# On creating new essential spectrum by self-adjoint extension of gapped
operators

Alessandro Michelangeli  Department of Mathematics and Natural Sciences,
Prince Mohammad Bin Fahd University
Al Khobar 31952 (Saudi Arabia)
and Hausdorff Center for Mathematics, University of Bonn
Endenicher Allee 60
D-53115 Bonn (Germany)
and TQT Trieste Institute for Theoretical Quantum Technologies, Trieste
(Italy<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Given a densely defined and gapped symmetric operator with infinite deficiency
index, it is shown how self-adjoint extensions admitting arbitrarily
prescribed portions of the gap as essential spectrum are identified and
constructed within a general extension scheme. The emergence of new spectrum
in the gap by self-adjoint extension is a problem with a long history and
recent deep understanding, and yet it remains topical in several recent
applications. Whereas it is already an established fact that, in case of
infinite deficiency index, any kind of spectrum inside the gap can be
generated by a suitable self-adjoint extension, the present discussion has the
virtue of showing the clean and simple operator-theoretic mechanism of
emergence of such extensions.

###### Key words and phrases:

self-adjoint operators on Hilbert space; Kreĭn-Višik-Birman extension theory;
point spectrum; essential spectrum, Kreĭn-von Neumann extension

###### 2010 Mathematics Subject Classification:

46C99, 47A10, 47B25, 47N20, 47N50

Partially supported by the Italian National Institute for Higher Mathematics
INdAM, the Alexander von Humboldt Foundation, Bonn, and Prince Mohammed Bin
Fahd University, Al Khobar. This note is strongly inspired by the very clean
and instructive presentations on closely related subjects delivered by M.
Holzmann and by K. Pankrashkin at the workshop _Spectral Theory of
Differential Operators in Quantum Theory_ (J. Behrndt, F. Gesztesy, A. Laptev,
and C. Tretter organisers) held at the ESI Vienna on 7-11 November 2022, and
by B. Benhellal at the workshop _Singular Perturbations and Geometric
Structures_ (M. Gallone, A. Michelangeli, and L. Rizzi organisers) held at
SISSA Trieste on 21-23 November 2022. The author is also most grateful to the
Erwin Schrödinger International Institute for Mathematics and Physics (ESI),
Vienna, and to the Mathematical Institute at the Silesian University in Opava
(and in particular to N. A. Caruso and K. Hasík), for the kind hospitality
during the period in which this project was set up.

##  1\. Introduction and background

The general problem of producing prescribed portions and types of additional
spectrum by constructing suitable self-adjoint extensions of certain symmetric
operators, or also, on a closely related footing, by taking singular
perturbations of self-adjoint operators, is a long lasting one in operator
theory, both in abstract terms and in application, typically, to differential
operators. This is witnessed, beside many fundamental precursors that are
going to be briefly surveyed in a moment, by the recent advances made by the
group of Prof J. Behrndt and the group of Prof K. Pankrashkin, which provided
the main inspiration and motivation for the present note and will be
illustrated below.

The playground for such a problem are _gapped_ operators, namely densely
defined symmetric operators $S$ on a complex Hilbert space $\mathcal{H}$
admitting an open (finite or semi-infinite) interval $J=(a,b)$ as a ‘gap’, in
the precise sense that

(1.1) $\begin{split}\langle
f,Sf\rangle\;&\geqslant\;b\,\|f\|^{2}\qquad\qquad\forall
f\in\mathcal{D}(S)\qquad\textrm{if }a\,=\,-\infty\,,\\\
\Big{\|}\Big{(}S-\frac{a+b}{2}\mathbbm{1}\Big{)}f\Big{\|}\;&\geqslant\;\frac{b-a}{2}\,\|f\|\qquad\;\,\forall
f\in\mathcal{D}(S)\qquad\textrm{if }a\,>\,-\infty\,.\end{split}$

Here and in the following, as customary, the notation
$\langle\cdot,\cdot\rangle$ indicates the scalar product of $\mathcal{H}$ (and
$\|\cdot\|$ is the associated norm), with the convention that the anti-linear
entry is the left one, and $\mathcal{D}(A)$ denotes the domain of an operator
$A$.

As a matter of fact, owing to classical results by Kreĭn [44, Theorem 21],
Stone [49, Theorem 9.21], Friedrichs [27], and Freudenthal [26], the gap
condition (1.1) is equivalent to the existence of a self-adjoint extension
$\widetilde{S}$ of $S$ having $J$ as a gap in its spectrum, i.e.,
$\sigma(\widetilde{S})\cap J=\emptyset$. Also, one sees that in the limit of
infinite gap the second condition of (1.1), equivalently re-written as

$\|Sf\|^{2}-(a+b)\langle f,Sf\rangle+ab\|f\|^{2}\;\geqslant\;0\,,$

takes the form of the first of (1.1) upon dividing by $a<0$ and letting
$a\to-\infty$.

Now, a gapped symmetric operator $S$ whose deficiency indices

$\dim(\ker(S^{*}-z\mathbbm{1}))\qquad\textrm{and}\qquad\dim(\ker(S^{*}-\overline{z}\mathbbm{1}))$

(that, owing to the existence of a gap, are necessarily equal for any
$z\in\mathbb{C}^{+}\cup J$) are _infinite_ allows for such a vast variety of
inequivalent self-adjoint extensions to make it rather plausible to expect
that there _are_ extensions of $S$ having any sort of prescribed spectrum
within the gap $J$.

This picture has been actually confirmed, with many related aspects still
under investigation, from multiple perspectives:

  * •

by considering the abstract existence or also abstract construction of self-
adjoint extensions with prescribed spectrum in the gap [20, 21, 18, 19, 1, 22,
17, 2, 13];

  * •

by considering abstract _singular perturbations_ of a self-adjoint operator
(in the standard sense, e.g., of [8]), which display prescribed eigenvalues
and eigenvectors [41, 6, 42, 3];

  * •

by focussing on explicit gapped differential operators on variable bounded
domains of $\mathbb{R}^{d}$ with given boundary condition of self-adjointness,
and identifying which shape of the domain reflects into spectrum of prescribed
type and position [24, 39, 38, 13];

  * •

by considering singular self-adjoint perturbations, supported on points or
more generally on manifolds of non-zero co-dimension, of an explicit
differential operator (Laplace, Dirac, etc.), and identifying which location,
shape, strength of the point-like perturbation reflect into spectrum of
prescribed type and position [5, 35, 11, 13, 14, 12, 15].

The works cited above provide extra background and older references: in fact,
all such mainstream ideally dates back to a very classical result by Kreĭn
[44, Theorem 23] on the existence, for gapped symmetric operators with (equal
and) _finite_ deficiency indices, of self-adjoint extensions with prescribed
finite point spectrum in the gap.

At the abstract operator-theoretic level it is today known that _any kind of
spectrum inside the gap of a gapped symmetric operator $S$ with _infinite_
deficiency indices on a separable complex Hilbert space $\mathcal{H}$ can be
generated by a suitable self-adjoint extension of $S$ in $\mathcal{H}$_. This
quite remarkable result is due to Brasche [17, Theorem 27] as the culmination
of a extensive and prolific investigation that includes the notable precursors
[20, Theorem 2.2], [19, Theorem 1], [1, Theorem 6.1], [22, Theorems 4.5 and
6.3] from collaborations with Albeverio, Neidhardt, and Weidmann, and also
from hints by Simon [22].

Despite the ‘ultimate’ character of such a general result, the focus remains
active both on the abstract operator-theoretic mechanisms for the appearance
of spectrum of prescribed type and location within the gap (on which further
important comments are going to follow later in this Introduction), and on a
variety of explicit differential operators of relevance in applications, self-
adjointly realised as extensions of gapped symmetric operators.

In this respect, it is instructive to highlight the recent activity by Behrndt
and collaborators, as well as by Pankrashkin and collaborators, concerning the
appearance of pre-definite parts of essential spectrum for self-adjoint
realisations of Schrödinger and Dirac operators with singular perturbation
supported on a suitable manifold of positive co-dimension in $\mathbb{R}^{d}$.
They provided valuable insight on such spectral phenomenon.

In a first significant model, a quantum particle scatters in one dimension
across an infinite sequence of delta-like impurities supported at points
$x_{1},x_{2},\dots$. The Hamiltonian is formally of the type

$H_{\Sigma,\gamma}\;=\;-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\sum_{k=1}^{\infty}\gamma_{k}\delta(\cdot-
x_{k})\,,$

for a given sequence
$\Sigma\equiv(x_{k})_{k\in\mathbb{N}}\subset(\ell_{-},\ell_{+})$ of
singularity locations, and a given sequence
$\gamma\equiv(\gamma_{k})_{k\in\mathbb{N}}\subset\mathbb{R}$ of interaction
magnitudes, and it is self-adjointly realised on $L^{2}(\ell_{-},\ell_{+})$ by
means of the boundary conditions

$\psi(x_{k}^{+})\;=\;\psi(x_{k}^{-})\,,\qquad\psi^{\prime}(x_{k}^{+})-\psi^{\prime}(x_{k}^{-})\;=\;\gamma_{k}\psi(x_{k})\,,\qquad\psi(\ell_{\pm})\;=\;0$

imposed on functions $\psi\in H^{1}((\ell_{-},\ell_{+})\cap
H^{2}((\ell_{-},\ell_{+})\setminus\bigcup_{k\in\mathbb{N}}\\{x_{k}\\})$. This
is by now a well-established construction, comprehensively discussed, e.g., in
[4, Chapters II.2 and III.2] as well as in [7, 43, 14]). Clearly, any such
$H_{\Sigma,\gamma}$ is a self-adjoint extension of the non-negative symmetric
operator

