11institutetext: Dale Frymark 22institutetext: Nuclear Physics Institute,
Department of Theoretical Physics, Czech Academy of Sciences, 25068 Řež, Czech
Republic.
22email<EMAIL_ADDRESS>33institutetext: Constanze Liaw 44institutetext:
University of Delaware, Department of Mathematical Sciences, 501 Ewing Hall,
Newark, DE 19716, USA, and
CASPER, Baylor University, One Bear Place #97328, Waco, TX 76798, USA.
44email<EMAIL_ADDRESS>
# Perspectives on General Left-Definite Theory††thanks: The authors would like
to congratulate Lance Littlejohn on the occasion of his 70th birthday, and
thank him for all of the help over the years. Lance provides the inspiration
and encouragement to dream big.

Dale Frymark and Constanze Liaw

###### Abstract

In 2002, Littlejohn and Wellman developed a celebrated general left-definite
theory for semi-bounded self-adjoint operators with many applications to
differential operators. The theory starts with a semi-bounded self-adjoint
operator and constructs a continuum of related Hilbert spaces and self-adjoint
operators that are intimately related with powers of the initial operator. The
development spurred a flurry of activity in the field that is still ongoing
today.
The main goal of this expository (with the exception of Proposition 1)
manuscript is to compare and contrast the complementary theories of general
left-definite theory, the Birman–Krein–Vishik (BKV) theory of self-adjoint
extensions and singular perturbation theory. In this way, we hope to encourage
interest in left-definite theory as well as point out directions of potential
growth where the fields are interconnected. We include several related open
questions to further these goals.

###### Acknowledgements.

Since August 2020, Liaw, C. has been serving as a Program Director in the
Division of Mathematical Sciences at the National Science Foundation (NSF),
USA, and as a component of this position, she received support from NSF for
research, which included work on this paper. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the NSF.

##  1 Introduction

The development of general left-definite theory by Littlejohn and Wellman had
vast repercussions for differential operators. It extended the important
concept of a left-definite Hilbert space to a continuum of associated Hilbert
spaces111All Hilbert spaces considered are assumed to be separable.. We appeal
to differential operators to further explain the motivation for this extension
from the classical left-definite space by following LW02  below. Note that
some knowledge of self-adjoint extension theory is assumed throughout the
manuscript, the reader is encouraged to consult the Appendix for some basic
definitions and notions. The references AG ; N  can be used for more details.
For some locally integrable positive weight function $w$, let

$\displaystyle\ell[y](x)={\lambda}w(x)y(x),$ (1)

where $\ell$ is Lagrangian symmetric differential expression of order $2n$
given by

$\displaystyle\ell^{n}[f](x)=\sum_{j=1}^{n}(-1)^{j}(a_{j}(x)f^{(j)}(x))^{(j)},\text{
}x\in(a,b),$ (2)

where $-\infty\leq a<b\leq\infty$. For simplicity, we assume each coefficient
$a_{j}(x)$ is smooth and positive on $(a,b)$. Due to the presence of $w(x)$ on
the right hand side of equation (1), $\mathcal{H}=L^{2}[(a,b),w(x)]$ is
referred to as the right-definite spectral setting for $w^{-1}\ell$. Two well-
known formulas play a role in defining the left-definite space.

For $f,g\in\mathcal{D}_{\scriptstyle\text{\rm max}}$, the maximal domain of
$\ell$, Green’s formula says that

$\displaystyle\int_{a}^{b}\ell[f](x)\overline{g}(x)dx=\int_{a}^{b}f(x)\overline{\ell[g]}(x)dx+[f,g](x)\big{|}_{a}^{b},$
(3)

where $[\,\cdot\,,\,\cdot\,]$ is the sesquilinear form for $\ell$. Dirichlet’s
formula is

$\displaystyle\int_{a}^{b}\ell[f](x)\overline{g(x)}dx=\sum_{j=0}^{n}\int_{a}^{b}a_{j}(x)$
$\displaystyle f^{(j)}(x)\overline{g}^{(j)}(x)dx$ (4)
$\displaystyle+\llbracket f,g\rrbracket(x)\big{|}_{a}^{b},$

where $\llbracket\,\cdot\,,\,\cdot\,\rrbracket$ is a bilinear form closely
related to the sesquilinear form $[\,\cdot\,,\,\cdot\,]$. Let ${\bf A}$ be a
self-adjoint operator acting via $\ell$ on a domain $\mathcal{D}({\bf A})$
such that for all $f,g\in\mathcal{D}({\bf A})$ the bilinear form $\llbracket
f,g\rrbracket(x)\big{|}_{a}^{b}=0$. In this case, equation (4) simplifies to

$\displaystyle\langle{\bf
A}f,g\rangle_{\mathcal{H}}=\int_{a}^{b}\ell[f](x)\overline{g(x)}dx=\sum_{j=0}^{n}\int_{a}^{b}a_{j}(x)$
$\displaystyle f^{(j)}(x)\overline{g}^{(j)}(x)dx.$ (5)

Such an operator ${\bf A}$ is then semi-bounded (see eq. (8)) thanks to the
positivity of the coefficient functions $a_{j}(x)$. The expression $\ell$ thus
generates an inner product defined by the right hand side of equation (5).
Explicitly, for $f,g\in\mathcal{D}({\bf A})$,

$\displaystyle\langle{\bf A}f,g\rangle_{\mathcal{H}}=\langle f,g\rangle_{1}.$
(6)

The closure of $\mathcal{D}({\bf A})$ in the topology generated by the norm
$\|\,\cdot\,\|_{1}=\langle\,\cdot\,,\,\cdot\,\rangle_{1}^{1/2}$ is then
denoted by $H_{1}$ and referred to as the left-definite setting for
$w^{-1}\ell$, with $H_{1}$ the (first) left-definite space. This is due to the
inner product $\langle\,\cdot\,,\,\cdot\,\rangle_{1}$ originating from the
left hand side of equation (1). The terminology itself can be traced back to
Weyl in 1910 W .

This first left-definite space presented above is identified by Littlejohn and
Wellman as $\mathcal{D}({\bf A}^{1/2})$, along with a continuum of other
associated left-definite spaces in Definition 1. Their new characterization
makes connections to other areas of self-adjoint operator theory more clear.
The core definitions and results of left-definite theory (we omit the term
‘general’ from now on) are included in Section 3. A key aspect of the theory
is that it includes stability of spectral type, see Theorem 3.1, which can be
viewed as an important advantage over the other theories discussed. Since
left-definite theory was described in 2002, it has been the focus of many
works. Here, we focus on techniques and results recently developed by the
authors in order to present several related conjectures and open questions. It
is the our hope that researchers may take inspiration from these problems to
make contributions to the field.

The Birman–Krein–Vishik (BKV) theory of semi-bounded forms, which puts closed
semi-bounded forms into a one-to-one correspondence with self-adjoint
operators, agrees with general left-definite theory. In many ways, this theory
could also be thought of as a generalization of equation (5): identifying
$\langle{\bf A}f,g\rangle_{\mathcal{H}}$ as an inner product in another
Hilbert space for general self-adjoint operators ${\bf A}$. BKV theory and its
connections to left-definite theory are the subject of Section 4. Proposition
1 proves that a continuum of semi-bounded closed forms are actually associated
with each semi-bounded self-adjoint operator via left-definite theory. Such
freedom in choosing a form to work with may provide valuable flexibility in
applications.

