mathbb{1})\Psi$
as

$\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})+\mathbb{1}\big{)}\widehat{\Psi}(y^{-})=\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})-\mathbb{1}\big{)}\widehat{\Psi}(y^{+})\,,$

where
$\widehat{\Psi}\equiv(\psi_{1},\psi^{\prime}_{1},\psi_{2},\psi^{\prime}_{2})$
and $\widehat{\Theta}_{\omega,\alpha,\beta}$ is defined by (3.10), i.e.,

$\widehat{\Theta}_{\omega,\alpha,\beta}=\begin{bmatrix}0&\omega\sqrt{1+\alpha\beta}&0&-i\alpha\\\
\omega\sqrt{1+\alpha\beta}&\beta&i\beta&0\\\
0&-i\beta&0&-\omega\sqrt{1+\alpha\beta}\\\
i\alpha&0&-\omega\sqrt{1+\alpha\beta}&-\alpha\end{bmatrix}\,,$

one can check that

$\det\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})-\mathbb{1}\big{)}=0=\det\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})+\mathbb{1}\big{)}$

and so, proceeding as above,

$\text{\rm
ran}\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})+\mathbb{1}\big{)}\cap\text{\rm
ran}\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})-\mathbb{1}\big{)}=\\{0\\}\,.$

Thus the separating boundary conditions

(4.11)
$\displaystyle\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})+\mathbb{1}\big{)}\widehat{\Psi}(y^{-})=0$
(4.12)
$\displaystyle\big{(}2i\widehat{\Theta}_{\omega,\alpha,\beta}(\sigma_{2}\oplus\sigma_{2})-\mathbb{1}\big{)}\widehat{\Psi}(y^{+})=0\,.$

hold for
$\widehat{\mathsf{H}}_{\omega,\alpha,\beta}:=\widehat{\mathsf{H}}_{\widehat{\Theta}_{\omega,\alpha,\beta}}$
and

$\widehat{\mathsf{H}}_{\omega,\alpha,\beta}=\widehat{\mathsf{H}}_{\omega,\alpha,\beta}^{-}\oplus\widehat{\mathsf{H}}_{\omega,\alpha,\beta}^{+}\,,$

where the self-adjoint operators
$\widehat{\mathsf{H}}_{\omega,\alpha,\beta}^{-}$ and
$\widehat{\mathsf{H}}_{\omega,\alpha,\beta}^{+}$ denote the Schrödinger
operator $-\frac{d^{2}}{dx^{2}}\,\mathbb{1}$ in
$L^{2}((-\infty,y);\mathbb{C}^{2})$ and $L^{2}((y,+\infty);\mathbb{C}^{2})$,
with boundary conditions (4.11) and (4.12) respectively. Furtermore,

$(\mathsf{D}_{\omega,\alpha,\beta}^{\pm})^{2}=\widehat{\mathsf{H}}_{\omega,\alpha,\beta}^{\pm}+\frac{\mathbb{1}}{4}\,.$

In the case $n=1$, $\Pi\not=\mathbb{1}$, there are no separating boundary
conditions since the boundary conditions in $\text{\rm
dom}(\mathsf{D}_{\Pi,\Theta})$ rewrites as

$(\Pi-\mathbb{1})\sigma_{1}\Psi(y^{-})=(\Pi-\mathbb{1})\sigma_{1}\Psi(y^{+})\,,\qquad(2i\Theta\sigma_{1}+\Pi)\Psi(y^{-})=(2i\Theta\sigma_{1}-\Pi)\Psi(y^{+})$

and the linear operator in the relation on the left has a trivial kernel.

By the $n=1$ case, one immediately gets the family of separating and local
boundary conditions: it suffices to take
$\Theta_{\underline{\omega},\underline{\alpha},\underline{\beta}}:=\Theta_{\omega_{1},\alpha_{1},\beta_{1}}\oplus\dots\oplus\Theta_{\omega_{n},\alpha_{n},\beta_{n}}$.
Then, using the abbreviated notations
$\mathsf{D}_{\underline{\omega},\underline{\alpha},\underline{\beta}}\equiv\mathsf{D}_{\Theta_{\underline{\omega},\underline{\alpha},\underline{\beta}}}$
and
$\widehat{\mathsf{H}}_{\underline{\omega},\underline{\alpha},\underline{\beta}}\equiv\widehat{\mathsf{H}}_{\widehat{\Theta}_{\underline{\omega},\underline{\alpha},\underline{\beta}}}$,
where
$\widehat{\Theta}_{\underline{\omega},\underline{\alpha},\underline{\beta}}:=U^{*}\big{(}\widehat{\Theta}_{\omega_{1},\alpha_{1},\beta_{1}}\oplus\dots\oplus\widehat{\Theta}_{\omega_{n},\alpha_{n},\beta_{n}}\big{)}U$
and $\widehat{\Theta}_{\omega_{k},\alpha_{k},\beta_{k}}$ is defined by (3.10),
i.e.,
$\widehat{\Theta}_{\omega_{k},\alpha_{k},\beta_{k}}:=M_{1}(\Theta_{\omega_{k},\alpha_{k},\beta_{k}}\oplus\Theta_{\omega,\alpha,\beta})M_{2}^{-1}$,
one obtains

$\mathsf{D}_{{\underline{\omega}},\underline{\alpha},\underline{\beta}}=\mathsf{D}^{-}_{\omega_{1},\alpha_{1},\beta_{1}}\oplus\mathsf{D}_{\omega_{1,2},\alpha_{1,2}\beta_{1,2}}\oplus\mathsf{D}_{\omega_{2,3},\alpha_{2,3}\beta_{2,3}}\oplus\dots\oplus\mathsf{D}_{\omega_{n-1,n},\alpha_{n-1,n}\beta_{n-1,n}}\oplus\mathsf{D}^{+}_{\omega_{n},\alpha_{n}\beta_{n}}$

and

$\widehat{\mathsf{H}}_{{\underline{\omega}},\underline{\alpha},\underline{\beta}}=\widehat{\mathsf{H}}^{-}_{\omega_{1},\alpha_{1},\beta_{1}}\oplus\widehat{\mathsf{H}}_{\omega_{1,2},\alpha_{1,2}\beta_{1,2}}\oplus\widehat{\mathsf{H}}_{\omega_{2,3},\alpha_{2,3}\beta_{2,3}}\oplus\dots\oplus\widehat{\mathsf{H}}_{\omega_{n-1,n},\alpha_{n-1,n}\beta_{n-1,n}}\oplus\widehat{\mathsf{H}}^{+}_{\omega_{n},\alpha_{n}\beta_{n}}\,.$