$\mathring{H}\;=\;\Big{(}-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\Big{)}\bigg{|}_{C^{\infty}_{c}((\ell_{-},\ell_{+})\setminus\bigcup_{k\in\mathbb{N}}\\{x_{k}\\})}\,.$

It has been recently proved [14, Theorems 1.1 and 4.4] that for bounded
$(\ell_{-},\ell_{+})$ any pre-definite closed and lower semi-bounded subset
$S_{\mathrm{ess}}\subset\mathbb{R}$ is the _essential spectrum_ of one of the
$H_{\Sigma,\gamma}$’s, for a suitable choice of $(x_{k})_{k\in\mathbb{N}}$ and
$(\gamma_{k})_{k\in\mathbb{N}}$. An analogous result [13, Theorems 3.1 and
3.3], under the constraint $\\{0\\}\subset
S_{\mathrm{ess}}\subset[0,+\infty)$, has been also proved for the counterpart
model formally described by

$\widetilde{H}_{\Sigma,\gamma}\;=\;-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\sum_{k=1}^{\infty}\gamma_{k}\langle\delta^{\prime}_{x_{k}},\cdot\rangle\,\delta^{\prime}_{x_{k}}\,,$

where $\delta^{\prime}_{x_{k}}\equiv\delta^{\prime}(\cdot-x_{k})$ is the
distributional derivative of $\delta(\cdot-x_{k})$ and
$\langle\delta^{\prime}_{x_{k}},\cdot\rangle$ denotes its action
$\phi\mapsto\langle\delta^{\prime}_{x_{k}},\phi\rangle=-\phi^{\prime}(x_{k})$
on test functions $\phi$: the $\widetilde{H}_{\gamma}$’s constitute another
class of self-adjoint extensions of $\mathring{H}$ and are self-adjointly
realised with action $-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}$ and boundary
conditions of self-adjointness

$\psi^{\prime}(x_{k}^{+})\;=\;\psi^{\prime}(x_{k}^{-})\,,\qquad\psi(x_{k}^{+})-\psi(x_{k}^{-})\;=\;\gamma_{k}\psi^{\prime}(x_{k})\,.$

Either result relies on a similar idea as in the preceding construction [39]
of bounded domains of ‘room-and-passages’ type, where the self-adjoint Neumann
Laplacian is proved to have also essential spectrum – somewhat contrarily to
the intuition that self-adjoint Laplace-type operators on bounded domains only
have purely discrete spectrum (or, equivalently, a compact resolvent).

A second type of model under deep scrutiny, from the same perspective of
emergence of essential spectrum in a gap by self-adjoint extension, concerns
the three-dimensional Dirac operator with critical combination of
electrostatic and Lorenz scalar shell interaction supported at compact
surfaces. In this case the Hamiltonian governing the evolution of the quantum
particle is the operator $H_{\Sigma,\varepsilon,\mu}$ in
$L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$ formally acting as

$H_{\Sigma,\varepsilon,\mu}\;=\;-\mathrm{i}\sum_{j=1}^{3}\alpha_{j}\partial_{x_{j}}+m\beta+(\varepsilon\mathbbm{1}+\mu\beta)\delta_{\Sigma}$

for given mass, electrostatic shell, and Lorenz shell parameters,
respectively, $m\geqslant 0$ and $\varepsilon,\mu\in\mathbb{R}$, where
$\alpha_{1},\alpha_{2},\alpha_{3},\beta$ are four anti-commuting $4\times 4$
Hermitian matrices squaring to the identity – for concreteness, the usual
choice of the Pauli matrices, $x\equiv(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$,
and $\delta_{\Sigma}$ is the delta distribution along the normal to $\Sigma$
at each point of a given sufficiently smooth compact surface
$\Sigma\subset\mathbb{R}^{3}$. Thus, $H_{\Sigma,\varepsilon,\mu}$ is self-
adjointly realised by means of $\delta$-type boundary conditions along the
normal direction at each point of $\Sigma$, and is by construction a self-
adjoint extension of

$\mathring{H}\;=\;\bigg{(}\mathrm{i}\sum_{j=1}^{3}\alpha_{j}\partial_{x_{j}}+m\beta\bigg{)}\bigg{|}_{C^{\infty}_{c}(\mathbb{R}^{3}\setminus\Sigma,\mathbb{C}^{4})}\,,$

or also, from another point of view, $H_{\Sigma,\varepsilon,\mu}$ is a
_singular perturbation_ (in the sense of, e.g., [8]) of the self-adjoint
operator $H_{0}$ defined by

$\mathcal{D}(H_{0})\;=\;H^{1}(\mathbb{R}^{3},\mathbb{C}^{4})\,,\qquad
H_{0}\psi\;=\;-\mathrm{i}\sum_{j=1}^{3}\alpha_{j}\partial_{x_{j}}\psi+m\beta\psi\,.$

As a matter of fact, $H_{0}$ is gapped with

$\sigma(H_{0})\;=\;\sigma_{\mathrm{ess}}(H_{0})=(-\infty,-m]\cup[m,+\infty)\,,$

and for $\varepsilon^{2}-\mu^{2}\neq 4$ each singular perturbations
$H_{\Sigma,\varepsilon,\mu}$ preserve the essential spectrum and only produces
at most finitely many eigenvalues of finite multiplicity within the gap [9,
12, 48]. This is analogous to the more classical problem (see [30, 28] and
references therein) of the self-adjoint realisations of the Dirac-Coulomb
Hamiltonian as extensions of

$\bigg{(}\mathrm{i}\sum_{j=1}^{3}\alpha_{j}\partial_{x_{j}}+m\beta+\frac{\nu}{\,|x|\,}\bigg{)}\bigg{|}_{C^{\infty}_{c}(\mathbb{R}^{3}\setminus\\{0\\},\mathbb{C}^{4})}$

in the relevant regime $\nu\in\mathbb{R}$, $\frac{\sqrt{3}}{2}<|\nu|<1$ of
existence of multiple extensions: each such self-adjoint realisation comes
with the same essential spectrum as $\sigma_{\mathrm{ess}}(H_{0})$ and with
additional point spectrum consisting of an infinity of eigenvalues of finite
multiplicity, accumulating to one of the two edges of the gap $(-m,m)$
(depending on the sign of $\nu$). Instead, in the critical regime
$\varepsilon^{2}-\mu^{2}=4$ it has been recently revealed that
$H_{\Sigma,\varepsilon,\mu}$ above may have additional essential spectrum (as
compared to $H_{0}$) in the gap $(-m,m)$: in particular [12, Theorem 6.6], if
$\Sigma$ is partially flat, then the point $-\frac{m\mu}{\varepsilon}$ belongs
to $\sigma_{\mathrm{ess}}(H_{\Sigma,\varepsilon,\mu})$. Even more strikingly
[15, Theorem 1], in terms of the constant

$A_{\Sigma}\;:=\;\max_{x\in\Sigma}|\kappa_{1}(x)-\kappa_{2}(x)|\,,$

where $\kappa_{1}$ and $\kappa_{2}$ are the two principal curvatures of the
smooth and compact surface $\Sigma$, one has

$\sigma_{\mathrm{ess}}(H_{\Sigma,\varepsilon,\mu})\;=\;\sigma_{\mathrm{ess}}(H_{0})\cup\Big{[}-\frac{m\mu}{\varepsilon}-\frac{A_{\Sigma}}{\,2\varepsilon}\,,-\frac{m\mu}{\varepsilon}+\frac{A_{\Sigma}}{\,2\varepsilon}\,\Big{]}\,.$

When $\Sigma$ is a sphere (or a union of disjoint spheres), and only in that
case, $\kappa_{1}\equiv\kappa_{2}$ [25, Prop. 4, Sect. 3.2], thereby implying
that for such $\Sigma$ the operator $H_{\Sigma,\varepsilon,\mu}$ produces one
single point of additional essential spectrum.

As a matter of fact, a common trait of the recent activity illustrated above
is that the emergence of additional essential spectrum in the gap is detected
through various types of analyses that are all particularly laborious, to say
the least, both in explicit models of singular perturbations of certain
Schrödinger or Dirac operators, and in abstract settings.

Thus, for instance, the spectral analysis [15] of the above-mentioned three-
dimensional Dirac operator with critical electrostatic-Lorenz scalar shell
interaction goes first through a reformulation in terms of matrix singular
integral operators over the surface $\Sigma$ and a subsequent reduction, by
means of the theory of block operator matrices, to the analysis of the
essential spectrum of a particular pencil (in fact, a Schur complement) of
integral operators over $\Sigma$. In turn, this analysis is performed in terms
of principal symbols of suitable auxiliary operators, and it relies on a
subtle toolbox concerning the spectrum of classical pseudo-differential
operators of order zero on compact Riemannian manifolds.

As for the emergence of pre-definite regions of essential spectrum of one-
dimensional singular Schrödinger operators with countably infinite point-like
$\delta$-interactions, this is obtained in [14] by choosing location and
magnitude of the singular interactions so as to partition
$(\ell_{-},\ell_{+})$ into adjacent intervals on which one builds an auxiliary
operator of singular point interaction in the very same spirit of the room-
and-passage construction [39] for Neumann Laplacians: such auxiliary operator
is constructed so that its eigenvalues constitute a dense of the prescribed
spectral set $S_{\mathrm{ess}}$, and the final $H_{\Sigma,\gamma}$ is then
built as a compact perturbation, in the resolvent sense, of the auxiliary
operator; it therefore displays the same essential spectrum. (The singular
$\delta^{\prime}$-interaction case is treated along the same ideas [13].)