There is also a scale of Hilbert spaces that is used in singular self-adjoint
perturbation theory, when the perturbation does not lie in the Hilbert space
but can be shown to belong to an associated Hilbert space. This scale of
Hilbert spaces is generated from a self-adjoint operator, and if this operator
is semi-bounded they are equivalent to the continuum of left-definite spaces
of the operators. However, the scale of Hilbert spaces is focused on linear
bounded functionals which act on these left-definite spaces, thereby
corresponding to left-definite spaces with negative indices that are defined
through duality. Although independently defined, there are instances in which
the definitions of the spaces occurring in left-definite theory are identical
to those in the scale of Hilbert spaces. We find this intriguing, especially
due to the fact that they are used for different, complementary purposes. The
connection between this scale of spaces and left-definite theory is described
in Section 5.

One of the main obstacles in both BKV theory and singular perturbation theory
is that the associated spaces are difficult to describe with explicit boundary
conditions. This is another area where connections to left-definite theory can
be of benefit, as there are now clear descriptions even in some hard to handle
examples, i.e. Sturm–Liouville operators with limit-circle endpoints.

Self-adjoint extensions of symmetric Sturm–Liouville operators with limit-
circle endpoints were recently characterized in terms of a singular
perturbation by the authors in BFL . The perturbation heavily exploited the
connection between the scale of Hilbert spaces and BKV theory, along with
boundary triples and boundary pairs (see e.g. BdS ). In particular, we mention
that boundary triples are partially defined by the Green’s formula in equation
(3) and boundary pairs essentially operate on two key spaces from the scale of
Hilbert spaces.

The basics of this perturbation setup are described in Section 6 in order to
formulate a conjecture about how left-definite theory may assist with
extending the construction to powers of such operators.

While the applications in this manuscript are focused on ordinary differential
operators, it should be pointed out that left-definite theory does apply to
self-adjoint elliptic partial differential operators too. However, to the best
knowledge of the authors this fact has not been thoroughly explored.

Overall, the three theories: left-definite, BKV and singular perturbation, are
found to be complementary to each other. We hope this manuscript helps readers
find new and interesting connections between fields they may or may not have
encountered before and sparks their interest in building upon these
connections.

###  1.1 Notation

The notation $L_{\scriptstyle\text{\rm
max}}=\\{\ell,\mathcal{D}_{\scriptstyle\text{\rm max}}\\}$ is used to say that
the operator $L_{\scriptstyle\text{\rm max}}$ acts via $\ell$ on the domain
$\mathcal{D}_{\scriptstyle\text{\rm max}}$. Self-adjoint operators are written
in bold face, e.g. ${\bf A}$, for emphasis while all other operators are
typeset normally.

Due to the three different theories in the manuscript, we clarify the notation
for each here to help avoid any confusion. In left-definite theory, the inner
product is denoted with a single subscript $r$, i.e.
$\langle\,\cdot\,,\,\cdot\,\rangle_{r}$, and the Hilbert space associated with
this inner product is written as $\mathcal{H}_{r}$ (see Definition 1). In the
BKV theory of semi-bounded forms, the inner product and associated Hilbert
space are both denoted with the form as a subscript, i.e.
$\langle\,\cdot\,,\,\cdot\,\rangle_{{}_{\scriptstyle\mathfrak{t}_{{\bf
S}-\gamma}}}$ and $\mathcal{H}_{{}_{\scriptstyle\mathfrak{t}_{{\bf
S}-\gamma}}}$ (see equation (9)). Finally, when discussing the scale of
Hilbert spaces associated with a self-adjoint operator ${\bf A}$, we use
double subscript $\langle\,\cdot\,,\,\cdot\,\rangle_{s,-s}$ to denote the
duality pairing and, e.g. $\mathcal{H}_{-s}({\bf A})$ to denote the desired
space (see Definition 3).

##  2 Sturm–Liouville Operators

Many of the theories and conjectures in this manuscript, especially those in
Section 3 and 6, are illustrated by applications to Sturm–Liouville operators
because they are so well studied, see e.g. AG ; BEZ ; E ; Z . We briefly
introduce the central concepts of these operators here.

Consider the classical Sturm–Liouville differential equation

$\displaystyle\dfrac{d}{dx}\left[p(x)\dfrac{df}{dx}(x)\right]+q(x)f(x)=-\lambda
w(x)f(x),$

where $p(x),w(x)>0$ a.e. on $(a,b)$ and $q(x)$ real-valued a.e. on $(a,b)$,
with $a<b$ and $a,b\in\mathbb{R}\cup\\{\pm\infty\\}$. Furthermore, let
$1/p(x),q(x),w(x)\in L^{1}_{\scriptstyle\text{\rm loc}}(a,b)$. The
differential expression can be viewed as a linear operator, mapping a function
$f$ to the function

$\displaystyle\ell[f](x):=-\dfrac{1}{w(x)}\left(\dfrac{d}{dx}\left[p(x)\dfrac{df}{dx}(x)\right]+q(x)f(x)\right).$
(7)

This unbounded operator acts on the Hilbert space $L^{2}[(a,b),w]$, endowed
with the inner product $\langle
f,g\rangle:=\int_{a}^{b}f(x)\overline{g(x)}w(x)dx.$ In this setting, the
eigenvalue problem $\ell[f](x)=\lambda f(x)$ can be considered. However, the
operator acting via $\ell[\,\cdot\,]$ on $L^{2}[(a,b),w]$ is not self-adjoint
a priori; additional boundary conditions may be required to ensure this
property. The definition here is slightly different than that of the
differential operator in equation (2), where to explain the different left and
right-definite settings the weight function was not included. Hence, the
operator of interest ended up arising from $w^{-1}\ell$, which agrees with the
expression here.

Endpoints are either in the limit-point or limit-circle case depending on
whether one or two solutions are in $L^{2}[(a,b),w]$, respectively. When in
the limit-circle case, we assume that endpoints are non-oscillatory, see e.g.
BEZ ; Z  for more.

Furthermore, the operator $\ell^{n}[\,\cdot\,]$ is defined as the order two
operator $\ell[\,\cdot\,]$ composed with itself $n$ times, creating a
differential operator of order $2n$. Every formally symmetric differential
expression $\ell^{n}[\,\cdot\,]$ of order $2n$ with coefficients
$a_{k}:(a,b)\to\mathbb{R}$ and $a_{k}\in C^{k}(a,b)$, for $k=0,1,\dots,n$ and
$n\in\mathbb{N}$, has the Lagrangian symmetric form given in equation (2).
Further details can be found in DS ; ELT .

The classical differential expressions of Jacobi, Laguerre and Hermite are all
semi-bounded and admit such a representation. Semi-boundedness is defined as
the existence of a constant $k\in\mathbb{R}$ such that for all $x$ in the
domain of the operator $A$ the following inequality holds:

$\displaystyle\langle Ax,x\rangle\geq k\langle x,x\rangle.$ (8)

This additional property, combined with self-adjointness, allows for a
continuum of nested Hilbert spaces to be defined within $L^{2}[(a,b),w]$ via
non-negative real powers of the expression $\ell$. Indeed, this continuum will
provide a Hilbert scale, and many facts about the spectrum and the operators
(see e.g. DHS ; LW13 ) can be deduced using this point of view. More details
about Hilbert scales can be found in AK ; KP . This particular Hilbert scale
with self-adjoint operators that are semi-bounded is the topic of left-
definite theory LW02 .

##  3 Left-definite theory

General left-definite theory uses powers of a semi-bounded self-adjoint
differential operator to create a continuum of operators whereupon spectral
properties can be studied. This spectral information is invariant in the sense
that knowledge about the spectrum of one of the operators in this continuum
allows us to obtain insight into all the other operators. We begin by
reviewing some of the main definitions and results from the landmark paper by
Littlejohn and Wellman LW02  in order to introduce the theory.