Here $\mathsf{D}_{\omega_{k-1,k},\alpha_{k-1,k}\beta_{k-1,k}}$ denotes the
self-adjoint Dirac operator in $L^{2}((y_{k-1},y_{k});\mathbb{C}^{2})$ with
boundary conditions of the kind (4.6) at $y_{k-1}$ (with parameters
$\omega_{k-1},\alpha_{k-1},\beta_{k-1}$) and of the kind (4.5) at $y_{k}$
(with parameters $\omega_{k},\alpha_{k},\beta_{k}$);
$\widehat{\mathsf{H}}_{\omega_{k-1,k},\alpha_{k-1,k},\beta_{k-1,k}}$ is
defined in a similar way, using the boundary conditions (4.11) and (4.12).
Furthermore,

$(\mathsf{D}_{\omega_{k-1,k},\alpha_{k-1,k},\beta_{k-1,k}})^{2}=\widehat{\mathsf{H}}_{\omega_{k-1,k},\alpha_{k-1,k},\beta_{k-1,k}}+\frac{\mathbb{1}}{4}\,,\qquad
1\leq k\leq n\,.$

###  4.4. Supersymmetry

Since

$\sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{1}=0=\sigma_{3}\sigma_{2}+\sigma_{2}\sigma_{3}\,,$

one has

$\sigma_{2}\mathsf{D}_{\mathbb{R}\backslash
Y}+\mathsf{D}_{\mathbb{R}\backslash Y}\sigma_{2}=0\,.$

Therefore, if $(\Pi,\Theta)$ is such that

(4.13) $\begin{cases}\rho\Psi\in\text{\rm ran}(\Pi)\\\
\Pi\tau\Psi=\Theta\rho\Psi\end{cases}\quad\Longrightarrow\quad\begin{cases}\rho\sigma_{2}\Psi\in\text{\rm
ran}(\Pi)\\\ \Pi\tau\sigma_{2}\Psi=\Theta\rho\sigma_{2}\Psi\,,\end{cases}$

then $\sigma_{2}$ anti-commutes with $\mathsf{D}_{\Pi,\Theta}$ and so, by
(3.1), the system

(4.14)
$\left(\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}+\frac{\mathbb{1}}{4},\sigma_{2},\mathsf{D}_{\Pi,\Theta}\right)$

has supersymmetry (see, e.g., [4, Chapter 1], [15, Section 6.3]).

By

$\langle\sigma_{2}\Psi\rangle_{y}=\sigma_{2}\langle\Psi\rangle_{y}\,,\qquad[\sigma_{2}\Psi]_{y}=\sigma_{2}[\Psi]_{y}\,,$

and by $\sigma_{1}\sigma_{2}=-\sigma_{2}\sigma_{1}$, one gets

$\tau\sigma_{2}\Psi=\sigma_{2}^{\oplus}\tau\Psi\,,\qquad\rho\sigma_{2}\Psi=-\sigma_{2}^{\oplus}\rho\Phi\,,\qquad\sigma_{2}^{\oplus}:=\sigma_{2}\oplus\dots\oplus\sigma_{2}\,.$

Therefore, (4.13) holds whenever

(4.15)
$\Pi\sigma^{\oplus}_{2}-\sigma^{\oplus}_{2}\Pi=0=\Theta\sigma^{\oplus}_{2}+\sigma_{2}^{\oplus}\Theta\,.$

Given a couple $(\Pi,\Theta)$ which satisfies (4.15), let us further suppose
that

(4.16) $\det(\Theta+\Pi\tau G_{0}\Pi)\not=0\,.$

Then, by (3.5), zero is not an eigenvalue of $\mathsf{D}_{\Pi,\Theta}$, i.e.,
the system (4.14) has no supersymmetric state and there is a spontaneous
supersymmetry breaking (see, e.g., [4, Section 1.8]).

In the case $n=1$, the solutions of (4.15) are found immediately.

If $\Pi\not=\mathbb{1}$, then $\Pi=\Pi_{\pm}:=v_{\pm}\otimes v_{\pm}$ and
$\Theta=0$, where $v_{\pm}$, $|v_{\pm}|=1$, solves $\sigma_{2}v_{\pm}=\pm
v_{\pm}$.

If $\Pi=\mathbb{1}$, then $\Theta=\Theta_{a,b}:=b\sigma_{1}+a\sigma_{3}$,
$a,b\in\mathbb{R}$.

Since, by (2.7), $\tau G_{0}=-\frac{1}{2}\,\sigma_{3}$,
$\det(\Theta_{a,b}+\tau G_{0})=0$ if and only if $b=0$ and $a=\frac{1}{2}$.
Therefore, for any $(a,b)\in\mathbb{R}^{2}\backslash\\{(\frac{1}{2},0)\\}$ the
system (4.14) with $\Pi=\mathbb{1}$ and $\Theta=\Theta_{a,b}$ has no
supersymmetric state and there is a spontaneous supersymmetry breaking.

Notice that once the solution of (4.15) in known in the $n=1$ case, then the
set of solutions for the case of $n>1$ local point interactions is readily
obtained: $\Pi=\Pi_{1}\oplus\dots\oplus\Pi_{n}$ and
$\Theta=\Theta_{1}\oplus\dots\oplus\Theta_{n}$, where $(\Pi_{k},\Theta_{k})$
is equal either to $(\Pi_{\pm},\mathbb{0})$ or to
$(\mathbb{1},\Theta_{a_{k},b_{k}})$.

###  4.5. Quantum Graphs.

Since $\mathsf{D}_{\Pi,\Theta}$ is a generic self-adjoint extension of
$S=\mathsf{D}|C^{\infty}_{comp}(\mathbb{R}\backslash
Y;\mathbb{C}^{2})=(\mathsf{D}|C^{\infty}_{comp}(I_{0};\mathbb{C}^{2}))\oplus\cdots\oplus(\mathsf{D}|C^{\infty}_{comp}(I_{n};\mathbb{C}^{2}))$,
the nonlocal extensions of $S$ provide the self-adjoint realizations of the
Dirac operator on a quantum graph with the two ends
$\overline{I}_{0}=(-\infty,y_{1}]$ and $\overline{I}_{n}=[y_{n},+\infty)$ and
the $(n-1)$ edges
$\overline{I}_{1}=[y_{1},y_{2}],\dots,\overline{I}_{n-1}=[y_{n-1},y_{n}]$; the
boundary conditions corresponding to the couple $(\Pi,\Theta)$ specify the
connectivity of the graph. The case of a compact graph can be obtained by
imposing separating boundary conditions at the two ends. Likewise, the
nonlocal extensions of
$\mathsf{H}\mathbb{1}|C^{\infty}_{comp}(\mathbb{R}\backslash
Y;\mathbb{C}^{2})$ provide self-adjoint realizations of the Schrödinger
operator on a quantum graph with two ends and $(n-1)$ edges. For an
introduction to the theory of quantum graphs we refer to the book [9] and the
many references there; however, let us point out that our way of building the
self-adjoint realizations on the graph is not the standard one.