The abstract operator-theoretic analysis that culminated with the above-
mentioned result by Brasche is no cheap business either, as the subtle works
[19, 1, 22, 17] show. In such investigations, starting with a symmetric
operator $S$ with infinite deficiency indices and with gap $J$, the self-
adjoint extension of $S$ which provides the pre-definite region and type of
spectrum inside the gap is eventually identified ‘indirectly’ up to unitary
equivalence, in the sense that for any arbitrary auxiliary self-adjoint
operator $A_{\mathrm{aux}}$ an operator $A=A^{*}\supset S$ is shown to exists
such that $AE_{A}(J)$ is unitarily equivalent to
$A_{\mathrm{aux}}E_{A_{\mathrm{aux}}}(J)$ (the notation $E_{H}(\cdot)$
standing for the operator-valued measure associated with the self-adjoint
$H$). This line of reasoning, as far as the existence of extensions with pre-
definite absolutely continuous or singular continuous spectrum in the gap is
concerned, employs ingenious arguments from measure theory and block
operators, and in a sense eventually leaves the ‘explicit mechanism’ of
construction of $A$ from $S$ somewhat obfuscated.

It is instead cleaner, in its basic mechanisms, how to abstractly construct
self-adjoint extensions with only prescribed _point spectrum_ within the gap –
hence also with prescribed _essential spectrum_ , as one may select a
countable dense in the assigned region of new essential spectrum and build the
desired self-adjoint extension so that it admits the previously selected
points as eigenvalues. The idea comes from the already mentioned ‘starting’
result [44, Theorem 23] by Kreĭn for finite deficiency indices: in case of
infinite deficiency indices the same idea was refined and adapted in [20,
Theorem 2.2], [1, Lemma 2.1 and Proposition 3.1], [17, Lemma 29 and Corollary
30], [6, Theorem 3.3]. This approach will be further discussed in Sections 2
and 3 in comparison to the construction presented in the present work.

In fact, it is precisely _the construction, in the infinite deficiency index
case, of self-adjoint extensions with pre-definite point spectrum
(respectively, essential spectrum) within the gap_ which is at the core of
this note, both in view of the abstract results cited here above, and of the
previously reviewed recent analysis of explicit models.

In particular, _the perspective here is to identify the mechanism of emergence
of the desired extension with respect to general self-adjoint extension
schemes_.

That is: how does the prescribed set of point spectrum, or of essential
spectrum, select the desired operator, out of the general family of self-
adjoint extensions produced and classified by abstract extension schemes?

Recently, Behrndt and Khrabustovskyi [13, Theorem 4.3] have partially explored
this direction, producing an elegant, clean and short proof of Brasche’s
theorem in the case of new essential spectrum, which has the value of
characterising the desired extension as a convenient restriction of $S^{*}$
with respect to the decomposition

$\mathcal{D}(S^{*})\;=\;\mathcal{D}(\overline{S})\dotplus\ker(S^{*}-\mu\mathbbm{1})$

for $\mu$ in the gap of $S$, the latter identity being the starting point of
any abstract self-adjoint extension scheme (see (2.2) below). Their argument,
however, relies on the crucial additional assumption that $S$ admit a self-
adjoint extension with _compact resolvent_ : this condition is exploited in
multiple steps, in particular showing that from an arbitrary self-adjoint
operator $\Xi$ on the infinite-dimensional Hilbert space
$\ker(S^{*}-\mu\mathbbm{1})$ one can build a self-adjoint restriction $A$ of
$S^{*}$ which is, upon identifying $\Xi$ as an operator on the whole
$\mathcal{H}=\ker(S^{*}-\mu\mathbbm{1})\oplus\mathrm{ran}(\overline{S}-\mu\mathbbm{1})$,
a compact perturbation in the resolvent sense of $\Xi$ itself, thereby
implying that $\sigma_{\mathrm{ess}}(A)=\sigma_{\mathrm{ess}}(\Xi)$.

In this note, the emergence of arbitrary new point spectrum and essential
spectrum in the gap when the deficiency index is infinite is shown in its
simple and clean operator-theoretic mechanism within the general scheme of
self-adjoint extensions, with no restriction of separability, compactness of
the resolvent, and the like.

Useful preparations are discussed in Section 2, including a crucial
construction in the prototypical case of _unital_ deficiency index.

The general mechanism when the deficiency index is _infinite_ is then
discussed in Section 3.

##  2\. Preparation: unital deficiency index

Let us start by recalling the following general construction and
classification of _all_ the self-adjoint extensions of a given gapped
symmetric operator. Up to an additive shift by a suitable multiple of the
identity, it is clearly non-restrictive to assume the gap to be around zero,
which will be done throughout.

This is a classical result that emerged about seven decades ago as the
culmination of seminal works by Kreĭn [44], Višik [50], and Birman [16], and
is therefore fair to refer to as the Kreĭn-Višik-Birman theory, or also Kreĭn-
Višik-Birman-Grubb theory in view of the subsequent adaptation by Grubb [36]
to the more general, but technically similar problem of the closed extensions
of closed operators (see, e.g., [33] and [32, Chapter 2] and the references
therein, as well as [37, Chapter 13] and [40]). In more recent times it has
become customary to re-derive Theorem 2.1 below within the theory of boundary
triplets (one updated reference to which is [10]), a modern reformulation of
the approach by Višik and Birman. The original classification a la Birman is
particularly efficient and informative in the mathematical analysis of several
quantum mechanical models of current interest – see, e.g., [46, 47, 30, 28,
29, 34, 31, 45].

######  Theorem 2.1.

Let $S$ be a densely defined symmetric operator in a complex Hilbert space
$\mathcal{H}$, which admits a self-adjoint extension $S_{\mathrm{D}}$ that has
everywhere defined bounded inverse on $\mathcal{H}$ – equivalently, assume
that $S$ is a densely defined gapped symmetric operator with zero in the gap.
There is a one-to-one correspondence between the family of the self-adjoint
extensions of $S$ in $\mathcal{H}$ and the family of the self-adjoint
operators in Hilbert subspaces of $\ker S^{*}$. If $T$ is any such operator,
in the correspondence $T\leftrightarrow S_{T}$ each self-adjoint extension
$S_{T}$ of $S$ is given by

(2.1) $\begin{split}S_{T}\;&:=\;S^{*}\upharpoonright\mathcal{D}(S_{T})\,,\\\
\mathcal{D}(S_{T})\;&:=\;\left\\{f+S_{\mathrm{D}}^{-1}(Tv+w)+v\left|\begin{array}[]{c}f\in\mathcal{D}(\overline{S})\,,\;v\in\mathcal{D}(T)\\\
w\in\ker
S^{*}\cap\mathcal{D}(T)^{\perp}\end{array}\right.\right\\}.\end{split}$

Observe that indeed
$\mathcal{D}(S)\subset\mathcal{D}(S_{T})\subset\mathcal{D}(S^{*})$, as

(2.2) $\mathcal{D}(S^{*})\;=\;\mathcal{D}(\overline{S})\dotplus
S_{\mathrm{D}}^{-1}\ker S^{*}\dotplus\ker S^{*}$

(see, e.g., [32, Proposition 2.2]).

It is convenient and customary to refer to the operator $T$ indexing the self-
adjoint extension $S_{T}$ as the _(Birman) extension parameter_ of $S_{T}$.

It is known that quite an amount of information on $S_{T}$ may be read out
from the often ‘easier’ labelling operator $T$: one, in particular, is
invertibility (see, e.g., [32, Theorem 2.22]):

######  Theorem 2.2.

Under the assumptions Theorem 2.1:

  1. (i)

$\ker S_{T}=\ker T$, and therefore $S_{T}$ is injective $\Leftrightarrow$ $T$
is injective;

  2. (ii)

$S_{T}$ is surjective $\Leftrightarrow$ $T$ is surjective;

  3. (iii)

$S_{T}$ is invertible in the whole $\mathcal{H}$ $\Leftrightarrow$ $T$ is
invertible in the whole $\overline{\mathcal{D}(T)}$.

Now, as a first application of the general results above, consider a gapped
symmetric operator around zero with _unital deficiency index_ , i.e.,

(2.3) $\begin{array}[]{l}\textrm{$S$ is a densely defined and symmetric
operator in $\mathcal{H}$}\\\ \textrm{with a gap $(a,b)\ni 0$ in the sense of
(1.1),}\\\ \textrm{\emph{equivalently},}\\\ \textrm{with distinguished
$S_{\mathrm{D}}=S_{\mathrm{D}}^{*}\supset S$ such that
$S_{\mathrm{D}}^{-1}\in\mathcal{B}(\mathcal{H})$.}\end{array}$

and

(2.4) $d(S)\,=\,1\,,$

where, under assumption (2.3),

(2.5) $d(S)\;:=\;\dim\ker S^{*}\;=\;\dim\ker(S^{*}-z\mathbbm{1})\qquad\forall
z\in\mathbb{C}\setminus\mathbb{R}\,.$

In this context it is meant that the gap $(a,b)$ may be finite (namely, finite
$a$ and $b$) or semi-infinite (say, $a=-\infty$ and $b>0$ finite). Recall that
the fact that the gap condition (2.3) ensures the validity of the identity in
(2.5) for any complex non-real $z$ makes it legitimate to refer to $d(S)$ as
_the_ deficiency index of $S$.

Observe also that (2.3) implies
$0\notin\sigma_{\mathrm{point}}(\overline{S})$, i.e.,
$\ker\overline{S}=\\{0\\}$: $\overline{S}$ is thus invertible on its range.