Let $\mathcal{V}$ be a vector space over $\mathbb{C}$ with inner product
$\langle\,\cdot\,,\,\cdot\,\rangle$ and norm $\|\,\cdot\,\|$. The resulting
inner product space is denoted
$(\mathcal{V},\langle\,\cdot\,,\,\cdot\,\rangle)$.

######  Definition 1 ((LW02, , Theorem 3.1))

Suppose ${\bf A}$ on the Hilbert space
$\mathcal{H}=(\mathcal{V},\langle\,\cdot\,,\,\cdot\,\rangle)$ is a self-
adjoint operator that is bounded below by $kI$, where $k>0$. Let
$0<r\in\mathbb{R}$. Define
$\mathcal{H}_{r}=(\mathcal{V}_{r},\langle\,\cdot\,,\,\cdot\,\rangle_{r})$ with

$\mathcal{V}_{r}=\mathcal{D}({\bf A}^{r/2})$

and

$\langle x,y\rangle_{r}=\langle{\bf A}^{r/2}x,{\bf
A}^{r/2}y\rangle\quad\text{for}\quad x,y\in\mathcal{V}_{r}.$

Then $\mathcal{H}_{r}$ is said to be the $r$-th left-definite space associated
with the pair $(\mathcal{H},{\bf A})$.

It was proved in (LW02, , Theorem 3.1) that
$\mathcal{H}_{r}=(\mathcal{V}_{r},\langle\,\cdot\,,\,\cdot\,\rangle_{r})$ is
also described as the left-definite space associated with the pair
$(\mathcal{H},{\bf A}^{r})$. Specifically, we have:

  1. 1.

$\mathcal{H}_{r}$ is a Hilbert space,

  2. 2.

$\mathcal{D}({\bf A}^{r})$ is a subspace of $\mathcal{V}_{r}$,

  3. 3.

$\mathcal{D}({\bf A}^{r})$ is dense in $\mathcal{H}_{r}$,

  4. 4.

$\langle x,x\rangle_{r}\geq k^{r}\langle x,x\rangle$ for
$x\in\mathcal{V}_{r}$, and

  5. 5.

$\langle x,y\rangle_{r}=\langle{\bf A}^{r}x,y\rangle$ for
$x\in\mathcal{D}({\bf A}^{r})$, $y\in\mathcal{V}_{r}$.

The left-definite domains are defined as the domains of compositions of the
self-adjoint operator ${\bf A}$, but the operator acting on this domain is
slightly more difficult to define.

######  Definition 2 ((LW02, , Definition 2.2/2.3))

Let $\mathcal{H}=(\mathcal{V},\langle\,\cdot\,,\,\cdot\,\rangle)$ be a Hilbert
space. Suppose ${\bf A}:\mathcal{D}({\bf A})\subset\mathcal{H}\to\mathcal{H}$
is a self-adjoint operator that is bounded below by $k>0$. Let $1\leq
r\in\mathbb{R}$. If there exists a self-adjoint operator ${\bf
A}_{r}:\mathcal{H}_{r}\to\mathcal{H}_{r}$ that is a restriction of ${\bf A}$
from the domain $\mathcal{D}({\bf A})$ to $\mathcal{D}({\bf A}_{r})$, we call
such an operator an $r$-th left-definite operator associated with
$(\mathcal{H},{\bf A})$. For $0<r<1$ we obtain an $r$-th left-definite
operator associated with $(\mathcal{H},{\bf A})$ analogously, but by taking
the closure of the domain $\mathcal{D}({\bf A})$ with respect to the norm
induced by the inner product $\langle\,\cdot\,,\,\cdot\,\rangle_{r}$.

The connection between the $r$-th left-definite operator and the $r$-th
composition of the self-adjoint operator ${\bf A}$ can be made explicit:

######  Corollary 1 ((LW02, , Corollary 3.3))

Suppose ${\bf A}$ is a self-adjoint operator in the Hilbert space
$\mathcal{H}$ that is bounded below by $k>0$. For each $r>0$, let
$\mathcal{H}_{r}=(\mathcal{V}_{r},\langle\,\cdot\,,\,\cdot\,\rangle_{r})$ and
${\bf A}_{r}$ denote, respectively, the $r$-th left-definite space and the
$r$-th left-definite operator associated with $(\mathcal{H},{\bf A})$. Then

  1. 1.

$\mathcal{D}({\bf A}^{r})=\mathcal{V}_{2r}$, in particular, $\mathcal{D}({\bf
A}^{1/2})=\mathcal{V}_{1}$ and $\mathcal{D}({\bf A})=\mathcal{V}_{2}$;

  2. 2.

$\mathcal{D}({\bf A}_{r})=\mathcal{D}({\bf A}^{(r+2)/2})$, in particular,
$\mathcal{D}({\bf A}_{1})=\mathcal{D}({\bf A}^{3/2})$ and $\mathcal{D}({\bf
A}_{2})=\mathcal{D}({\bf A}^{2})$.

The left-definite theory is particularly important for self-adjoint
differential operators that are bounded below, as they are generally
unbounded. The theory is trivial for bounded operators, as shown in (LW02, ,
Theorem 3.4).

Our applications of left-definite theory will be focused on differential
operators which possess a complete orthogonal set of eigenfunctions in
$\mathcal{H}$. In (LW02, , Theorem 3.6) it was proved that the point spectrum
of ${\bf A}$ coincides with that of ${\bf A}_{r}$, and similarly for the
continuous spectrum and for the resolvent set. It is possible to say more, a
complete set of orthogonal eigenfunctions will persist throughout each space
in the Hilbert scale.

######  Theorem 3.1 ((LW02, , Theorem 3.7))

If $\\{{\varphi}_{n}\\}_{n=0}^{\infty}$ is a complete orthogonal set of
eigenfunctions of ${\bf A}$ in $\mathcal{H}$, then for each $r>0$,
$\\{{\varphi}_{n}\\}_{n=0}^{\infty}$ is a complete set of orthogonal
eigenfunctions of the $r$-th left-definite operator ${\bf A}_{r}$ in the
$r$-th left-definite space $\mathcal{H}_{r}$.

Another perspective on the last theorem is that it gives us a valuable
indicator for when a space is a left-definite space for a specific operator.
Also, we note that left-definite theory can be extended to bounded below
operators by applying shifts. Uniqueness is then given up to the chosen shift.

One of the main applications of the theory is to Sturm–Liouville operators. In
particular, if the operator has a complete system of orthogonal eigenfunctions
then proving boundedness from below is easier and it is possible to invoke
Theorem 3.1. An example of how left-definite theory can be applied to such an
operator is now briefly described, full details can be found in (LW02, ,
Section 12).

###### Example 1

For ${\alpha}>-1$, consider the classical self-adjoint Laguerre differential
operator ${\bf A}$ acting on
$\mathcal{H}=L^{2}\left[(0,\infty),x^{{\alpha}}e^{-x}\right]$ via

$\displaystyle\ell[f](x)=\dfrac{1}{x^{{\alpha}}e^{-x}}\left[-x^{{\alpha}+1}e^{-x}f^{\prime}(x)\right]^{\prime},$

such that $\operatorname{dom}({\bf A})$ possesses the Laguerre polynomials as
a complete set of orthogonal eigenfunctions. The $n$th left-definite Hilbert
space associated with the pair $(\mathcal{H},{\bf A})$, also possessing this
complete set of eigenfunctions, is defined as
$\mathcal{H}_{n}=(\mathcal{V}_{n},\langle\,\cdot\,,\,\cdot\,\rangle_{n})$,
where