As an explicit example, let us consider the Dirac operator on the eye graph
(see [13, Section III.D]). Therefore, we choose the subclass of boundary
conditions for the Dirac operator in $L^{2}(G;\mathbb{C}^{2})$,
$G=(-\infty,y_{1}]\sqcup[y_{1},y_{2}]\sqcup[y_{2},y_{3}]\sqcup[y_{3},+\infty)$,
connecting $\Psi(y_{1}^{-})$ with both $\Psi(y_{1}^{+})$ and $\Psi(y_{2}^{+})$
and connecting $\Psi(y_{3}^{+})$ with both $\Psi(y_{2}^{-})$ and
$\Psi(y_{3}^{-})$. Such kind of boundary conditions give to $G$ the topology
of a circle with two ends.

$y_{1}$$y_{1}$$y_{2}$$y_{3}$$y_{3}$$y_{2}$ $\approx$

Furthermore, we restrict to Kirchhoff-type boundary conditions, meaning that
we select the ones which, in the non relativistic limit, correspond to
Kirchhoff (or free) boundary conditions for the Schrödinger operator in
$L^{2}(G)$ (see [10], [11], [12]). Therefore, we require, for
$\Psi\equiv(\psi_{1},\psi_{2})$ in the self-adjointness domain,

(4.17)
$\begin{cases}\psi_{1}(y_{1}^{-})=\psi_{1}(y_{1}^{+})=\psi_{1}(y_{2}^{+})&\\\
\psi_{2}(y_{1}^{-})-\psi_{2}(y_{1}^{+})-\psi_{2}(y_{2}^{+})=0&\\\
\psi_{1}(y_{2}^{-})=\psi_{1}(y_{3}^{-})=\psi_{1}(y_{3}^{+})&\\\
\psi_{2}(y_{2}^{-})+\psi_{2}(y_{3}^{-})-\psi_{2}(y_{3}^{+})=0\,.&\end{cases}$

These boundary conditions rewrite as

(4.18) $\begin{cases}[\psi_{1}]_{y_{1}}=[\psi_{1}]_{y_{3}}=0&\\\
[\psi_{2}]_{y_{1}}=-\psi_{2}(y_{2}^{+})&\\\
[\psi_{2}]_{y_{3}}=\psi_{2}(y_{2}^{-})&\\\
\langle\psi_{1}\rangle_{y_{1}}=\psi_{1}(y_{2}^{+})&\\\
\langle\psi_{1}\rangle_{y_{3}}=\psi_{1}(y_{2}^{-})&\end{cases}\equiv\quad\begin{cases}[\psi_{1}]_{y_{1}}=[\psi_{1}]_{y_{3}}=0&\\\
[\psi_{2}]_{y_{1}}+[\psi_{2}]_{y_{2}}+[\psi_{2}]_{y_{3}}=0&\\\
\langle\psi_{1}\rangle_{y_{1}}-\langle\psi_{1}\rangle_{y_{3}}=[\psi_{1}]_{y_{2}}&\\\
\langle\psi_{1}\rangle_{y_{1}}-2\langle\psi_{1}\rangle_{y_{2}}+\langle\psi_{1}\rangle_{y_{3}}=0&\\\
2\langle\psi_{2}\rangle_{y_{2}}=[\psi_{2}]_{y_{3}}-[\psi_{2}]_{y_{1}}\,.&\end{cases}$

The relations
$[\psi_{1}]_{y_{1}}=[\psi_{1}]_{y_{3}}=[\psi_{2}]_{y_{1}}+[\psi_{2}]_{y_{2}}+[\psi_{2}]_{y_{3}}=0$
in (4.18) coincide with $\rho\Psi\in\text{\rm ran}(\Pi)$, where the orthogonal
projector $\Pi:\mathbb{C}^{6}\to\mathbb{C}^{6}$ is represented by the matrix

(4.19) $\Pi=\frac{1}{3}\begin{bmatrix}2&0&-1&0&-1&0\\\ 0&0&0&0&0&0\\\
-1&0&2&0&-1&0\\\ 0&0&0&3&0&0\\\ -1&0&-1&0&2&0\\\ 0&0&0&0&0&0\end{bmatrix}.$

Then, one can easily check that the other relations in (4.18) are equivalent
to $\Pi\tau\Psi=\Theta\rho\Psi$ whenever $\Theta$, a symmetric operator in the
$3$-dimensional subspace $\text{\rm ran}(\Pi)$, is represented, as a symmetric
linear operator in $\mathbb{C}^{6}$ preserving $\text{\rm ran}(\Pi)$, by the
Hermitian matrix

(4.20) $\Theta=\frac{i}{2}\begin{bmatrix}0&0&0&-1&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 1&0&0&0&-1&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&0&0\end{bmatrix}.$

Therefore, using $L^{2}(G;\mathbb{C}^{2})\equiv
L^{2}(\mathbb{R};\mathbb{C}^{2})$, one gets $\mathsf{D}_{\rm
K}\equiv\mathsf{D}_{\Pi,\Theta}$, where $\Pi$ and $\Theta$ are as in (4.19)
and (4.20) respectively and where $\mathsf{D}_{\rm K}$ denotes the Dirac
operator on the eye graph with the Kirchhoff boundary conditions at the two
vertices. Then, by Remark 3.4 and by (4.17), the Schrödinger operator
$\widehat{\mathsf{H}}_{\widehat{\Pi},\widehat{\Theta}}$ satisfies both the
boundary conditions ${\rm K}$ and ${\rm K}_{*}$, where

(4.21) ${\rm
K}\equiv\begin{cases}\psi_{1}(y_{1}^{-})=\psi_{1}(y_{1}^{+})=\psi_{1}(y_{2}^{+})&\\\
\psi_{1}^{\prime}(y_{1}^{-})-\psi_{1}^{\prime}(y_{1}^{+})-\psi_{1}^{\prime}(y_{2}^{+})=0&\\\
\psi_{1}(y_{2}^{-})=\psi_{1}(y_{3}^{-})=\psi_{1}(y_{3}^{+})&\\\
\psi_{1}^{\prime}(y_{2}^{-})+\psi_{1}^{\prime}(y_{3}^{-})-\psi_{1}^{\prime}(y_{3}^{+})=0&\end{cases}\quad{\rm
K}_{*}\equiv\begin{cases}\psi_{2}^{\prime}(y_{1}^{-})=\psi_{2}^{\prime}(y_{1}^{+})=\psi_{2}^{\prime}(y_{2}^{+})&\\\
\psi_{2}(y_{1}^{-})-\psi_{2}(y_{1}^{+})-\psi_{2}(y_{2}^{+})=0&\\\
\psi_{2}^{\prime}(y_{2}^{-})=\psi_{2}^{\prime}(y_{3}^{-})=\psi_{2}^{\prime}(y_{3}^{+})&\\\
\psi_{2}(y_{2}^{-})+\psi_{2}(y_{3}^{-})-\psi_{2}(y_{3}^{+})=0\,.&\end{cases}$