An instructive special case of (2.3), ubiquitous in applications, is that of a
lower semi-bounded and densely defined $S$, meaning

(2.6) $\mathfrak{m}(S)\;:=\;\inf_{\begin{subarray}{c}\psi\in\mathcal{D}(S)\\\
\psi\neq
0\end{subarray}}\frac{\langle\psi,S\psi\rangle}{\;\|\psi\|^{2}}\;>\;-\infty\,,$

with strictly positive lower bound, that is, $\mathfrak{m}(S)>0$.

For generic $\lambda\in(a,b)$, set

(2.7)
$\mathcal{D}_{\lambda}\,:=\,\mathcal{D}(\overline{S})\dotplus\ker(S^{*}-\lambda\mathbbm{1})\,,\qquad
S_{\lambda}\,:=\,S^{*}\upharpoonright\mathcal{D}_{\lambda}\,.$

Observe that
$\mathcal{D}(S^{*})=\mathcal{D}(S^{*}-\lambda\mathbbm{1})=\mathcal{D}((S-\lambda\mathbbm{1})^{*})$
($\lambda$ being real), and moreover having taken $\lambda$ in the gap
guarantees that $S-\lambda\mathbbm{1}$ too has a distinguished self-adjoint
extension $(S-\lambda\mathbbm{1})_{\mathrm{D}}$ which has everywhere defined
and bounded inverse on $\mathcal{H}$. Owing to (2.2) above one has

(2.8)
$\mathcal{D}(S^{*}-\lambda\mathbbm{1})\,=\,\mathcal{D}(\overline{S}-\lambda\mathbbm{1})\dotplus(S-\lambda\mathbbm{1})_{\mathrm{D}}^{-1}\ker(S^{*}-\lambda\mathbbm{1})\dotplus\ker(S^{*}-\lambda\mathbbm{1})\,.$

Thus, since
$\mathcal{D}(\overline{S})=\mathcal{D}(\overline{S}-\lambda\mathbbm{1})$, it
is indeed confirmed that the sum in (2.7) is direct.

Actually, (2.7) is a way of introducing the crucial operator $S_{\lambda}$
that is _not_ the convenient one needed for exporting our argument to the case
of infinite deficiency case, and for this reason it will be replaced in a
moment by a more informative, equivalent characterisation (Lemma 2.6 below).
We simply started with (2.7) for its direct connection with several analogous
operators exploited in past analyses from the above-mentioned literature.

######  Lemma 2.3.

$S_{\lambda}$ is a self-adjoint extension of $S$.

###### Proof.

By construction, $S_{\lambda}$ is a densely defined extension of $S$. In fact,
$S_{\lambda}$ is self-adjoint; for,

(2.9)
$S_{\lambda}\,=\,S^{*}\upharpoonright\mathcal{D}_{\lambda}\,=\,(S^{*}-\lambda\mathbbm{1})\upharpoonright\mathcal{D}_{\lambda}+\lambda\mathbbm{1}$

and $(S^{*}-\lambda\mathbbm{1})\upharpoonright\mathcal{D}_{\lambda}$ is a
self-adjoint extension of $S-\lambda\mathbbm{1}$. The latter claim is checked
by means of Theorem 2.1 applied to $S-\lambda\mathbbm{1}$, choosing now the
labelling operator $T$ to be the zero operator on the whole
$\ker(S^{*}-\lambda\mathbbm{1})$. ∎

######  Remark 2.4.

When in particular the operator $S$ from (2.3) satisfies $\mathfrak{m}(S)>0$,
the above construction for $S_{\lambda}$ requires $\lambda<\mathfrak{m}(S)$.
In this case, $\mathfrak{m}(S-\lambda\mathbbm{1})=\mathfrak{m}(S)-\lambda>0$,
and the distinguished extension in (2.8) can be taken to be the _Friedrichs
extension_ $(S-\lambda\mathbbm{1})_{\mathrm{F}}$ of $S-\lambda\mathbbm{1}$.
With this choice,
$(S^{*}-\lambda\mathbbm{1})\upharpoonright\mathcal{D}_{\lambda}$ is nothing
but the _Kreĭn-von Neumann extension_ $(S-\lambda\mathbbm{1})_{\mathrm{N}}$ of
$S-\lambda\mathbbm{1}$ (see, e.g., [32, Theorem 2.10]), namely its unique
smallest positive self-adjoint extension. Observe that, whereas
$(S-\lambda\mathbbm{1})_{\mathrm{F}}=S_{\mathrm{F}}-\lambda\mathbbm{1}$, in
general $(S-\lambda\mathbbm{1})_{\mathrm{N}}\neq
S_{\mathrm{N}}-\lambda\mathbbm{1}$.

Based on the latter remark, even when going back to the general assumptions
(2.3)-(2.4) it is informative to refer to $S_{\lambda}$ as the sum of
$\lambda\mathbbm{1}$ plus the ‘extension of Kreĭn-von Neumann type’ of
$S-\lambda\mathbbm{1}$ – see (2.9) above.

######  Lemma 2.5.

Under the conditions (2.3)-(2.4) the gapped operator $S$ admits a self-adjoint
extension with eigenvalue given by an arbitrary $\lambda$ in the gap.

###### Proof.

For any such $\lambda$, one such operator is precisely $S_{\lambda}$. It is
indeed a self-adjoint extension of $S$ (Lemma 2.3) and moreover (2.9) shows
that on any non-zero
$u\in\ker(S^{*}-\lambda\mathbbm{1})\subset\mathcal{D}(S_{\lambda})$ (such $u$
certainly existing, owing to (2.4)) one has $S_{\lambda}u=\lambda u$. ∎

For arbitrary $\lambda$ in the spectral gap of $S$, $S_{\lambda}$ is a self-
adjoint extension of $S$ with eigenvalue $\lambda$. As such, it _must_
correspond to an extension parameter $T$ in the sense of the general
classification of Theorem 2.1. Since $S$ has by assumption unital deficiency
index, $T$ is in this case the self-adjoint operator acting in the one-
dimensional space $\ker(S^{*}-\lambda\mathbbm{1})$ as the multiplication by a
real number $\beta$. Thus, in the notation therein, $S_{\lambda}$ must have
the form $S_{T}$ for some $T$ of that sort, for which it is convenient to use
the ad hoc notation $S^{[\beta]}$ (meaning: the extension $S_{T}$ where $T$ is
the multiplication by $\beta$). Explicitly, specialising Theorem 2.1 for
unital deficiency index,

(2.10) $\mathcal{D}(S^{[\beta]})\;=\;\;\left\\{f+S_{\mathrm{D}}^{-1}(\beta
u)+u\left|\begin{array}[]{c}f\in\mathcal{D}(\overline{S})\,,\\\ u\in\ker
S^{*}\end{array}\\!\\!\right.\right\\}$

The choice $\beta=\infty$ is tacitly included, and corresponds to the
Friedrichs extension $S_{\mathrm{F}}$ of $S$.

######  Proposition 2.6.

$S_{\lambda}=S^{[\beta]}$ for

(2.11)
$\beta\,=\,\lambda\,\big{\langle}w_{0},S_{\mathrm{D}}(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}w_{0}\big{\rangle}\,,$

where $w_{0}$ is a unit vector (i.e., $\|w_{0}\|=1$) spanning $\ker S^{*}$
(formula (2.11) is clearly insensitive of the phase of $w_{0}$).

###### Proof.

As argued above, for given $S$ and $\lambda$ (and, a priori, for given
$S_{\mathrm{D}}$ and $(S-\lambda\mathbbm{1})_{\mathrm{D}}$ as well), there
exists one and only one $\beta\in\mathbb{R}\cup\\{\infty\\}$, depending on
such data, for which $S_{\lambda}=S^{[\beta]}$, or, equivalently,
$\mathcal{D}_{\lambda}=\mathcal{D}(S^{[\beta]})$. This means, in view of
(2.10), that for arbitrary $\phi\in\mathcal{D}(\overline{S})$ and
$v\in\ker(S^{*}-\lambda\mathbbm{1})$, the element
$\phi+v\in\mathcal{D}_{\lambda}$ can be re-written as

$\phi+v\;=\;f+S_{\mathrm{D}}^{-1}(\beta u)+u$

for uniquely determined $f\equiv
f_{\phi,v,\lambda}\in\mathcal{D}(\overline{S})$ and $u\equiv
u_{\phi,v,\lambda}\in\ker S^{*}$.

Now, decompose

$v\,=\,\overline{S}x+z$

with $x\in\mathcal{D}(\overline{S})$ and $z\in\ker S^{*}$, uniquely determined
via the orthogonal direct sum
$\mathcal{H}=\,\mathrm{ran}\,\overline{S}\oplus\ker S^{*}$ (recall that under
the current assumptions,
$\overline{\mathrm{ran}\,S}=\mathrm{ran}\,\overline{S}$).