$\displaystyle\mathcal{V}_{n}:=\left\\{f:(0,\infty)\to\mathbb{C}~{}~{}\bigg{|}~{}~{}f\in\text{AC}_{\scriptstyle\text{\rm
loc}}^{(n-1)}(0,\infty);~{}f^{(n)}\in
L^{2}\left[(0,\infty),t^{{\alpha}+n}e^{-t}\right]\right\\}$

and

$\displaystyle\langle
p,q\rangle_{n}:=\sum_{j=0}^{n}b_{j}(n,k)\int_{0}^{\infty}p^{(j)}(t)\overline{q^{(j)}(t)}t^{{\alpha}+j}e^{-t}dt~{}~{}\text{
for }(p,q\in\mathcal{P}),$

where $\mathcal{P}$ is the space of all (possibly complex-valued) polynomials.
The constants $b_{j}(n,k)$ are defined as

$\displaystyle
b_{j}(n,k):=\sum_{i=0}^{j}\dfrac{(-1)^{i+j}}{j!}\binom{j}{i}(k+i)^{n}.$

For several years after the discovery of the general left-definite theory,
descriptions of left-definite spaces were similar to those of Example 1, i.e.
the boundary conditions were not classically expressed by GKN theory. Similar
results for specific (mostly classical) operators can be found in BLTW ; EKLWY
; ELT  and their references. Some progress towards expressing left-definite
spaces in terms of these standard boundary conditions was made much later in
LW15  and then expanded upon in FFL .

In order to present the main result from FFL , we let ${\bf L}^{n}$ be a self-
adjoint operator defined by left-definite theory on $L^{2}[(a,b),w]$ with
domain $\mathcal{D}_{\bf L}^{n}$ that includes a complete system of orthogonal
eigenfunctions. Enumerate the orthogonal eigenfunctions as
$\\{P_{k}\\}_{k=0}^{\infty}$. Let ${\bf L}^{n}$ operate on its domain via
$\ell^{n}[\,\cdot\,]$, a differential operator of order $2n$,
$n\in\mathbb{N}$, generated by composing a Sturm–Liouville differential
expression with itself $n$ times. Furthermore, let ${\bf L}^{n}$ be an
extension of the minimal operator $L^{n}_{\scriptstyle\text{\rm min}}$ that
has deficiency indices $(n,n)$, and the associated maximal domain be denoted
by $\mathcal{D}_{\scriptstyle\text{\rm max}}^{n}$. See the Appendix for the
general definitions of these domains.

This allows us to compare several different potential descriptions of the
left-definite domain. Consider

$\displaystyle\mathcal{A}_{n}$
$\displaystyle:=\Big{\\{}f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}^{n}~{}:~{}f,f^{\prime},\dots,f^{(2n-1)}\in AC_{\scriptstyle\text{\rm
loc}}(a,b);$ $\displaystyle\hskip 136.5733pt(p(x))^{n}f^{(2n)}\in
L^{2}[(a,b),w]\Big{\\}},$ $\displaystyle\mathcal{B}_{n}$
$\displaystyle:=\left\\{f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}^{n}~{}:~{}[f,P_{j}]_{n}\Big{|}_{a}^{b}=0\text{ for
}j=0,1,\dots,n-1\right\\},$ $\displaystyle\mathcal{C}_{n}$
$\displaystyle:=\left\\{f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}^{n}~{}:~{}[f,P_{j}]_{n}\Big{|}_{a}^{b}=0\text{ for any }n\text{ distinct
}j\in\mathbb{N}\right\\},\text{ and}$ $\displaystyle\mathcal{F}_{n}$
$\displaystyle:=\left\\{f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}^{n}~{}:~{}\left[a_{j}(x)f^{(j)}(x)\right]^{(j-1)}\Big{|}_{a}^{b}=0\text{
for }j=1,2,\dots,n\right\\}.$

The function $p(x)$ above is from the standard definition of a Sturm-Liouville
differential operator, given in equation (7), and the $a_{j}(x)$’s are from
the Lagrangian symmetric form of the operator in equation (2). The following
conjecture about the equality of these domains is found in FFL , which was in
turn adapted from LW15 .

###### Conjecture 1

Let ${\bf L}^{n}$ be a self-adjoint operator defined by left-definite theory
on $L^{2}[(a,b),w]$ with domain $\mathcal{D}_{\bf L}^{n}$ that includes a
complete system of orthogonal polynomial eigenfunctions, that is, we use
$\mathcal{D}_{\bf L}^{n}=\mathcal{A}_{n}$. Let ${\bf L}^{n}$ operate on its
domain via the expression $\ell^{n}[\,\cdot\,]$, a differential operator of
order $2n$, where $n\in\mathbb{N}$, generated by composing a Sturm–Liouville
differential operator with itself $n$ times. Furthermore, let ${\bf L}^{n}$ be
an extension of the minimal operator $L^{n}_{\scriptstyle\text{\rm min}}$,
which has deficiency indices $(n,n)$. Then
$\mathcal{A}_{n}=\mathcal{B}_{n}=\mathcal{C}_{n}=\mathcal{F}_{n}=\mathcal{D}_{\bf
L}^{n}$, $\forall n\in\mathbb{N}$.

The conjecture was partially answered in (FFL, , Theorem 6.5) under several
extra assumptions: that $\mathcal{A}_{n}=\mathcal{B}_{n}$ and that
$f\in\mathcal{F}_{n}$ implies that $f^{\prime\prime},\dots,f^{(2n-2)}\in
L^{2}(a,b)$. The two primary ideas of the proof were a careful analysis of the
sesquilinear form for the operator ${\bf L}^{n}$ and the introduction of a
matrix of boundary values to help determine when GKN conditions were satisfied
or not. The conjecture was answered in the affirmative for the Jacobi
differential operator in FL .

###### Theorem 3.2

Let ${\bf L}^{n}$, $n\in\mathbb{N}$, be a self-adjoint operator defined by
left-definite theory on $L^{2}[(a,b),w]$ with the left-definite domain
$\mathcal{D}_{\bf L}^{n}$. Let ${\bf L}^{n}$ operate on its domain via
$\ell^{n}[\,\cdot\,]$, a classical Jacobi differential expression of order
$2n$, with parameters ${\alpha},\beta>0$, generated by composing the
Sturm–Liouville operator with itself $n$ times. Furthermore, let ${\bf L}^{n}$
be an extension of the minimal operator $L^{n}_{\scriptstyle\text{\rm min}}$,
which has deficiency indices $(2n,2n)$. Then $\mathcal{D}_{\bf
L}^{n}=\mathcal{A}_{n}=\mathcal{B}_{n}=\mathcal{C}_{n}=\mathcal{F}_{n}$,
$\forall n\in\mathbb{N}$.

Note that the apparent discrepancy between deficiency indices here is merely
due to the use of separated boundary conditions instead of connected ones. It
is also worth pointing out that the methods of FL  differ greatly from those
of FFL . In particular, solutions to the Jacobi differential equation are
written as infinite sums and only certain terms are shown to belong to the
maximal domain modulo the minimal domain. This finite, explicit decomposition
of the deficiency spaces result in easier manipulations of the sesquilinear
form to prove the equality of the domains. Conjecture 1 remains open for other
operators; the methods of FL  are almost certainly applicable to wider classes
of operators and may be helpful. In particular, operators which do not possess
a complete system of orthogonal polynomial eigenfunctions have not been
considered. A possible weakening of this hypothesis would be to replace the
orthogonal polynomials with principal solutions. Principal solutions are
intimately related to the Friedrichs extension of a symmetric operator, which
is the subject of our last conjecture.

The Friedrichs extension is usually defined through the closed semi-bounded
form associated with a self-adjoint operator, see Section 4 (specifically
equation (10)) for more about these forms or BdS ; BFL ; MZ  for complete
details. However, it suffices to think of the extension as the “smallest”
self-adjoint extension among all other self-adjoint extensions (in the sense
that it has the smallest form domain); it is often called the “soft” extension
for this reason.