This gives

(4.22) $(\mathsf{D}_{\rm K})^{2}=\left(\mathsf{H}_{\rm
K}+\frac{1}{4}\right)\oplus\left(\mathsf{H}_{{\rm
K}_{*}}+\frac{1}{4}\right)\,,$

where $\mathsf{H}_{\rm K}$ is the Schrödinger operator in $L^{2}(G)$ with the
boundary conditions ${\rm K}$ and $\mathsf{H}_{{\rm K}_{*}}$ is the
Schrödinger operator in $L^{2}(G)$ with the boundary conditions ${\rm K}_{*}$.
The boundary conditions ${\rm K}$ coincide with the usual Kirchhoff ones (see
[9, eq. (1.4.4)]) while the boundary conditions ${\rm K}_{*}$, are as sort of
reversed Kirchhoff ones (named ”homogeneous $\delta^{\prime}$ vertex
conditions” in [11]) given by the exchange $\psi\leftrightarrow\psi^{\prime}$.
The boundary conditions ${\rm K_{*}}$, like the ${\rm K}$ ones, give, in the
case of the real line, the free Schrödinger operator; thus, (4.22) is
consistent with (1.1). Furthermore, the Schrödinger operator $\mathsf{H}_{\rm
K}\oplus\mathsf{H}_{{\rm K}_{*}}$ appears in the nonrelativistic limit of
$\mathsf{D}_{K}$, see [11, Proposition 1.3].

The arguments in the previous example extend to any graph: by Remark 3.4, to
the Kirchhoff-type boundary conditions for the Dirac operator $\mathsf{D}_{\rm
K}$ on the graph, i.e., to

$\begin{cases}\text{$\psi_{1}$ continuous at any vertex $v$}&\\\
\sum_{v}^{\pm}\psi_{2}(v)=0\ \text{for any vertex $v$,}&\end{cases}$

correspond, for the Schrödinger operators $\mathsf{H}_{\rm K}$ and
$\mathsf{H}_{{\rm K}_{*}}$ such that (4.22) holds, the boundary conditions

${\rm K}\equiv\begin{cases}\text{$\psi$ continuous at any vertex $v$}&\\\
\sum_{v}^{\pm}\psi^{\prime}(v)=0\,\ \text{for any vertex
$v$}&\end{cases}\quad{\rm K}_{*}\equiv\begin{cases}\text{$\psi^{\prime}$
continuous an any vertex $v$}&\\\ \sum_{v}^{\pm}\psi(v)=0\,\ \text{for any
vertex $v$}.&\end{cases}$

Here, $\sum_{v}^{\pm}f(v)$ means the sum over all the points $y_{k}\in Y$
corresponding to the vertex $v$ with the sign convention

$f(v):=\begin{cases}-f(y_{k}^{+})&\text{$y_{k}$ is at the left end of the
interval/half-line}\\\ +f(y_{k}^{-})&\text{$y_{k}$ is at the right end of the
interval/half-line.}\end{cases}$

## References

  * [1] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden: Solvable Models in Quantum Mechanics, 2nd Edition, AMS Chelsea Publ., 2005.
  * [2] S. Albeverio, W. Karwowski, V. Koshmanenko: Square powers of singularly perturbed operators. Math. Nachr. 173 (1995), 5-24.
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  * [6] J. Behrndt, P. Exner, M. Holzmann, V. Lotoreichik: On Dirac operators in $\mathbb{R}^{3}$ with electrostatic and Lorentz scalar $\delta$-shell interactions. _Quantum Stud. Math. Found._ , 6, 295-314, 2019.
  * [7] J. Behrndt, M. Langer, V. Lotoreichik: Schrödinger operators with $\delta$ and $\delta^{\prime}$-potentials supported on hypersurfaces. Ann. Henri Poincaré 14 (2013), 385-423.
  * [8] S. Benvegnù, L. Da̧browski: Relativistic point interaction, _Lett. in Math. Physics_ , 30 (1994), 159-167.
  * [9] G. Berkolaiko, P. Kuchment: Introduction to Quantum Graphs. American Mathematical Society, 2013.
  * [10] W. Borrelli, R. Carlone, L. Tentarelli: Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit. SIAM J. Math. Anal. 51 (2019), 1046-1081.
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  * [12] W. Borrelli, R. Carlone, L. Tentarelli: A note on the Dirac operator with Kirchhoff-type vertex conditions on metric graphs. In A. Michelangeli (ed.): Mathematical challenges of zero-range physics - models, methods, rigorous results, open problems, 81-104, Springer, 2021.
  * [13] W. Bulla, T. Trenkler: The free Dirac operator on compact and noncompact graphs. J. Math. Phys. 31 (1990), 1157-1163.
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  * [16] L. Da̧browski, H. Grosse: On nonlocal point interactions in one, two, and three dimensions. J. Math. Phys. 26 (1985), 2777-2780.
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# On a conjecture of Ramírez Alfonsín and Skałba III

Yuchen Ding and Takao Komatsu  (Yuchen Ding) School of Mathematical Science,
Yangzhou University, Yangzhou 225002, People’s Republic of China
<EMAIL_ADDRESS>(Takao Komatsu) Department of Mathematical Sciences, School
of Science, Zhejiang Sci-Tech University, Hangzhou 310018 China, People’s
Republic of China<EMAIL_ADDRESS>
###### Abstract.

Let $1<c<d$ be two relatively prime integers. For a non-negative integer
$\ell$, let $g_{\ell}(c,d)$ be the largest integer $n$ such that $n=cx+dy$ has
at most $\ell$ non-negative solutions $(x,y)$. In this paper we prove that

$\pi_{\ell,c,d}\sim\frac{\pi\bigl{(}g_{\ell}(c,d)\bigr{)}}{2\ell+2}\quad(\text{as}~{}c\to\infty)\,,$

where $\pi_{\ell,c,d}$ is the number of primes $n$ having more than $\ell$
distinct non-negative solutions to $n=cx+dy$ with $n\leq g_{\ell}(c,d)$, and
$\pi(x)$ denotes the number of all primes up to $x$. The case where $\ell=0$
has been proved by Ding, Zhai and Zhao recently, which was conjectured
formerly by Ramírez Alfonsín and Skałba.

###### Key words and phrases:

Frobenius number, $\ell$-numerical semigroup, Primes in residue classes,
Hardy–Littlewood method, Siegel–Walfisz theorem.

###### 2010 Mathematics Subject Classification:

Primary 11N05; Secondary 11D07.

##  1\. Introduction

Let $c_{1},\dots,c_{k}$ ($k\geq 2$) be a set of distinct integers with
$c_{i}>1$ ($i=1,\dots,k$). For a given non-negative integer $\ell$, let
$S_{\ell}(c_{1},\dots,c_{k})$ (written as $S_{\ell}$ for shorthand), which is
called the $\ell$-numerical semigroup, with $\gcd(c_{1},\dots,c_{k})=1$ be the
set of all the elements $n$ whose number of solutions to
$c_{1}x_{1}+\cdots+c_{k}x_{k}=n$ is more than $\ell$. Then for the set of non-
negative integer $\mathbb{N}_{0}$, $\mathbb{N}_{0}\backslash S_{\ell}$ is the
set of all the elements of $n$ whose number of solutions to
$c_{1}x_{1}+\cdots+c_{k}x_{k}=n$ ($x_{1},\dots,x_{k}\in\mathbb{N}_{0}$) is
less than or equal to $\ell$. In [3, 4, 5], the properties of the
$\ell$-numerical semigroups and explicit forms of crucial numbers are
discussed111In [3, 4, 5] and more references, the terminology of $p$-numerical
semigroup is frequently used. However, in this paper, we mainly deal with
prime numbers, so we do not use $p$ or $q$, but instead use $\ell$..