Applying $S^{*}$ on both sides of (2) yields
$\overline{S}\phi+\lambda\overline{S}x+\lambda z=\overline{S}f+\beta u$,
whence

$\overline{S}\phi+\lambda\overline{S}x-\overline{S}f\,=\,\beta u-\lambda z\,.$

As the l.h.s. above belongs to $\mathrm{ran}\,{\overline{S}}$ and the r.h.s.
belongs to $\ker S^{*}$, necessarily both sides vanish and

$f\,=\,\phi+\lambda x\,,\qquad\beta u\,=\,\lambda z\,.$

By means of (2) above, (2) is re-written as

$\begin{split}\phi+v\,&=\,\phi+\lambda x+S_{\mathrm{D}}^{-1}(\lambda z)+u\\\
&=\,\phi+\lambda S_{\mathrm{D}}^{-1}(\overline{S}x+z)+u\\\ &=\,\phi+\lambda
S_{\mathrm{D}}^{-1}v+u\,,\end{split}$

yielding

$u\,=\,(\mathbbm{1}-\lambda
S_{\mathrm{D}}^{-1})v\,=\,(S_{\mathrm{D}}-\lambda\mathbbm{1})S_{\mathrm{D}}^{-1}v\,.$

Formulas (2) and (2) provide the expression for
$f\in\mathcal{D}(\overline{S})$ and $u\in\ker S^{*}$, for given
$\phi+v\in\mathcal{D}_{\lambda}$, satisfying (2). In order to find the
dependence of $\beta$ on $\lambda$ it is convenient to combine (2) with the
second of (2): this gives

$\lambda z\,=\,\beta
u\,=\,\beta(S_{\mathrm{D}}-\lambda\mathbbm{1})S_{\mathrm{D}}^{-1}v\,,$

whence (using also (2))

$\lambda(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}z\,=\,\beta
S_{\mathrm{D}}^{-1}v\,=\,\beta S_{\mathrm{D}}^{-1}(\overline{S}x+z)\,.$

Recall, indeed, that since $\lambda$ belongs to the gap of $S$ then
$S_{\mathrm{D}}-\lambda\mathbbm{1}$ has everywhere defined and bounded inverse
in $\mathcal{H}$. Applying
$S^{*}=(S^{*}-\lambda\mathbbm{1})+\lambda\mathbbm{1}$ to the latter identity
yields

$\lambda
z+\lambda^{2}(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}z\,=\,\beta\overline{S}x+\beta
z\,.$

Taking the scalar product of each side with $z$ itself gives

$(\lambda-\beta)\|z\|^{2}+\lambda^{2}\big{\langle}z,(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}z\big{\rangle}\,=\,0\,,$

as $z\perp\overline{S}x$, whence (in the non-trivial case $z\neq 0$)

$\beta\,=\,\lambda+\lambda^{2}\,\frac{\,\big{\langle}z,(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}z\big{\rangle}\,}{\|z\|^{2}}\,=\,\lambda\,\frac{\,\big{\langle}z,S_{\mathrm{D}}(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}z\big{\rangle}\,}{\|z\|^{2}}\,.$

Clearly, (2) above is the desired formula (2.11). ∎

######  Remark 2.7.

  1. (1)

$S_{\lambda}$ is non-invertible when $\lambda=0$, as it admits the eigenvalue
zero. Through the identification $S_{\lambda}=S^{[\lambda]}$ (Proposition 2.6)
this is consistent with the fact (Theorem 2.2) that $S^{[\beta]}$ loses its
invertibility precisely when $\beta=0$.

  2. (2)

In Proposition 2.6’s proof above, the implication
$\lambda=0\Rightarrow\beta=0$ is already deduced from conditions (2)-(2).
Indeed, if $\lambda=0$, then $\beta u=\lambda z=0$ irrespective of the choice
of $\phi+v\in\mathcal{D}_{\lambda}$; the option $\beta\neq 0$ would imply that
all the $u\equiv u_{\phi,v,\lambda}$’s must vanish, whereby
$\phi+v=f+S_{\mathrm{D}}^{-1}(\beta u)+u=f\in\mathcal{D}(\overline{S})$, i.e.,
$\mathcal{D}_{\lambda}\subset\mathcal{D}(\overline{S})$, a contradiction.

######  Example 2.8.

It is instructive to visualise the main identities of Proposition 2.6’s proof
with a concrete example. In $\mathcal{H}:=L^{2}(\mathbb{R}^{+})$ consider

$\mathcal{D}(S)\,:=\,C^{\infty}_{c}(\mathbb{R}^{+})\,,\qquad
Sf\,:=\,-f^{\prime\prime}+f\,.$

Such $S$ is a densely defined and lower semi-bounded symmetric operator, with
$\mathfrak{m}(S)=1$, thus with gap $(-\infty,1)$. $S$ then admits the
Friedrichs extension, $S_{\mathrm{F}}$, which is the distinguished extension
$S_{\mathrm{D}}$ for this example. It is standard to argue (see, e.g., [33,
Section 7.1]) that $S$ has unital deficiency index and

$\begin{split}\mathcal{D}(\overline{S})\,&=\,H^{2}_{0}(\mathbb{R}^{+})\,,\\\
\mathcal{D}(S_{\mathrm{F}})\,&=\,H^{2}(\mathbb{R}^{+})\cap
H^{1}_{0}(\mathbb{R}^{+})\,,\\\
\mathcal{D}(S^{*})\,&=\,H^{2}(\mathbb{R}^{+})\,,\\\ \ker
S^{*}\,&=\,\mathrm{span}\\{e^{-t}\\}\,,\end{split}$

all operators $\overline{S}$, $S_{\mathrm{F}}$, $S^{*}$ acting as
$f\mapsto-f^{\prime\prime}+f$. Thus, the boundary condition for any
$f\in\mathcal{D}(\overline{S})$ is $f(0)=f^{\prime}(0)=0$, and the boundary
condition for any $f\in\mathcal{D}(S_{\mathrm{F}})$ is $f(0)=0$. For $\lambda$
in the gap of $S$, namely $\lambda<1$,

$\ker(S^{*}-\lambda\mathbbm{1})\,=\,\mathrm{span}\big{\\{}e^{-t\sqrt{1-\lambda}}\big{\\}}\,.$

Consider $v=p\,e^{-t\sqrt{1-\lambda}}\in\ker(S^{*}-\lambda\mathbbm{1})$ for
arbitrary $p\in\mathbb{C}$. In the decomposition

$v\,=\,\overline{S}x+z\,,\qquad x\in\mathcal{D}(\overline{S})\,,\;z\in\ker
S^{*}\,,$

necessarily $z=q\,e^{-t}$ for some $q\in\mathbb{C}$, and the identity $\langle
z,v\rangle_{L^{2}}=\|z\|^{2}_{L^{2}}$ yields

$q\,=\,p\,\frac{2}{\,\sqrt{1-\lambda}+1\,}\,.$

Let us now identify the element
$u=(S_{\mathrm{F}}-\lambda\mathbbm{1})S_{\mathrm{F}}^{-1}v\in\ker S^{*}$. With
$w:=S_{\mathrm{F}}^{-1}v$, one writes
$u=(S_{\mathrm{F}}-\lambda\mathbbm{1})w$. Here $w$ is determined by the ODE
$v=S^{*}w=-w^{\prime\prime}+w$ with boundary condition $w(0)=0$: easy
computations show that the only $L^{2}$-solution to this problem is
$w=-\frac{p}{\lambda}(e^{-t\sqrt{1-\lambda}}-e^{-t})$. In turn,

$u\,=\,(S_{\mathrm{F}}-\lambda\mathbbm{1})w\,=\,-w^{\prime\prime}+(1-\lambda)w\,=\,p\,e^{-t}\,.$

Since one must have $\beta u=\lambda z$, the above findings for $u$ and $z$
yield

$\beta\,p\,e^{-t}\,=\,\lambda\,q\,e^{-t}\,,$

whence finally

$\beta\,=\,\lambda\,\frac{q}{p}\,=\,\frac{2\lambda}{\,\sqrt{1-\lambda}+1\,}\,.$

This is the actual dependence $\beta\equiv\beta(\lambda)$. Last, let us show
that such expression is precisely the one given by (2.11). It is non-
restrictive to assume $\lambda\neq 0$ henceforth, as the implication
$\lambda=0\Rightarrow\beta=0$ is already guaranteed by general arguments
(Remark 2.7(ii)). One has

$\begin{split}\|z\|_{L^{2}}^{2}\,&=\,\frac{\>|q|^{2}}{2}\,,\\\
\big{\langle}z,S_{\mathrm{F}}(S_{\mathrm{F}}-\lambda\mathbbm{1})^{-1}z\big{\rangle}_{L^{2}}\,&=\,|q|^{2}\langle
e^{-t},S_{\mathrm{F}}(S_{\mathrm{F}}-\lambda\mathbbm{1})^{-1}e^{-t}\rangle_{L^{2}}\,.\end{split}$

The element
$h:=(S_{\mathrm{F}}-\lambda\mathbbm{1})^{-1}e^{-t}\in\mathcal{D}(S_{\mathrm{F}})$
is the $L^{2}$-solution to the problem
$-h^{\prime\prime}+(1-\lambda)h=e^{-t}$, $h(0)=0$, which is easily found to be
$h=\frac{1}{\lambda}(e^{-t\sqrt{1-\lambda}}-e^{-t})$. Then,
$S_{\mathrm{F}}h=e^{-t\sqrt{1-\lambda}}$, and

$\begin{split}\beta\,&=\,\lambda\,\frac{\,\big{\langle}z,S_{\mathrm{F}}(S_{\mathrm{F}}-\lambda\mathbbm{1})^{-1}z\big{\rangle}_{L^{2}}\,}{\;\|z\|_{L^{2}}^{2}}\\\
&=\,2\lambda\,\langle
e^{-t},S_{\mathrm{F}}h\rangle_{L^{2}}\,=\,2\lambda\,\langle
e^{-t},e^{-t\sqrt{1-\lambda}}\rangle_{L^{2}}\,=\,\frac{2\lambda}{\,\sqrt{1-\lambda}+1\,}\,,\end{split}$

as found before. It is worth observing that $\lambda\mapsto\beta(\lambda)$ is
strictly monotone increasing, with
$\beta(\lambda)\xrightarrow{\;\lambda\to-\infty\;}-\infty$ and
$\beta(\lambda)\xrightarrow{\;\lambda\to 1^{-}\;}+\infty$. Thus, the
collection of the $S_{\lambda}$’s exhaust the whole family of self-adjoint
extensions of $S$, as $\lambda$ runs in the gap of $S$.