###### Conjecture 2

Let $A$ be a closed semi-bounded symmetric operator and ${\bf A}_{F}$ be its
Friedrichs self-adjoint extension. Then the $2r$-th left-definite space of
${\bf A}_{F}$ coincides with the domain of the Friedrichs extension of the
$r$-th power of ${\bf A}$. Explicitly,

$\displaystyle\operatorname{dom}(({\bf A}_{F})^{r})=\operatorname{dom}(({\bf
A}^{r})_{F}).$

In other words, the conjecture is suggesting that the action of taking powers
of an operator commutes with the action of taking the Friedrichs extension. In
every computed case the conjecture seems to hold but in the stated generality
the status is unclear. Verification in the case where $A$ is the Jacobi
differential operator can be found in (F, , Cor. 5.1). Left-definite theory
need not be mentioned in the statement of the conjecture, clearly, but it is
the authors’ opinion that left-definite theory can nonetheless be of great
help in proving the statement.

A related conjecture (that would build upon the spectral stability results of
Theorem 3.1 if confirmed) posits that multiplicity of eigenvalues are
invariant under left-definite theory.

###### Conjecture 3

Let ${\bf A}$ be a semi-bounded self-adjoint operator in a Hilbert space
$\mathcal{H}$. Let ${\lambda}$ be an eigenvalue of ${\bf A}$ with multiplicity
$m$. Then ${\lambda}$ is also an eigenvalue for the $r$-th left-definite
operator ${\bf A}_{r}$ in the $r$-th left-definite space $\mathcal{H}_{r}$ of
multiplicity $m$.

The difficulty of determining multiplicity of eigenvalues restricts the amount
of evidence available to support the conjecture. In the case when ${\bf A}$ is
the Jacobi differential operator, left-definite operators and Weyl
$m$-functions for their extensions can be found in F  and the spectral
analysis of BFL  could be applied to potentially verify the conjecture for the
example.

Conjecture 3 also has implications for the intertwining of eigenvalues between
distinct extensions. Usually this intertwining takes place between the
Friedrichs extension and a transversal extension, which can be seen e.g. in a
comparison of Neumann and Dirichlet eigenvalues for Sturm–Liouville operators
with two regular endpoints. Essentially, it would be interesting to see if
eigenvalue intertwining between two self-adjoint extensions implied that the
$r$-th left-definite operators of the two extensions also had eigenvalue
intertwining.

##  4 Comparison with BKV Semi-Bounded Form Theory

We first introduce the basics of the so-called Birman–Krein–Vishik (BKV)
theory of semi-bounded forms. A collection of results from the theory along
with original references can be found in e.g. AS . The brief presentation here
mostly follows that of (BdS, , Chapter 5) and (Kato, , Chapter 6), which can
also be consulted for more details.

Let ${\bf A}$ be a semi-bounded self-adjoint operator with lower bound $m({\bf
A})<\infty$. There is a natural way to identify ${\bf A}$ with a closed semi-
bounded form $\mathfrak{t}$ in $\mathcal{H}$ with the same lower bound
$m(\mathfrak{t})=m({\bf A})$ via the First and Second Representation Theorems,
see e.g. (BdS, , Theorem 5.1.18) and (BdS, , Theorem 5.1.23). Namely, let
${\varphi}\in\operatorname{dom}({\bf A})$,
$\psi\in\operatorname{dom}(\mathfrak{t})$, $\gamma<m({\bf A})$ and define

$\displaystyle\operatorname{dom}(\mathfrak{t}_{{\bf A}})$
$\displaystyle=\operatorname{dom}({\bf A}-\gamma)^{1/2},$
$\displaystyle\mathfrak{t}_{{\bf A}}[{\varphi},\psi]$
$\displaystyle=\langle({\bf
A}-\gamma){\varphi},\psi\rangle+\gamma\langle{\varphi},\psi\rangle.$

Equivalently, if ${\varphi},\psi\in\operatorname{dom}(\mathfrak{t}_{{\bf
A}})$, then

$\displaystyle\mathfrak{t}_{{\bf A}}[{\varphi},\psi]=\langle({\bf
A}-\gamma)^{1/2}{\varphi},({\bf
A}-\gamma)^{1/2}\psi\rangle+\gamma\langle{\varphi},\psi\rangle.$

The space $\operatorname{dom}(\mathfrak{t}_{{\bf A}})$ endowed with the inner
product

$\displaystyle\langle{\varphi},\psi\rangle_{{}_{\scriptstyle\mathfrak{t}_{{\bf
A}-\gamma}}}:=\mathfrak{t}_{{\bf
A}}[{\varphi},\psi]-\gamma\langle{\varphi},\psi\rangle,\text{ for
}{\varphi},\psi\in\operatorname{dom}(\mathfrak{t}),$ (9)

is a Hilbert space, denoted $\mathcal{H}_{{}_{\scriptstyle\mathfrak{t}_{{\bf
A}-\gamma}}}$.

More general semi-bounded symmetric operators $S$ (i.e. the minimal operator
of a Sturm–Liouville expression) also have a form associated with them via

$\displaystyle\mathfrak{t}_{S}[f,g]=\langle Sf,g\rangle,\text{ for
}f,g\in\operatorname{dom}(S).$ (10)

The semi-bounded self-adjoint operator ${\bf S}_{F}$ associated with the
closure of the form $\mathfrak{t}_{S}$ in equation (10) is called the
Friedrichs extension of $S$, see (BdS, , Definition 5.3.2).

This theory is often used to distinguish or construct specific self-adjoint
extensions from a symmetric operator, as it is often more convenient to define
a closed semi-bounded form than a self-adjoint operator. An ordering of closed
semi-bounded forms then corresponds to an ordering of self-adjoint extensions,
is also useful for this purpose. Details can be found in e.g. (BdS, , Section
5.2) and (BFL, , Remark 3.5).

It is also possible to put densely defined, closed, sectorial forms
$\mathfrak{t}_{T}$ into one-to-one correspondence with $m$-sectorial operators
$T$, but this falls outside the scope of the current manuscript (see e.g. Arl
; BT ; C ). However, this shows that the correspondence between forms and
operators is usable in a wide variety of contexts.

In comparison with Section 3, it should be clear that given a semi-bounded
self-adjoint operator ${\bf A}$ the first left-definite space automatically
coincides with the domain of the associated closed semi-bounded form after an
appropriate shift to make the operator positive, i.e.
$\mathcal{V}_{1}=\operatorname{dom}(\mathfrak{t}_{\bf A})$. Indeed, for
$f,g\in\mathcal{V}_{1}$ the action of $\mathfrak{t}_{\bf A}[f,g]$ is equal to
$\langle f,g\rangle_{1}$, the inner product in the first left-definite space.

If we restrict our attention to closed semi-bounded forms, left-definite
theory is a natural extension of BKV theory; instead of associating a single
closed semi-bounded form with a self-adjoint operator, it is possible to
associate a whole continuum of closed semi-bounded forms.

###### Proposition 1

Let ${\bf A}$ be a semi-bounded self-adjoint operator with lower bound $m({\bf
A})$. For each $r\in\mathbb{N}$, the form (we suppress the dependence on ${\bf
S}$ here), with $f,g\in\operatorname{dom}(\mathfrak{t}_{r})$ and
$\gamma<m({\bf A})$, given by

$\displaystyle\operatorname{dom}(\mathfrak{t}_{r})$
$\displaystyle=\operatorname{dom}({\bf A}-\gamma)^{r/2},$
$\displaystyle\mathfrak{t}_{r}[f,g]$ $\displaystyle=\langle(({\bf
A}-\gamma)^{r/2}f,({\bf A}-\gamma)^{r/2}g\rangle+\gamma\langle f,g\rangle,$

is closed and semi-bounded.