The set $\mathbb{N}_{0}\backslash S_{\ell}$ is finite if and only if
$\gcd(c_{1},\dots,c_{k})=1$. Then, there exists the largest element
$g_{\ell}(c_{1},\dots,c_{k})$, which is called the $\ell$-Frobenius number.
When $\ell=0$, $g(c_{1},\dots,c_{k})=g_{0}(c_{1},\dots,c_{k})$ is the original
Frobenius number, which is the central topic on the classically well-known
linear Diophantine problem of Frobenius. In general, to find an explicit
closed form of $g_{\ell}(c_{1},\dots,c_{k})$ is very difficult for $k\geq 3$,
but when $k=2$, the $\ell$-Frobenius number can be given explicitly: for any
non-negative integer $\ell$

$g_{\ell}(c,d)=(\ell+1)cd-c-d$ (1.1)

(see [3, 4, 5] for more general formulas and related concepts).

For integers $c,d$ with $1<c<d$ and $\gcd(c,d)=1$, let $\pi_{\ell,c,d}$ be the
number of primes in

$\mathcal{N}_{\ell}(c,d):=\\{n\in\mathbb{N}_{0}|n\leq g_{\ell}(c,d),~{}n\in
S_{\ell}(c,d)\\}\,,$

and $\pi(x)$ denotes the number of all primes up to $x$.

Ramírez Alfonsín and Skałba [6] proved that for any $\varepsilon>0$, there is
a constant $k_{\varepsilon}>0$ such that

$\pi_{0,c,d}\geq k_{\varepsilon}\frac{g_{0}(c,d)}{\bigl{(}\log
g_{0}(c,d)\bigr{)}^{2+\epsilon}}\,,$

and conjectured the following:

$\pi_{0,c,d}\sim\frac{\pi(g_{0}\bigl{(}c,d)\bigr{)}}{2}\quad(\text{as}~{}c\to\infty)\,.$
(1.2)

Ding [1] made some progress on the above conjecture (1.2). More precisely, for
all but at most

$O\bigl{(}N(\log N)^{1/2}(\log\log N)^{1/2+\varepsilon}\bigr{)}$

pairs $c$ and $d$, one has

$\pi_{0,c,d}=\frac{\pi\bigl{(}g_{0}(c,d)\bigr{)}}{2}+O\left(\frac{\pi\bigl{(}g_{0}(c,d)\bigr{)}}{\bigl{(}\log\log(cd)\bigr{)}^{\varepsilon}}\right)\,.$

Since

$\frac{\pi\bigl{(}g_{0}(c,d)\bigr{)}}{2}+O\left(\frac{\pi\bigl{(}g_{0}(c,d)\bigr{)}}{\bigl{(}\log\log(cd)\bigr{)}^{\varepsilon}}\right)\sim\frac{\pi\bigl{(}g_{0}(c,d)\bigr{)}}{2}\quad(\text{as}~{}c\to\infty)$

and the total number of the pairs $(c,d)$ with $1<c<d$, $\gcd(c,d)=1$ and
$cd\ll N$ is $\gg N\log N$, Ding’s result provided an almost all version of
(1.2).

Though it seemed to be out of reach [6], very recently Ding, Zhai and Zhao [2]
completely proved (1.2).

The main purpose of this paper is to show a more general result of (1.2) as
follows:

######  Theorem 1.1.

Let $\ell$ be a non-negative integer. For integers $c$ and $d$ with $1<c<d$
and $\gcd(c,d)=1$, we have

$\pi_{\ell,c,d}\sim\frac{\pi\bigl{(}g_{\ell}(c,d)\bigr{)}}{2\ell+2}\quad(\text{as}~{}c\to\infty)\,.$

Remark. When $\ell=0$, this is reduced to the main result in [2, Theorem 1.1].
The result itself is still never obvious because the speed of convergence is
very slow.

##  2\. Preliminaries

From now on, $c$ and $d$ will always denote two positive integers satisfying
that $1<c<d$ and $\gcd(c,d)=1$. Below, the first lemma is quite trivial.

######  Lemma 2.1.

Suppose that $cx+dy=cx^{\prime}+dy^{\prime}$ with
$x,y,x^{\prime},y^{\prime}\in\mathbb{N}_{0}$, then

$c|(y-y^{\prime})\quad\text{and}\quad d|(x-x^{\prime}).$

######  Lemma 2.2.

Suppose that the number of solutions to

$n=cx+dy\quad(x,y\in\mathbb{N}_{0})$

is exactly $\ell$, then we have

$n=(\ell-1)cd+cx_{0}+dy_{0}$

for some $0\leq x_{0}\leq d-1$ and $0\leq y_{0}\leq c-1$.

###### Proof.

Suppose that $\ell$ solutions are given as

$n=cx_{1}+dy_{1}=cx_{2}+dy_{2}=\cdots=cx_{\ell}+dy_{\ell}\,.$

Without loss of generality, assume that

$x_{1}<x_{2}<\dots<x_{\ell}.$

So, $y_{1}>y_{2}>\dots>y_{\ell}$. Now, by Lemma 2.1, we have

$x_{2}=x_{1}+d,\quad x_{3}=x_{1}+2d,\quad\dots,\quad x_{\ell}=x_{1}+(\ell-1)d$

with $0\leq x_{1}\leq d-1$ and $0\leq y_{\ell}\leq c-1$. Thus,

$n=c\bigl{(}x_{1}+(\ell-1)d\bigr{)}+dy_{\ell}=(\ell-1)cd+cx_{1}+dy_{\ell}.$

This completes the proof of Lemma 2.2. ∎

Let $\mathcal{P}$ be the set of primes.