The above line of reasoning places under the scope of the general
classification scheme of Theorem 2.1 an operator, the above $S_{\lambda}$,
whose construction already appeared, with ad hoc or implicit arguments, in
[20, 1, 17, 3].

This is instructive per se, and also paves the way for the analysis of the
actual case of interest, a gapped operator $S$ with _infinite_ deficiency
index, which is the object of the next Section.

In fact, there is already one distinguished case with infinite deficiency
index, crucially present in many applications (a recent topical example is
[34, 31]), which can be treated by exploiting the preparatory arguments
presented so far: symmetric operators defined by an infinite orthogonal sum of
gapped operators with unital deficiency index.

######  Theorem 2.9.

Let $\mathcal{H}=\bigoplus_{n\in\mathbb{N}}\mathcal{H}_{n}$ be a (countably
infinite) orthogonal direct sum of Hilbert spaces, and let
$S=\bigoplus_{n\in\mathbb{N}}S_{n}$ be the operator acting in $\mathcal{H}$
given by the orthogonal sum of operators $S_{n}$’s, each of which is a densely
defined, gapped, symmetric operator acting in the corresponding Hilbert space
$\mathcal{H}_{n}$ and having unital deficiency index. Assume furthermore that
all gaps have a common intersection $(a,b)\subset\mathbb{R}$ which, without
loss of generality, is assumed to contain $0$. Then:

  1. (i)

$S$ is a densely defined, gapped, symmetric operator with infinite deficiency
index and admitting self-adjoint extensions;

  2. (ii)

for any finite or countably infinite collection $(\lambda_{k})_{k}$ in $(a,b)$
there is a self-adjoint extension of $S$ having the $\lambda_{k}$’s as
eigenvalues; if $\lambda$ is any of such values, then its multiplicity as an
eigenvalue equals the number of the $\lambda_{k}$’s being equal to $\lambda$;

  3. (iii)

for any arbitrary closed subset $K\subset[a,b]$ there is a self-adjoint
extension of $S$ whose essential spectrum contains $K$.

Here $[a,b]$ denotes the closure of $(a,b)$ also in the case when $(a,b)$ is
semi-infinite.

###### Proof of Theorem 2.9.

Part (i) is obvious, and part (iii) is a direct consequence of part (ii), by
choosing the collection $(\lambda_{k})_{k}\subset K\cap(a,b)$ to be a dense of
$K$.

Concerning part (ii), consider non-restrictively the _countable_ collection
$(\lambda_{n})_{n\in\mathbb{N}}$ in $(a,b)$: then, one example of the claimed
self-adjoint extension is

$\widetilde{S}\,=\,\bigoplus_{n\in\mathbb{N}}\,S_{n,\lambda_{n}}\,,$

where each $S_{n,\lambda_{n}}$ is the self-adjoint extension of $S_{n}$ in
$\mathcal{H}_{n}$ defined by

$\mathcal{D}(S_{n,\lambda_{n}})\,:=\,\mathcal{D}(\overline{S_{n}})\dotplus\ker(S_{n}^{*}-\lambda\mathbbm{1})\,,\qquad
S_{n,\lambda_{n}}\,:=\,S_{n}^{*}\upharpoonright\mathcal{D}(S_{n,\lambda_{n}})\,.$

In other words, $S_{n,\lambda_{n}}$ is for $S_{n}$ the analogue of the
operator $S_{\lambda}$ defined in (2.7) with respect to $S$ therein. The fact
that $S_{n,\lambda_{n}}$ extends self-adjointly $S_{n}$ in $\mathcal{H}_{n}$
follows from Lemma 2.3. Thus, $\widetilde{S}$ is indeed a self-adjoint
extension of $S$ in $\mathcal{H}$. Furthermore, as seen in the proof of Lemma
2.5, given a (certainly existing)
$v_{n}\in\ker(S_{n}^{*}-\lambda_{n}\mathbbm{1})\setminus\\{0\\}\subset\mathcal{D}(S_{n,\lambda_{n}})\subset\mathcal{D}(S^{*})$,
one has $\widetilde{S}v_{n}=S_{n,\lambda_{n}}v_{n}=\lambda_{n}v_{n}$. This
shows that all the $\lambda_{n}$’s are eigenvalues of $\widetilde{S}$. The
conclusion on the multiplicity of such eigenvalues is then straightforward. ∎

For completeness of reference, infinite orthogonal sums of operators with
common gap, each with strictly positive deficiency index, were already
analysed in [1, Theorems 6.1 and 6.2] and [23] in the context of self-adjoint
extensions producing prescribed spectrum in the gap. This was turn based on
the the operator-theoretic toolbox devised in [20, Theorem 2.2] and [1, Lemma
2.1 and Proposition 3.1], which is _not_ explicitly modelled on the extension
scheme revisited here in Theorem 2.1.

##  3\. General case of gapped operators with infinite deficiency index

Let us finally consider in this Section the general and actual case of
interest, namely the creation by self-adjoint extension of arbitrary (point
and) essential spectrum within the gap of a densely defined symmetric operator
with infinite deficiency index.

This means that from now on we consider an _infinite-dimensional_ Hilbert
space $\mathcal{H}$ and, acting in it, a densely defined symmetric operator
$S$ satisfying the general gap condition (2.3) supplemented with the
additional assumption

(3.1) $d(S)\,=\,\dim\ker S^{*}\,=\,\infty$

that replaces (2.4) above. As is evident by the arguments that follow, the
case of non-zero finite deficiency index is trivially covered as well.

As emphasised in Section 1, the existence of self-adjoint extensions of $S$
with arbitrary point spectrum in the gap is an already established result: the
virtue of the new discussion presented now is to re-obtain and re-understand
it by _identifying_ the actual operator-theoretic _mechanism_ of emergence of
new essential spectrum _within the general self-adjoint extension scheme of
Theorem 2.1_.

It is instructive for comparison purposes to briefly revisit the classical
line of reasoning. It originates from ideas and stimuli by Albeverio, leading
to the first complete result [21, Theorem 2.2] in the infinite deficiency
index case by Brasche, Neidhardt, and Weidmann, and later refinements and
reformulations by Albeverio, Brasche, and Neidhardt [1, Proposition 3.1], by
Brasche [17, Corollary 30], and by Albeverio, Konstantinov, and Koshmanenko
[6, Theorem 3.5].

The idea is:

  1. (i)

thanks to the infinity of the deficiency index of $S$, one shows that it is
possible to iteratively pick countably many orthonormal eigenvectors
$v_{1},v_{2},v_{3},\dots$ of $S^{*}$ with eigenvalues, respectively,
$\lambda_{1},\lambda_{2},\lambda_{3},\dots$ arbitrarily chosen within the gap
of $S$, and correspondingly to define

(3.2)
$\begin{split}\mathcal{D}^{\prime}\,&:=\,\mathcal{D}(\overline{S})\dotplus\bigoplus_{n}\mathrm{span}\\{v_{n}\\}\,,\\\
S^{\prime}\,&:=\,S^{*}\upharpoonright\mathcal{D}^{\prime}\,;\end{split}$

  2. (ii)

by construction, $S^{\prime}$ is a closed symmetric extension of $S$;
furthermore, $\bigoplus_{n}\mathrm{span}\\{v_{n}\\}$ is a reducing subspace
for $S^{\prime}$ and therefore, with respect to the Hilbert space orthogonal
direct sum

(3.3)
$\mathcal{H}\,=\,\mathrm{span}\\{v_{1},v_{2},v_{3},\dots\\}^{\perp}\>\oplus\>\Big{(}\bigoplus_{n}\mathrm{span}\\{v_{n}\\}\Big{)}\,,$

one may re-write

(3.4)
$S^{\prime}\,=\,\widehat{S}\oplus\Big{(}\bigoplus_{n}S_{n}\Big{)}\,,\quad
S_{n}\,:=\,S^{*}\upharpoonright\mathrm{span}\\{v_{n}\\}\,,$

where $\widehat{S}$ is the densely defined and closed symmetric operator in
the Hilbert subspace $\mathrm{span}\\{v_{1},v_{2},v_{3},\dots\\}^{\perp}$
defined by

(3.5)
$\begin{split}\mathcal{D}(\widehat{S})\,&:=\,\mathcal{D}(S^{\prime})\cap\mathrm{span}\\{v_{1},v_{2},v_{3},\dots\\}^{\perp}\,=\,\mathcal{D}(\overline{S})\cap\mathrm{span}\\{v_{1},v_{2},v_{3},\dots\\}^{\perp}\,,\\\
\widehat{S}&:=\,S^{*}\upharpoonright\mathcal{D}(\widehat{S})\,;\end{split}$

  3. (iii)

last, one can prove that $\widehat{S}$ is gapped, with at least the same gap
as $S$, hence it admits a distinguished self-adjoint extension
$\widehat{S}_{\mathrm{D}}$ in
$\mathrm{span}\\{v_{1},v_{2},v_{3},\dots\\}^{\perp}$, with everywhere defined
and bounded inverse, and with spectral gap given by the gap of $\widehat{S}$,
thus a spectral gap containing all the $\lambda_{n}$’s; since each $S_{n}$ is
obviously self-adjoint in $\mathrm{span}\\{v_{n}\\}$, the operator

(3.6) $\widehat{S}_{\mathrm{D}}\oplus\Big{(}\bigoplus_{n}S_{n}\Big{)}$

is self-adjoint in $\mathcal{H}$, it extends $S$, and it admits all the
$\lambda_{n}$’s as eigenvalues.