###### Proof

Here, we consider the left-definite spaces associated with the shifted
operator ${\bf A}-\gamma$. Semi-boundedness of $\mathfrak{t}_{r}$ is assured
by item (4) after Definition 1. Lemma 5.1.9 of BdS  says that the semi-bounded
form $\mathfrak{t}_{r}$ is closed if and only if the space
$\mathcal{H}_{{}_{\scriptstyle({\bf A}-\gamma)^{r}}}$ is in fact a Hilbert
space. Endow the space $\operatorname{dom}(\mathfrak{t}_{r})$ with the inner
product $\langle\cdot,\cdot\rangle_{r}$ from Definition 1 and notice that this
coincides with the definition of the space $\mathcal{H}_{{}_{\scriptstyle({\bf
A}-\gamma)^{r}}}$ above. Since $\mathcal{H}_{r}$ is a Hilbert space, this
immediately implies that $\mathfrak{t}_{r}$ is closed.

While the BKV theory of semi-bounded forms gives valuable knowledge about
self-adjoint extensions, in practice the domain of the operator ${\bf
A}^{1/2}$ is often difficult to determine, even for elementary choices of
${\bf A}$. Explicit domains can be determined when ${\bf S}$ is a
Sturm–Liouville operator, see (BdS, , Section 6.9). Left-definite domains have
some advantages and disadvantages in this area.

As far as advantages go, for even $r$, left-definite domains do not involve
fractional powers of operators and therefore are somewhat natural to consider.
The square of a self-adjoint operator is sometimes useful in applications and
falls into this category. There is also some spectral stability inherent in
left-definite operators, see Theorem 3.1, that is, to the best knowledge of
the authors, not available in BKV form theory. Left-definite theory thus
provides alternatives to the classical form used in BKV theory and these form
domains can be expressed with classical boundary conditions, see e.g. Theorem
3.2.

The main disadvantage of the theory is that it is unclear whether the ordering
of closed semi-bounded forms is preserved under left-definite theory. This is
made more difficult by the fact that explicit boundary conditions are somewhat
elusive, see Theorem 3.2 for an example where they were determined. A proof of
Conjecture 2 would go a long way towards solving this discrepancy. BKV theory,
as previously mentioned, is also applicable to a wider range of operators.

It is also important to note that, in contrast to Section 3, the discussion
here was not concerned with differential operators but holds in the wider
context of semi-bounded self-adjoint operators.

BKV theory was exploited in BFL  to build a boundary pair for Sturm–Liouville
operators with limit-circle endpoints that was compatible with a boundary
triple. This allows the set up of a perturbation problem that describes all
possible self-adjoint extensions of the minimal operator by using a scale of
spaces in the following section.

##  5 Scale of Spaces from Singular Perturbation Theory

Let ${\bf A}$ and ${\bf T}$ be operators on a Hilbert space $\mathcal{H}$. In
perturbation theory, we are interested in the following question: If we know
the properties of operator ${\bf A}$ well, what can we say about the formal
operator ${\bf A}+{\bf T}$?

When the Hilbert space $\mathcal{H}$ is infinite, an immediate question is the
rigorous definition what the sum of two operators ${\bf A}+{\bf T}$. To
briefly illustrate the severity of potential problems, recall that, by the
closed graph theorem, we know that the unbounded operators ${\bf A}$ and ${\bf
T}$ are only defined on a dense subset of $\mathcal{H}$. Noticing this, we
realize that it can easily happen that the intersection of their domains is
empty. In this case, ${\bf A}+{\bf T}$ would not be interesting as its domain
equals the empty set. Of course, less severe scenarios can also lead to
serious issues with the meaning of ${\bf A}+{\bf T}$.

Throughout perturbation theory, this problem is dealt with by making situation
or application dependent assumptions on ${\bf A}$ and/or ${\bf T}$. Most
frequently, some smallness hypothesis on the perturbation ${\bf T}$ is
imposed. In our application to Sturm–Liouville operators, both ${\bf A}$ and
${\bf T}$ will be unbounded self-adjoint operators (making their sum self-
adjoint as well), though ${\bf A}$ will be bounded from below and ${\bf T}$
will be of finite rank, i.e. has finite dimensional range. The fact that ${\bf
T}$ is of finite rank, will allows us to formulate this perturbation problem
rather concretely. With this setup, the operator ${\bf A}$ will be one self-
adjoint extensions of a Sturm–Liouville operator and the perturbed operators
${\bf A}+{\bf T}$ will stand in bijection to all possible self-adjoint
extensions of the minimal operator with ${\bf T}$ encoding those boundary
conditions.

Further, in our application the following key property holds: the range of
${\bf T}$ is contained in a Hilbert space generated by operator ${\bf A}$. It
will turn out that ${\bf T}$ is relatively bounded with respect to ${\bf A}$,
so that the domain of ${\bf A}+{\bf T}$ equals that of ${\bf A}$. To carry out
this plan, we now define this scale of Hilbert spaces and then discuss the
meaning of ${\bf A}+{\bf T}$ in our situation.

The following definition of these finite rank singular form bounded
perturbations roughly follows that in AK . Let ${\bf A}$ be a self-adjoint
operator on $\mathcal{H}$. Consider the non-negative operator $|{\bf A}|=({\bf
A}^{*}{\bf A})^{1/2}$, whose domain coincides with the domain of ${\bf A}$. We
introduce a scale of Hilbert spaces.

######  Definition 3 (see, e.g. (AK, , Section 1.2.2))

For $s\geq 0$, define the space $\mathcal{H}_{s}(A)$ to consist of
${\varphi}\in\mathcal{H}$ for which the $s$-norm

$\displaystyle\|{\varphi}\|_{s}:=\|(|{\bf
A}|+I)^{s/2}{\varphi}\|_{\mathcal{H}},$ (11)

is bounded. The space $\mathcal{H}_{s}({\bf A})$ equipped with the norm
$\|\cdot\|_{s}$ is complete. The adjoint spaces, formed by taking the linear
bounded functionals on $\mathcal{H}_{s}({\bf A})$, are used to define these
spaces for negative indices, i.e. $\mathcal{H}_{-s}({\bf
A}):=\mathcal{H}_{s}^{*}({\bf A})$. The corresponding norm in the space
$\mathcal{H}_{-s}({\bf A})$ is thus defined by (11) as well. The collection of
these $\mathcal{H}_{s}({\bf A})$ spaces will be called the _scale of Hilbert
spaces associated with the self-adjoint operator ${\bf A}$_.

Alternatively, if ${\bf A}$ is semi-bounded with lower bound $m({\bf A})$,
then we can choose to consider ${\bf A}-\gamma$ for $\gamma<m({\bf A})$
instead of $|{\bf A}|+I$. In particular, both options generate the same spaces
with equivalent norms.