######  Lemma 2.3.

Let $\pi_{\ell,c,d}$ be defined as the one in introduction, then we have

$\pi_{\ell,c,d}:=\sum_{\begin{subarray}{c}\ell cd<n\leq g_{\ell}(c,d)\\\
n=\ell cd+cx_{0}+dy_{0}\in\mathcal{P}\\\ 0\leq x_{0}\leq d;\,0\leq y_{0}\leq
c\end{subarray}}1\,.$

###### Proof.

If $n$ has more than $\ell+1$ solutions with the form $cx+dy$
($x,y\in\mathbb{N}_{0}$), then by Lemma 2.2, we have

$n=(\ell+k)cd+cx_{0}+dy_{0}\qquad(0\leq x_{0}\leq d-1;\,0\leq y_{0}\leq
c-1)\,.$

for some $k\in\mathbb{Z}^{+}$. For these $n$, we will have

$n\geq(\ell+1)cd>g_{\ell}(c,d).$

So, if $n\leq g_{\ell}(c,d)$ with more than $\ell$ solutions, then $n$ has
exactly $\ell+1$ solutions (expressions). Employing again Lemma 2.2, we obtain
that

$n=\ell cd+cx_{0}+dy_{0}\qquad(0\leq x_{0}\leq d-1;\,0\leq y_{0}\leq c-1)\,,$

provided that $n\leq g_{\ell}(c,d)$.

Furthermore, if $n=\ell cd+cx_{0}+dy_{0}\leq g_{\ell}(c,d)$ with

$0\leq x_{0}\leq d\quad\text{and}\quad 0\leq y_{0}\leq c,$

then we clearly have $x_{0}\leq d-1$ and $y_{0}\leq c-1$.

Now, the lemma follows from the trivial fact that $\ell cd\not\in\mathcal{P}$.
∎

As usual, the von Mangoldt function $\Lambda(n)$ is defined as

$\Lambda(n)=\begin{cases}\log p&\text{if $n=p^{\alpha}$ ($\alpha>0$)};\\\
0&\text{otherwise}.\end{cases}$

In order to prove Theorem 1.1, we shall firstly provide the following weighted
version whose proof will be put on the next section.

######  Proposition 2.1.

Let $\ell$ be a non-negative integer. For integers $c$ and $d$ with $1<c<d$
and $\gcd(c,d)=1$, we have

$\psi_{\ell,c,d}\sim\frac{g_{0}(c,d)}{2}\quad(\text{as}~{}c\to\infty)\,,$

where

$\psi_{\ell,c,d}=\sum_{\begin{subarray}{c}\ell cd<n\leq g_{\ell}(c,d)\\\
n=\ell cd+cx_{0}+dy_{0}\\\ 0\leq x_{0}\leq d;\,0\leq y_{0}\leq
c\end{subarray}}\Lambda(n)\,.$

##  3\. Proof of Proposition 2.1

###### Proof of Proposition 2.1.

Our proof follows from the argument of [2, Theorem 1.1] with some adjustments.
By the same shorthand in [2], we write $g_{0}(c,d)=cd-c-d$ briefly as $g$.
Throughout the proof, the integer $c$ is supposed to be sufficiently large.

By definitions of $\psi_{\ell,c,d}$, we have

$\displaystyle\psi_{\ell,c,d}=\sum_{\begin{subarray}{c}\ell cd<n\leq
g_{\ell}(c,d)\\\ n=\ell cd+cx_{0}+dy_{0}\\\ 0\leq x_{0}\leq d;\,0\leq
y_{0}\leq c\end{subarray}}\Lambda(n)=\sum_{\begin{subarray}{c}n\leq g\\\
n=cx_{0}+dy_{0}\\\ 0\leq x_{0}\leq d;\,0\leq y_{0}\leq
c\end{subarray}}\Lambda(\ell cd+n)+O_{\ell}(\log g)\,.$

For any real $\alpha$, let

$\displaystyle f(\alpha)$ $\displaystyle=\sum_{0\leq n\leq g}\Lambda(\ell
cd+n)e(\alpha n)\,,$ $\displaystyle h(\alpha)$ $\displaystyle=\sum_{0\leq
x\leq d\atop 0\leq y\leq c}e\bigl{(}\alpha(cx+dy)\bigr{)}\,,$

where $e(t)$ denotes $e^{2\pi it}$ for any number $t$ as usual. Then by the
orthogonal relation, we have

$\psi_{\ell,c,d}=\int_{0}^{1}f(\alpha)h(-\alpha)d\alpha\,.$ (3.1)

We are in a position to introduce the Hardy–Littlewood method to evaluate the
above integral.

Let $Q<c^{1/3}$ denote a parameter depending only on $c$ and $d$ which shall
be decided later. Define the major arcs to be

$\displaystyle\mathfrak{M}(Q)=\bigcup_{1\leq q\leq
Q}\bigcup_{\begin{subarray}{c}1\leq a\leq q\\\
\gcd(a,q)=1\end{subarray}}\left\\{\alpha:\left|\alpha-\frac{a}{q}\right|\leq\frac{Q}{qg}\right\\}.$

By our assumption, we have $Q<g^{1/6}$, from which it follows trivially that
the above subsets are pairwise disjoint (see e.g. [2, Section 2]). In
addition, we note that

$\mathfrak{M}(Q)\subseteq\left[\frac{1}{Q}-\frac{Q}{qg},1+\frac{Q}{qg}\right]\subseteq\left[\frac{Q+1}{g},1+\frac{Q+1}{g}\right].$

The minor arcs are then defined to be

$\displaystyle\mathfrak{m}(Q)=\left[\frac{Q+1}{g},1+\frac{Q+1}{g}\right]\setminus\mathfrak{M}(Q).$

From Eq. (3.1), it is plain that

$\displaystyle\psi_{\ell,c,d}$
$\displaystyle=\int_{\frac{Q+1}{g}}^{1+\frac{Q+1}{g}}f(\alpha)h(-\alpha)d\alpha$
$\displaystyle=\int_{\mathfrak{M}(Q)}f(\alpha)h(-\alpha)d\alpha+\int_{\mathfrak{m}(Q)}f(\alpha)h(-\alpha)d\alpha.$
(3.2)

###  3.1. Estimates of the minor arcs

Note that

$\displaystyle\left|f(\alpha)\right|$ $\displaystyle=\left|\sum_{0\leq n\leq
g}\Lambda(\ell cd+n)e(\alpha n)\right|$ $\displaystyle=\left|\sum_{\ell cd\leq
m\leq\ell cd+g}\Lambda(m)e(\alpha m)e(-\alpha\ell cd)\right|\quad(m=\ell
cd+n)$ $\displaystyle\leq\left|\sum_{\ell cd\leq m\leq\ell
cd+g}\Lambda(m)e(\alpha m)\right|.$

By a remarkable theorem of Vinogradov (see e.g. [7, Theorem 3.1]) as well as
the Dirichlet approximation theorem (see e.g. [7, Lemma 2.1]), we can obtain
that

$\sup_{\alpha\in\mathfrak{m}(Q)}\left|\sum_{m}\Lambda(m)e(\alpha
m)\right|\ll_{\ell}\frac{g(\log g)^{4}}{Q^{1/2}}+g^{4/5}(\log g)^{4}\,$

(see [2, Lemma 3.1]), which means that

$\sup_{\alpha\in\mathfrak{m}(Q)}|f(\alpha)|\ll_{\ell}\frac{g(\log
g)^{4}}{Q^{1/2}}+g^{4/5}(\log g)^{4}\,.$

By using the estimate in [2, Lemma 3.2]

$\int_{0}^{1}|h(-\alpha)|d\alpha\ll(\log g)^{2}\,,$

we have

$\displaystyle\int_{\mathfrak{m}(Q)}f(\alpha)h(-\alpha)$
$\displaystyle\leq\sup_{\alpha\in\mathfrak{m}(Q)}|f(\alpha)|\int_{\mathfrak{m}(Q)}|h(-\alpha)|d\alpha$
$\displaystyle\ll_{\ell}\left(\frac{g(\log g)^{4}}{Q^{1/2}}+g^{4/5}(\log
g)^{4}\right)\int_{0}^{1}|h(-\alpha)|d\alpha$
$\displaystyle\ll_{\ell}\frac{g(\log g)^{6}}{Q^{1/2}}+g^{4/5}(\log g)^{6}\,.$
(3.3)

###  3.2. Calculations of the major arc

We now calculate the integral on the major arcs.