(We refer to the above-mentioned literature for the somewhat laborious, and
occasionally also subtle, justifications of the above scheme.)

In the classical argument outlined above one exploits the possibility of
expressing the closed symmetric extension $S^{\prime}$ of $S$ in the reduced
form (3.4): the self-adjoint extension then only takes place in the Hilbert
subspace $\mathrm{span}\\{v_{1},v_{2},v_{3},\dots\\}^{\perp}$, and there the
distinguished extension is taken. This, however, obfuscates how the resulting
self-adjoint extension (3.6) of $S$ is classified as an extension $S_{T}$
within the standard extension scheme of Theorem 2.1. The reason for that is
further elucidated in the following Remark.

######  Remark 3.1.

Observe the difference, for concreteness in the prototypical case of unit
deficiency index – so, for the present Remark, let us return to the
assumptions (2.3)-(2.4). Following the scheme above, for arbitrary $\lambda$
in the gap of $S$ and for arbitrary eigenvector $v$ of $S^{*}$ with eigenvalue
$\lambda$ one considers the extension $S^{\prime}$ of $S$ defined by (3.2) as
the restriction of $S^{*}$ to

(3.7) $\mathcal{D}(\overline{S})\dotplus\ker(S^{*}-\lambda\mathbbm{1})\,,$

where $\ker(S^{*}-\lambda\mathbbm{1})=\mathrm{span}\\{v\\}$. This is precisely
the operator $S_{\lambda}$ of Section 2. There, the way to recognise
$S_{\lambda}$ as a member $S_{T}$ (in fact, $S^{[\beta]}$, in the notation
therein) of the family of self-adjoint extensions of $S$ was to operate on the
_second_ summand of the direct, non-orthogonal sum (3.7), decomposing
$v=\overline{S}x+z$ along $\mathrm{ran}\overline{S}$ and $\ker S^{*}$
respectively: this eventually led to the explicit identification of the Birman
operator $T$ acting on $\ker S^{*}$. In the scheme (3.2)-(3.6), instead, one
operates on the _first_ summand of (3.7), projecting
$\mathcal{D}(\overline{S})$ onto $\ker(S^{*}-\lambda\mathbbm{1})^{\perp}$, as
(3.5) clearly shows. This is not natural, though, if one aims at identifying
the Birman parameter $T$ that labels the considered self-adjoint extension.

The discussion laid down so far finally leads us to the announced
identification of the self-adjoint extension mechanism, within the extension
scheme of Theorem 2.1, for the construction of self-adjoint extensions of
gapped operators with arbitrary point spectrum in the gap (Theorem 3.4 below).

Observe that no separability restriction will be needed here on the Hilbert
space $\mathcal{H}$, unlike the above mentioned analyses [21, 1, 22, 17, 41,
2, 6, 42, 3, 13]. It will be not required either that the distinguished self-
adjoint extension $S_{\mathrm{D}}$ of $S$ referred to in assumption (2.3) has
compact resolvent, as instead crucially needed in [13].

The preparation is simple: under assumptions (2.3) and (3.1), let
$(\lambda_{n})_{n\in\mathcal{N}}$ be an arbitrary finite or countably infinite
collection of (possibly repeated) points inside the gap of $S$, and let
$(v_{n})_{n\in\mathcal{N}}$ be an orthonormal system in $\mathcal{D}(S^{*})$
with

(3.8) $S^{*}v_{n}\,=\,\lambda_{n}v_{n}\,,\qquad n\in\mathcal{N}\,.$

As mentioned already (see also [21, Theorem 2.2] or [1, Proposition 3.1]),
this choice is always possible by induction, owing to the fact that
$\dim\ker(S^{*}-\lambda_{n}\mathbbm{1})=\infty$. Then, in analogy to the
unital deficiency index case, set

(3.9)
$u_{n}\,:=\,(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})S_{\mathrm{D}}^{-1}v_{n}\,,\qquad
n\in\mathcal{N}\,,$

where $S_{\mathrm{D}}$ is the distinguished self-adjoint extension of $S$ with
everywhere defined and bounded inverse in $\mathcal{H}$. Clearly, the
$u_{n}$’s are all non-zero, as

(3.10)
$v_{n}\,=\,S_{\mathrm{D}}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})^{-1}u_{n}\,;$

indeed, since $\lambda_{n}$ belongs to the gap of $S$,
$S_{\mathrm{D}}-\lambda_{n}\mathbbm{1}$ has everywhere defined and bounded
inverse in $\mathcal{H}$.

######  Lemma 3.2.

$(u_{n})_{n\in\mathcal{N}}$ is a collection of linearly independent non-zero
vectors in $\ker S^{*}$. The linear map induced by the correspondence
$v_{n}\mapsto u_{n}$ is a bijection between the Hilbert subspaces

(3.11)
$V\,:=\,\bigoplus_{n\in\mathcal{N}}\mathrm{span}\\{v_{n}\\}\qquad\textrm{and}\qquad
U\,:=\,\overline{\,\mathrm{span}\\{u_{n}\,|\,n\in\mathcal{N}\\}}\,.$

###### Proof.

From (3.8) and (3.9) one has

$\begin{split}S^{*}u_{n}\,=\,S^{*}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})S_{\mathrm{D}}^{-1}v_{n}\,&=\,S^{*}(v_{n}-\lambda_{n}S_{\mathrm{D}}^{-1}v_{n})\\\
&=\,S^{*}v_{n}-\lambda_{n}v_{n}\,=\,0\,,\end{split}$

that is, $u_{n}\in\ker S^{*}$. As for the linear independence of the
$u_{n}$’s, observe first that for any selection of complex coefficients
$(c_{n})_{n\in\mathcal{N}}$ for which the following series converge as
elements of $\mathcal{D}(S^{*})$, one has

$\begin{split}\sum_{n\in\mathcal{N}}c_{n}u_{n}\,&=\,\sum_{n\in\mathcal{N}}c_{n}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})S_{\mathrm{D}}^{-1}v_{n}\\\
&=\,\sum_{n\in\mathcal{N}}c_{n}v_{n}-\sum_{n\in\mathcal{N}}c_{n}\lambda_{n}S_{\mathrm{D}}^{-1}v_{n}\,.\end{split}$

If $\sum_{n\in\mathcal{N}}c_{n}u_{n}=0$, then

$\sum_{n\in\mathcal{N}}c_{n}v_{n}\,=\,S_{\mathrm{D}}^{-1}\Big{(}\sum_{n\in\mathcal{N}}c_{n}\lambda_{n}v_{n}\Big{)}\,.$

The latter is an identity in
$\mathcal{D}(S_{\mathrm{D}})\cap\big{(}\bigoplus_{n\in\mathcal{N}}\mathrm{span}\\{v_{n}\\}\big{)}$.
In view of the orthonormality of the $v_{n}$’s one can re-express such
subspace as

$\begin{split}\mathcal{D}(S_{\mathrm{D}})\cap\Big{(}\bigoplus_{n\in\mathcal{N}}\mathrm{span}\\{v_{n}\\}\Big{)}\,&=\,\bigoplus_{n\in\mathcal{N}}\big{(}\mathcal{D}(S_{\mathrm{D}})\cap\mathrm{span}\\{v_{n}\\}\big{)}\\\
&\subset\,\bigoplus_{n\in\mathcal{N}}\big{(}\mathcal{D}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})\cap\ker(S^{*}-\lambda_{n}\mathbbm{1})\big{)}\,.\end{split}$

However,
$\mathcal{D}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})\cap\ker(S^{*}-\lambda_{n}\mathbbm{1})=\\{0\\}$
for each $n$, owing to the direct decomposition
$\mathcal{D}(S^{*})=\mathcal{D}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})\dotplus\ker(S^{*}-\lambda_{n}\mathbbm{1})$
(see (2.2) above). Therefore, each side in (3) must vanish. In particular,
$\sum_{n\in\mathcal{N}}c_{n}v_{n}=0$. As the $v_{n}$’s constitute an
orthonormal system, necessarily $c_{n}=0$ $\forall n\in\mathcal{N}$. This
establishes the linear independence of the $u_{n}$’s. The last part of the
thesis is now obvious, as (3.9) and (3.10) invert each other. ∎

Next, let us complete the general preparation by defining $\widetilde{T}$ as
the _self-adjoint_ operator in the Hilbert subspace $V$ obtained by linear
extension and operator closure from the matrix elements

(3.11) $\langle v_{m},\widetilde{T}v_{n}\rangle\,:=\,\lambda_{n}\langle
v_{m},v_{n}\rangle-\lambda_{m}\lambda_{n}\langle
v_{m},S_{\mathrm{D}}^{-1}v_{n}\rangle$

with respect to the orthonormal basis $(v_{n})_{n\in\mathcal{N}}$ of $V$, and
by defining $T$ as the corresponding self-adjoint operator in the Hilbert
subspace $U\subset\ker S^{*}$ via the bijection $V\cong U$ considered in Lemma
3.2. In practice,

$\widetilde{T}\,=\,\bigoplus_{n\in\mathcal{N}}\lambda_{n}P_{n}+{\sum_{n,m\in\mathcal{N}}}^{\\!\\!\\!\\!\oplus}\>\lambda_{m}\lambda_{n}\,P_{m}S_{\mathrm{D}}^{-1}P_{n}$

in the sense of operator direct sums, where the $P_{n}$’s are the orthogonal
projections, respectively, onto the $\mathrm{span}\\{v_{n}\\}$’s, and
correspondingly