It is not difficult to see that the spaces satisfy the nesting properties

$\displaystyle\ldots\subset\mathcal{H}_{2}({\bf A})\subset\mathcal{H}_{1}({\bf
A})\subset\mathcal{H}=\mathcal{H}_{0}({\bf A})\subset\mathcal{H}_{-1}({\bf
A})\subset\mathcal{H}_{-2}({\bf A})\subset\ldots,$

and that for every two $s,t$ with $s<t$, the space $\mathcal{H}_{t}({\bf A})$
is dense in $\mathcal{H}_{s}({\bf A})$ with respect to the norm
$\|\cdot\|_{s}$. Indeed, the operator $({\bf A}+I)^{t/2}$ defines an isometry
from $\mathcal{H}_{s}({\bf A})$ to $\mathcal{H}_{s-t}({\bf A})$. For
${\varphi}\in\mathcal{H}_{-s}({\bf A})$, $\psi\in\mathcal{H}_{s}({\bf A})$, we
define the duality pairing

$\displaystyle\langle{\varphi},\psi\rangle_{s,-s}:=\big{\langle}(|{\bf
A}|+I)^{-s/2}{\varphi},(|{\bf A}|+I)^{s/2}\psi\big{\rangle}.$

The main perspective we are choosing here is the relations of these Hilbert
scales with those spaces generated by left-definite theory, see Section 3.
Outside of this perspective we mention on the side that throughout the
literature of other fields similar constructions occur under different names.
For instance, the pairing of $\mathcal{H}_{1}({\bf A})$, $\mathcal{H}$, and
$\mathcal{H}_{-1}({\bf A})$ is sometimes referred to as a _Gelfand triple_ or
_rigged Hilbert space_. Also, when ${{\bf A}}$ is the derivative operator,
these scales are simply Hilbert–Sobolev spaces. When ${\bf A}$ is a general
differential operator, they are closely related. More details about Hilbert
scales can be found in KP .

Aside, we also mention that finite-rank perturbations of a given operator
${{\bf A}}$ arise most commonly when the vectors ${\varphi}$ are bounded
linear functionals on the domain of the operator ${\bf A}$; so, many
applications are focused on $\mathcal{H}_{-2}({\bf A})$. Here, we discuss the
case ${\varphi}\in\mathcal{H}_{-1}({\bf A})$ for the sake of simplicity, the
so-called form bounded singular case. However, AK  contains information on
extensions to ${\varphi}\in\mathcal{H}_{-2}({\bf A})$, and the case when
${\varphi}\notin\mathcal{H}_{-2}({\bf A})$ can be found in DKS ; Kurasov .

To define rank-$d$ form bounded perturbations of a self-adjoint operator
${{\bf A}}$ on a Hilbert space $\mathcal{H}$, consider a coordinate mapping
${\bf B}:\mathbb{C}^{d}\to\operatorname{Ran}({\bf
B})\subset\mathcal{H}_{-1}\left({\bf A}\right)$ that acts via multiplication
by the row vector

$\displaystyle\begin{pmatrix}f_{1},~{}~{}\ldots,f_{d}\\\
\end{pmatrix}\qquad\text{with }f_{1},\ldots,f_{d}\in\mathcal{H}_{-1}\left({\bf
A}\right).$

Formally, the mapping ${\bf B}^{*}:\operatorname{Ran}({\bf
B})\to\mathbb{C}^{d}$ acts by

$\displaystyle{\bf
B}^{*}\,\cdot\,=\begin{pmatrix}\langle\,\cdot\,,f_{1}\rangle_{s,-s}\\\
\vdots\\\ \langle\,\cdot\,,f_{d}\rangle_{s,-s}\end{pmatrix}.$

We say _formally_ , because the inner products occurring in ${\bf B}^{*}$ are
not defined on all of $\operatorname{Ran}({\bf B})$. However, in accordance
with the definition of $\mathcal{H}_{-1}\left({\bf A}\right)$, they do make
sense as a duality pairing on the quadratic form space of the unperturbed
operator ${{\bf A}}$. And that is all we need. Abusing notation slightly, we
use the same notation ${\bf B}^{*}$ for the operator restricted to this form
domain.

The quadratic form sense now gives rigorous meaning to the finite rank form
bounded singular perturbation

$\displaystyle{{\bf A}}_{{}_{\scriptstyle\Theta}}:={{\bf A}}+{\bf B}\Theta{\bf
B}^{*},$ (12)

where $\Theta:{\mathbb{C}}^{d}\to{\mathbb{C}}^{d}$ is an $d\times d$ matrix
(not a linear relation).

For the interpretation and application it is easiest to fix a coordinate map
${\bf B}$. The definition in equation (12) can be extended to linear relations
$\Theta$. It is well-known that boundary conditions then stand in bijection to
the self-adjoint linear relation $\Theta$. One way to access the explicit
translation between boundary conditions and $\Theta$ is via boundary triplets.
We decided not to include this information due to accessibility, see e.g. BFL
. More information on finite-rank perturbations can be found in e.g. FL2 ;
LT_JST ; LTSurvey .

Given this setup, the main problem usually becomes determining which space a
desired perturbation vector comes from. In practice, it can be very difficult
to actually compute the norm in Definition 3 so it does not appear there are
many tools available for this purpose. However, connections with left-definite
theory and BKV semi-bounded form theory can be exploited here.

For a semi-bounded self-adjoint operator ${\bf A}$ with lower bound $m({\bf
A})$, choose $\gamma<m({\bf A})$. Given the note after Definition 3, it is
then clear that $\mathcal{H}_{1}({\bf A}-\gamma)$ coincides with
$\mathcal{H}_{{}_{\scriptstyle{\bf A}-\gamma}}$ (inner product given in
equation (9)) so that the underlying spaces are the same. Hence, showing that
a perturbation vector belongs to the class $\mathcal{H}_{-1}({\bf A}-\gamma)$
is the same as showing it is a linear bounded functional on the first left-
definite space $\mathcal{V}_{1}$ or the form domain $\mathfrak{t}_{{\bf
A}-\gamma}$.

Here we find another limitation of BKV semi-bounded form theory, as it is
somewhat natural to consider what spaces are analogously related for more
singular perturbations. Fortunately, the connection with left-definite theory
remains valid in these cases and provides good indicators for what spaces
perturbation vectors might be in. For instance, a perturbation vector belongs
to the class $\mathcal{H}_{-2}({\bf A}-\gamma)$ if and only if it is a linear
bounded functional on $\mathcal{H}_{2}({\bf A}-\gamma)$, whose underlying
space is simply the second left-space $\mathcal{V}_{2}$, and so on and so
forth for higher values of $s$ in accordance with Corollary 1. Left-definite
theory is thus a fundamental subject in the study of singular perturbation
theory.

Singular perturbation theory is thus complementary to left-definite theory. It
provides a rigorous analysis of linear bounded functionals on left-definite
spaces, thereby expanding the framework to consider $-r\in\mathbb{N}$, and
many useful applications. However, because singular perturbation theory is for
general self-adjoint operators, it lacks the key spectral stability results of
Theorem 3.1 and explicit descriptions of the domains involved. If the self-
adjoint operator is strictly positive, both theories can be applied directly.

##  6 Perturbation setup

The connection between BKV semi-bounded form theory and singular perturbation
theory was recently used by the authors in BFL  to obtain a characterization
of all possible self-adjoint extensions of Sturm–Liouville differential
operators with one or two limit-circle endpoints. As explained in the previous
two subsections, the connection these theories have with left-definite theory
mean that the left-definite space $\mathcal{V}_{1}$ is also inherently
involved.

The perturbation setup itself is a bit technical, so we refer the reader to
the full manuscript BFL  for details and cover only the broad strokes here.

The operators ${\bf B}$ and ${\bf B}^{*}$ are as in Section 6 with the
perturbation vectors $f_{1}$ and $f_{2}$ chosen to have their duality pairing
act like the sesquilinear form for the Sturm–Liouville operator with one input
being the principal solution. Namely, $f_{1}$ and $f_{2}$ are elements from
$\mathcal{H}_{-1}({\bf A})$ defined so that

$\displaystyle\langle\,\cdot\,,f_{1}\rangle_{1,-1}:=[\,\cdot\,,u_{a}](x)\Big{|}_{x=a}\quad\text{and}\quad\langle\,\cdot\,,f_{2}\rangle_{1,-1}:=[\,\cdot\,,u_{b}](x)\Big{|}_{x=b},$
(13)

where $u_{a}$ and $u_{b}$ are principal solutions to the eigenvalue problem
for some ${\lambda}\in\mathbb{R}$ near the endpoints $x=a$ and $x=b$ of the
differential equation, respectively. The self-adjoint extension ${\bf A}_{0}$
of the minimal operator is chosen to be transversal to the domain of the
Friedrichs extension and defined via a boundary triple. Namely, if ${\bf
A}_{F}$ is the Friedrichs extension, then the span of $\operatorname{dom}({\bf
A}_{F})\cup\operatorname{dom}({\bf A}_{0})$ is the maximal domain and
$\operatorname{dom}({\bf A}_{F})\cap\operatorname{dom}({\bf A}_{0})$ is the
minimal domain.