$\displaystyle\int_{\mathfrak{M}(Q)}f(\alpha)h(-\alpha)$
$\displaystyle=\sum_{1\leq q\leq Q}\sum_{1\leq a\leq
q\atop\gcd(a,q)=1}\int_{\frac{a}{q}-\frac{Q}{qg}}^{\frac{a}{q}+\frac{Q}{qg}}f(\alpha)h(-\alpha)d\alpha$
$\displaystyle=\sum_{1\leq q\leq Q}\sum_{1\leq a\leq
q\atop\gcd(a,q)=1}\int_{-\frac{Q}{qg}}^{\frac{Q}{qg}}f\left(\theta+\frac{a}{q}\right)h\left(-\theta-\frac{a}{q}\right)d\theta$
$\displaystyle=\int_{-\frac{Q}{g}}^{\frac{Q}{g}}f(\theta)h(-\theta)d\theta+\mathcal{R}\,,$
(3.4)

where

$\mathcal{R}=\sum_{1<q\leq Q}\sum_{1\leq a\leq
q\atop\gcd(a,q)=1}\int_{-\frac{Q}{qg}}^{\frac{Q}{qg}}f\left(\theta+\frac{a}{q}\right)h\left(-\theta-\frac{a}{q}\right)d\theta.$

We shall later prove that $\mathcal{R}$ still contributes to the ‘error term’.

For any real $\theta$, we have

$\displaystyle f(\theta)$ $\displaystyle=\sum_{0\leq n\leq g}\Lambda(n+\ell
cd)e(\theta n)$ $\displaystyle=\sum_{\ell cd\leq m\leq g+\ell
cd}\Lambda(m)e\bigl{(}\theta(m-\ell cd)\bigr{)}\quad(m=n+\ell cd)$
$\displaystyle=e(-\theta\ell cd)\widetilde{f(\theta)}\,,$

where

$\widetilde{f(\theta)}=\sum_{\ell cd\leq m\leq g+\ell cd}\Lambda(m)e(\theta
m)\,.$

Let $\rho(m)=\Lambda(m)-1$. Then

$\displaystyle\widetilde{f(\theta)}-\sum_{\ell cd\leq m\leq g+\ell cd}e(\theta
m)=\sum_{\ell cd<m\leq g+\ell cd}\rho(m)e(\theta m)+O_{\ell}\left(\log
g\right)\,.$

By partial summation, we have

$\sum_{\ell cd<m\leq g+\ell cd}\rho(m)e(\theta m)=e\big{(}(\ell
cd+g)\theta\bigr{)}\sum_{m\leq\ell cd+g}\rho(m)\\\ -e(\ell
cd\theta)\sum_{m\leq\ell cd}\rho(m)-2\pi i\theta\int_{\ell cd}^{\ell
cd+g}\Bigg{(}\sum_{m\leq t}\rho(m)\Bigg{)}e(t\theta)dt\,.$ (3.5)

Using the prime number theorem, there exists some absolute constant
$\kappa_{1}>0$ so that

$\sum_{m\leq t}\rho(m)=\psi(t)-t\ll te^{-\kappa_{1}\sqrt{\log t}}.$

Inserting this into Eq. (3.5), it follows that

$\displaystyle\sum_{\ell cd<m\leq g+\ell cd}\rho(m)e(\theta m)$
$\displaystyle\ll_{\ell}ge^{-\kappa_{1}\sqrt{\log g}}+|\theta|\int_{\ell
cd}^{\ell cd+g}te^{-\kappa_{1}\sqrt{\log t}}dt$
$\displaystyle\ll_{\ell}ge^{-\kappa_{1}\sqrt{\log
g}}+|\theta|e^{-\kappa_{1}\sqrt{\log g}}\int_{\ell cd}^{\ell cd+g}t~{}dt$
$\displaystyle\ll_{\ell}g(1+|\theta|g)e^{-\kappa_{1}\sqrt{\log g}}\,.$

Thus,

$\widetilde{f(\theta)}=\sum_{\ell cd\leq m\leq g+\ell cd}e(\theta
m)+O_{\ell}\bigl{(}g(1+|\theta|g)e^{-\kappa_{1}\sqrt{\log g}}\bigr{)}\,.$

Hence,

$\displaystyle f(\theta)$ $\displaystyle=e(-\theta\ell cd)\sum_{\ell cd\leq
m\leq g+\ell cd}e(\theta
m)+O_{\ell}\bigl{(}g(1+|\theta|g)e^{-\kappa_{1}\sqrt{\log g}}\bigr{)}$
$\displaystyle=\sum_{0<n\leq g}e(\theta
n)+O_{\ell}\bigl{(}g(1+|\theta|g)e^{-\kappa_{1}\sqrt{\log g}}\bigr{)}\,.$

By the estimates of [2, Lemma 4.4], we have

$\displaystyle\int_{|\theta|\leq\frac{Q}{g}}f(\theta)h(-\theta)d\theta=\frac{g}{2}+O_{\ell}\left(\frac{g}{Q}(\log
g)^{2}+gQ^{2}e^{-\kappa_{1}\sqrt{\log g}}\right)\,.$ (3.6)

For $Q<c^{1/3}$, by [2, Lemma 4.5] we also have

$\displaystyle\mathcal{R}=\sum_{2\leq q\leq Q}\sum_{1\leq a\leq
q\atop\gcd(a,q)=1}\int_{|\theta|\leq\frac{Q}{qg}}f\left(\frac{a}{q}+\theta\right)h\left(-\frac{a}{q}-\theta\right)d\theta\ll_{\ell}dQ^{3}\,.$
(3.7)

Bringing together (3.4), (3.6) and (3.7), we conclude that for $Q<c^{1/3}$

$\displaystyle\int_{\mathfrak{M}(Q)}f(\alpha)h(-\alpha)=\frac{g}{2}+O_{\ell}\left(\frac{g}{Q}(\log
g)^{2}+gQ^{2}e^{-\kappa_{1}\sqrt{\log g}}+dQ^{3}\right).$ (3.8)

###  3.3. The asymptotic formula

It can be concluded from (3.2), (3.3) and (3.8) that for $Q<c^{1/3}$,

$\displaystyle\psi_{\ell,c,d}=\frac{g}{2}+O_{\ell}\left(\frac{g}{Q}(\log
g)^{2}+gQ^{2}e^{-\kappa_{1}\sqrt{\log g}}+dQ^{3}+\frac{g(\log
g)^{6}}{Q^{1/2}}+g^{4/5}(\log g)^{6}\right)\,.$

We now choose $Q=(\log g)^{14}$. Then we can obtain

$\displaystyle\psi_{\ell,c,d}=\frac{g}{2}+O_{\ell}\left(\frac{g}{\log
g}\right),$ (3.9)

provided that $c\geq(\log g)^{43}$.