$\langle u_{m},Tu_{n}\rangle\,=\,\langle v_{m},\widetilde{T}v_{m}\rangle\,,$

or, with any of the following equivalent expressions,

(3.12) $\begin{split}\langle u_{m},Tu_{n}\rangle\,&=\,\lambda_{n}\langle
v_{m},v_{n}\rangle-\lambda_{m}\lambda_{n}\langle
v_{m},S_{\mathrm{D}}^{-1}v_{n}\rangle\\\
&=\,\lambda_{n}\langle(S_{\mathrm{D}}-\lambda_{m}\mathbbm{1})S_{\mathrm{D}}^{-1}v_{m},v_{n}\rangle\\\
&=\,\lambda_{n}\langle
u_{m},S_{\mathrm{D}}(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})^{-1}u_{n}\rangle\\\
&=\,\lambda_{n}\langle u_{m},v_{n}\rangle\,.\end{split}$

######  Remark 3.3.

The present definition of $T$ reproduces formula (2.11) of the special case of
unital deficiency index (Proposition 2.6). Indeed, for that case (3.12) reads

$\langle u,Tu\rangle\,=\,\lambda\langle
u,S_{\mathrm{D}}(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}u\rangle$

for $\lambda$ in the gap of $S$ and
$u=(S_{\mathrm{D}}-\lambda\mathbbm{1})S_{\mathrm{D}}^{-1}v\in\ker S^{*}$ for
given _normalised_ $v\in\ker(S^{*}-\lambda\mathbbm{1})$; since $T$ is here the
multiplication by $\beta$, one then has

$\beta\,=\,\frac{\langle
u,Tu\rangle}{\>\|u\|^{2}}\,=\,\lambda\,\frac{\,\langle
u,S_{\mathrm{D}}(S_{\mathrm{D}}-\lambda\mathbbm{1})^{-1}u\rangle\,}{\>\|u\|^{2}}\,,$

consistently with (2.11).

######  Theorem 3.4.

Under the assumptions (2.3) and (3.1), and for an arbitrary finite or
countably infinite collection $(\lambda_{n})_{n\in\mathcal{N}}$ of points in
the spectral gap of $S$, the self-adjoint extension $S_{T}$ of $S$ labelled,
according to the general extension scheme (2.1), by the Birman parameter $T$
defined above has all the $\lambda_{n}$’s as eigenvalues. If $\lambda$ is any
of such values, then its multiplicity as an eigenvalue is no less than the
multiplicity of the $\lambda_{n}$’s being equal to $\lambda$.

###### Proof.

Let $(v_{n})_{n\in\mathcal{N}}$ be the orthonormal system selected in
$\mathcal{D}(S^{*})$ for the construction of $(u_{n})_{n\in\mathcal{N}}$, $U$,
$V$, and $T$ from (3.9), (3.11), and (3.12). Decompose

$v_{n}\,=\,\overline{S}x_{n}+z_{n}$

for $x_{n}\in\mathcal{D}(\overline{S})$ and $z_{n}\in\ker S^{*}$ uniquely
determined via the orthogonal direct sum
$\mathcal{H}=\,\mathrm{ran}\,\overline{S}\oplus\ker S^{*}$. Next, define

$\begin{split}f_{n}\,&:=\,\lambda_{n}x_{n}\,\in\,\mathcal{D}(\overline{S})\,,\\\
W\,&:=\,\ker S^{*}\ominus U\,,\quad\mathrm{i.e.,}\quad\ker S^{*}=U\oplus
W\,,\\\ w_{n}\,&:=\,\lambda_{n}P_{W}z_{n}\,\in\,W\,,\end{split}$

where $P_{W}$ denotes the orthogonal projection onto $W$. With these
definitions, consider the element
$f_{n}+S_{\mathrm{D}}^{-1}(Tu_{n}+w_{n})+u_{n}\in\mathcal{D}(S_{T})$, where
$S_{T}$ is the self-adjoint extension of $S$ parametrised by the present
Birman operator $T$, according to the general classification (2.1) of Theorem
2.1. One has

$\begin{split}f_{n}+&S_{\mathrm{D}}^{-1}(Tu_{n}+w_{n})+u_{n}\\\
&=\,\lambda_{n}x_{n}+S_{\mathrm{D}}^{-1}(Tu_{n}+w_{n})+(S_{\mathrm{D}}-\lambda_{n}\mathbbm{1})S_{\mathrm{D}}^{-1}v_{n}\\\
&=\,\lambda_{n}S_{\mathrm{D}}^{-1}(\overline{S}x_{n})+S_{\mathrm{D}}^{-1}(Tu_{n}+w_{n})+v_{n}-\lambda_{n}S_{\mathrm{D}}^{-1}v_{n}\\\
&=\,-\lambda_{n}S_{\mathrm{D}}^{-1}z_{n}+S_{\mathrm{D}}^{-1}(Tu_{n}+w_{n})+v_{n}\\\
&=\,S_{\mathrm{D}}^{-1}y_{n}+v_{n}\,,\end{split}$

where

$y_{n}\,:=\,Tu_{n}+w_{n}-\lambda_{n}z_{n}\,\in\,\ker S^{*}\,.$

By inspection, $y_{n}=0$. Indeed, along $U$ one has, for any
$m\in\mathcal{N}$,

$\begin{split}\langle u_{m},x_{n}\rangle\,&=\,\langle
u_{m},Tu_{n}\rangle-\lambda_{n}\langle u_{m},z_{n}\rangle\\\ &=\,\langle
u_{m},Tu_{n}\rangle-\lambda_{n}\langle u_{m},v_{n}\rangle\,=\,0\end{split}$

(having used that $u_{m}\perp w_{n}$ in the first identity,
$z_{n}=v_{n}-\overline{S}x_{n}$ and $u_{n}\perp\overline{S}x_{n}$ in the
second identity, and the last of (3.12) in the final step), whereas along $W$
one has

$P_{W}y_{n}\,=\,w_{n}-\lambda_{n}P_{W}z_{n}\,=\,0\,.$

Thus,

$v_{n}\,=\,f_{n}+S_{\mathrm{D}}^{-1}(Tu_{n}+w_{n})+u_{n}\,\in\,\mathcal{D}(S_{T})\,,$

and $S_{T}v_{n}=S^{*}v_{n}=\lambda_{n}v_{n}$, which proves that indeed the
self-adjoint extension $S_{T}$ admits all the $\lambda_{n}$’s as eigenvalues.
The statement on the multiplicity is then obvious, since the eigenvectors
$v_{n}$’s constitute an orthonormal system. ∎

######  Remark 3.5.

In the unital deficiency index case of Sect. 2 the present identification
$y_{n}=Tu_{n}+w_{n}-\lambda_{n}z_{n}$ reads $y=Tu-\lambda z=\beta u-\lambda z$
in the notation therein, since $\ker S^{*}=\mathrm{span}\\{u\\}=U$ and hence
$W=\ker S^{*}\ominus U=\\{0\\}$. As observed, $\beta u=\lambda z$, whence
$y=0$.

The above construction clearly covers also the case when the gapped operator
$S$ has _finite_ deficiency index and one produces a self-adjoint extension
with finitely many given eigenvalues within the gap. As mentioned, this
special case was the original result of Kreĭn [44, Theorem 23], but it is re-
obtained here in a considerably more economic way then Kreĭn’s ingenious but
involved proof. This cheapness of course relies on a highly sophisticated
toolbox, namely Theorem 2.1 providing the general classification of all self-
adjoint extensions of $S$, which was unavailable at the time of [44] and was
only completed by Višik [50] and Birman [16] in the following years.

For the same reason, Theorem 3.4 above provides a much more direct
construction of the desired self-adjoint extension also in the general case of
_infinite_ deficiency index, as compared to the classical reasoning outline at
the beginning of this Section.

######  Corollary 3.6.

Let $S$ be a densely defined, symmetric, gapped operator on an infinite-
dimensional Hilbert space $\mathcal{H}$, and assume that $S$ has infinite
deficiency index. For any arbitrary closed subset $K$ of the closure of the
gap of $S$ there is a self-adjoint extension of $S$ whose essential spectrum
contains $K$.

###### Proof.

It is clearly non-restrictive to assume the gap $(a,b)$ of $S$ to include $0$.
Conditions (2.3) and (3.1) are thus satisfied and it is possible to construct
the above self-adjoint extension $S_{T}$ of $S$ relative to a collection
$(\lambda_{n})_{n}$ of points that form a dense of $K\cap(a,b)$. Then $S_{T}$
admits all such $\lambda_{n}$’s as eigenvalues, implying that
$K\subset\sigma_{\mathrm{ess}}(S_{T})$. ∎

It is worth remarking, in view of the construction above, that the arbitrary
portion $K$ of new essential spectrum for $S_{T}$ inside the closure of the
gap $(a,b)$ can be of either type: points of $\sigma(S_{T})$ that are
accumulation points of actual eigenvalues, as well as isolated eigenvalues of
$S_{T}$ with infinite multiplicity.

As mentioned in the background discussion (Section 1), a very recent instance
of emergence, by self-adjoint extension, of one single isolated point of
essential spectrum within the gap is the three-dimensional Dirac operator with
critical shell interaction supported on the union of finitely many disjoint
spheres [15]. That is precisely the value $-m\mu/\varepsilon$ for given mass,
electrostatic shell, and Lorenz shell parameters, respectively, $m\geqslant 0$
and $\varepsilon,\mu\in\mathbb{R}$. In [15] such a value of essential spectrum
is identified by indirect means that do not elucidate its emergence within the
general self-adjoint extension scheme. It would be of valuable interest to
investigate whether the self-adjoint Dirac operator at criticality, with
singular perturbation on spherical shells, is indeed of the form $S_{T}$ of
Theorem 3.4.

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