For $c\leq(\log g)^{43}$, we have

$\displaystyle\psi_{\ell,c,d}$ $\displaystyle=\sum_{\begin{subarray}{c}\ell
cd<n\leq g_{\ell}(c,d)\\\ n=\ell cd+cx_{0}+dy_{0}\\\ 0\leq x_{0}\leq d;\,0\leq
y_{0}\leq c\end{subarray}}\Lambda(n)=\sum_{\begin{subarray}{c}0<m\leq g\\\
m=cx+dy\\\ x,y\in\mathbb{N}_{0}\end{subarray}}\Lambda(\ell cd+m)$
$\displaystyle=\sum_{1\leq y\leq c\atop\gcd(y,c)=1}\sum_{m+\ell cd\equiv
dy\\!\\!\\!\\!\\!\ ({\rm{mod}}\ c)\atop dy\leq m+\ell cd\leq g}\Lambda(m+\ell
cd)+O_{\ell}(\log g)$ $\displaystyle=\sum_{1\leq y\leq
c\atop\gcd(y,c)=1}\sum_{n\equiv dy\\!\\!\\!\\!\\!\ ({\rm{mod}}\ c)\atop
dy+\ell cd\leq n\leq g+\ell cd}\Lambda(n)+O_{\ell}(\log g)$
$\displaystyle=\sum_{1\leq y\leq c\atop\gcd(y,c)=1}\bigl{(}\psi(g+\ell
cd;c,dy)-\psi(dy+\ell cd;c,dy)\bigr{)}+O_{\ell}(\log g).$

Since $c\leq(\log g)^{43}\ll(\log d)^{43}$, by the Siegel–Walfisz theorem we
have

$\displaystyle\psi(g+\ell cd;c,dy)-\psi(dy+\ell
cd;c,dy)=\frac{g-dy}{\varphi(c)}+O_{\ell}\left(ge^{-\kappa_{2}\sqrt{\log
g}}\right)\,,$

where $\kappa_{2}>0$ is an absolute constant. Thus, for $c\leq(\log g)^{43}$
we have

$\displaystyle\psi_{\ell,c,d}$ $\displaystyle=\sum_{1\leq y\leq
c\atop\gcd(y,c)=1}\frac{g-dy}{\varphi(c)}+O_{\ell}\left(g(\log
g)^{43}e^{-\kappa_{2}\sqrt{\log g}}\right)$
$\displaystyle=g-\frac{1}{2}cd+O_{\ell}\left(g(\log
g)^{43}e^{-\kappa_{2}\sqrt{\log g}}\right)$
$\displaystyle=g/2+O_{\ell}\left(g/c+g(\log g)^{43}e^{-\kappa_{2}\sqrt{\log
g}}\right).$ (3.10)

Therefore, from (3.9) and (3.10) we know that

$\psi_{\ell,c,d}\sim g/2\quad(\text{as}~{}c\rightarrow\infty).$

This completes the proof of proposition 2.1. ∎

##  4\. Proof of Theorem 1.1

###### Proof of Theorem 1.1.

From now on, the symbol $p$ will always denote primes. For $\ell cd<p\leq
g_{\ell}(c,d)$, let

$\vartheta_{\ell,a,b}(t)=\sum_{\begin{subarray}{c}\ell cd<p\leq t\\\ p=\ell
cd+cx_{0}+dy_{0}\\\ 0\leq x_{0}\leq d;\,0\leq y_{0}\leq c\end{subarray}}\log
p$

and $\vartheta_{\ell,a,b}=\vartheta_{\ell,a,b}\bigl{(}g_{\ell}(c,d)\bigr{)}$.
From Lemma 2.3, we obtain that

$\displaystyle\pi_{\ell,a,b}=\sum_{\begin{subarray}{c}\ell cd<p\leq
g_{\ell}(c,d)\\\ p=\ell cd+cx_{0}+dy_{0}\\\ 0\leq x_{0}\leq d;\,0\leq
y_{0}\leq c\end{subarray}}1=\frac{\vartheta_{\ell,a,b}}{\log
g_{\ell}(c,d)}+\int_{\ell
cd}^{g_{\ell}(c,d)}\frac{\vartheta_{\ell,a,b}(t)}{t\log^{2}t}dt$ (4.1)

via partial summation. By the Chebyshev estimate, we have

$\vartheta_{\ell,a,b}(t)\leqslant\sum_{p\leqslant t}\log p\ll t,$

from which it follows that

$\int_{\ell
cd}^{g_{\ell}(c,d)}\frac{\vartheta_{\ell,a,b}(t)}{t\log^{2}t}dt\ll\int_{\ell
cd}^{g_{\ell}(c,d)}\frac{1}{\log^{2}t}dt\ll_{\ell}\frac{g}{(\log g)^{2}}.$
(4.2)

Again, using the Chebyshev estimate, we have

$\vartheta_{\ell,a,b}=\psi_{\ell,a,b}+O_{\ell}\bigl{(}\sqrt{g}\bigr{)}.$ (4.3)

Thus, by Proposition 2.1 and Eqs. (4.1), (4.2), (4.3), we conclude that

$\pi_{\ell,a,b}=\frac{\psi_{\ell,a,b}}{\log
g_{\ell}(c,d)}+O_{\ell}\left(\frac{\sqrt{g}}{\log g}+\frac{g}{(\log
g)^{2}}\right)\sim\frac{1}{2}\frac{g}{\log g_{\ell}(c,d)},$ (4.4)

as $c\rightarrow\infty$. Recall that

$\pi(g_{\ell}(c,d))\sim\frac{g_{\ell}(c,d)}{\log
g_{\ell}(c,d)}\sim\frac{(\ell+1)g}{\log
g_{\ell}(c,d)}\quad(\text{as~{}}c\rightarrow\infty).$ (4.5)

Now, Theorem 1.1 follows immediately from (4.4) and (4.5). ∎

## Acknowledgments

The first named author is supported by National Natural Science Foundation of
China (Grant No. 12201544), Natural Science Foundation of Jiangsu Province,
China (Grant No. BK20210784), China Postdoctoral Science Foundation (Grant No.
2022M710121). He is also supported by (Grant No. JSSCBS20211023 and
YZLYJF2020PHD051).

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