######  Theorem 6.1 ((BFL, , Theorem 3.6))

Let $\Theta$ be a self-adjoint linear relation in $\mathbb{C}^{2}$. Define
${\bf A}_{{}_{\scriptstyle\Theta}}$ as the singular rank-two perturbation:

$\displaystyle{\bf A}_{{}_{\scriptstyle\Theta}}:={\bf A}_{0}+{\bf B}\Theta{\bf
B}^{*}.$ (14)

Then every self-adjoint extension of the minimal operator
$L_{\scriptstyle\text{\rm min}}$ can be written as ${\bf
A}_{{}_{\scriptstyle\Theta}}$ for some $\Theta$.

In an application, the operator ${\bf A}_{{}_{\scriptstyle\Theta}}$ being
well-defined reduces to showing that the somewhat abstractly defined
$f_{1},f_{2}$ belong to $\mathcal{H}_{-1}({\bf A}_{0})$. It turns out that
this is essentially a consequence of constructing a boundary pair. In this
context, a boundary pair consists of

  * •

a bounded operator from $\mathcal{H}_{1}({\bf A}_{0})$ to $\mathbb{C}^{2}$
that is surjective and whose kernel is equal to the domain of the Friedrichs
extension,

  * •

and the space $\mathbb{C}^{2}$.

Such a map from $\mathcal{H}_{1}({\bf A}_{0})$ to $\mathbb{C}^{2}$ is then a
linear bounded functional on $\mathcal{H}_{1}({\bf A}_{0})$ and hence
generated by a function from $\mathcal{H}_{-1}({\bf A}_{0})$ via the duality
pairing. Indeed, the proof uses this fact, as $f_{1}$ and $f_{2}$ are chosen
to naturally fit this requirement. The other part of the theorem – that
equation (14) establishes a one-to-one relationship between self-adjoint
extensions and self-adjoint linear relations $\Theta$ – relies on a
parameterization stemming from the theory of boundary triples. Note that the
use of linear relations here is in contrast to the matrices used in Section 6.

We also note that the perturbation from Theorem 6.1 can only be rigorously
interpreted by appealing to BKV semi-bounded form theory, see (BFL, , Remark
3.7) for details.

Ideally, the perturbation setup could be adapted to work for powers of
Sturm–Liouville operators. This might provide a nice class of examples of
finite rank perturbations. The following conjecture states what a formulation
would look like.

###### Conjecture 4

Let ${\bf L}^{n}_{0}$, $n\in\mathbb{N}$, be a self-adjoint extension of the
minimal operator $L^{n}_{\scriptstyle\text{\rm min}}$ on $L^{2}[(a,b),w]$,
which has deficiency indices $(m,m)$ with $m\in\\{n,2n\\}$. Let ${\bf L}^{n}$
operate on its domain via $\ell^{n}[\,\cdot\,]$ (defined in equation (2)), a
semi-bounded Sturm–Liouville differential expression of order $2n$ generated
by composing the Sturm–Liouville operator with itself $n$ times. Let $\Theta$
be a self-adjoint relation in $\mathbb{C}^{m}$. Then there exist a choice of
${\bf L}^{n}_{0}$, ${\bf B}$ and ${\bf B}^{*}$ such that the singular rank-$m$
perturbation

$\displaystyle{\bf L}_{{}_{\scriptstyle\Theta}}:={\bf L}^{n}_{0}+{\bf
B}\Theta{\bf B}^{*}.$

is well-defined and every self-adjoint extension of the minimal operator
$L_{\scriptstyle\text{\rm min}}$ can be written as ${\bf
L}_{{}_{\scriptstyle\Theta}}$ for some $\Theta$.

The parametrization of self-adjoint linear relations again relates to the
boundary conditions imposed by an underlying boundary triple here, so
obtaining all self-adjoint extensions via the perturbation should not be an
overly difficult problem. The choice of the operator ${\bf L}^{n}_{0}$ allows
for some freedom; it was the result of a boundary triple construction in
Theorem 6.1 but these are not unique and the only requirement needed here is
that the chosen extension is transversal to the Friedrichs domain.

The real issue is finding perturbation vectors $f_{1},\dots,f_{m}$ from
$\mathcal{H}_{-1}({\bf L}_{0}^{n})$ so that ${\bf B}$ and ${\bf B}^{*}$ are
well-defined. In Theorem 6.1 these roles were essentially played by principal
solutions via equation (13), but an obvious analog does not immediately
present itself in this case. For any $\lambda\in\mathbb{R}$ there exist
general solutions to the eigenvalue problem for the uncomposed Sturm–Liouville
differential operator near an endpoint, one principal and one non-principal if
it is limit-circle. Hence, there are $n$ principal solutions to use in
equation (13), but it does not appear that these functions can create a
boundary triple like in the base case, see e.g. (BdS, , Chapter 6). This
problem was solved in F  for powers of the Jacobi differential operator by
applying a modified Gram–Schmidt process to these principal solutions, but a
generalization is outstanding.

A proof of Conjecture 2 would also be helpful here, as an analogous statement
may be true for the transversal self-adjoint extension ${\bf L}_{0}$. A
possible intermediary goal would then be to state the problem in terms of the
uncomposed operator ${\bf L}_{0}$: the operator ${\bf L}^{n}_{0}$ would be the
$2n$-th left-definite operator associated with ${\bf L}_{0}$. Hence, the main
problem in Conjecture 4 would be to show that the perturbation vectors in
${\bf B}^{*}$ are in $\mathcal{H}_{-n}({\bf L}_{0})$. This seemingly small
change could have a large impact.

## Appendix: Extension Theory

For readers not familiar with classical self-adjoint extension theory for
symmetric operators we include some basic definitions and notions as applied
to ordinary differential operators. The classical references AG ; N  can also
be consulted for further details.

Let $\ell$ be a Sturm–Liouville differential expression. It is important to
reiterate that the analysis of self-adjoint extensions does not at all involve
changing the differential expression associated with the operator, merely the
domain of definition by applying boundary conditions.

######  Definition 4 (see, e.g. (N, , Section 17.4))

The maximal domain of $\ell[\,\cdot\,]$ is given by

$\displaystyle\mathcal{D}_{\scriptstyle\text{\rm
max}}=\mathcal{D}_{\scriptstyle\text{\rm
max}}(\ell):=\big{\\{}f:(a,b)\to\mathbb{C}~{}:~{}f,pf^{\prime}$
$\displaystyle\in\text{AC}_{\scriptstyle\text{\rm loc}}(a,b);$ $\displaystyle
f,\ell[f]\in L^{2}[(a,b),w]\big{\\}}.$

The designation of “maximal” is appropriate in this case because
$\mathcal{D}_{\scriptstyle\text{\rm max}}(\ell)$ is the largest possible
subset of $L^{2}[(a,b),w]$ that $\ell$ maps back into $L^{2}[(a,b),w]$. For
$f,g\in\mathcal{D}_{\scriptstyle\text{\rm max}}(\ell)$ and $a<a_{0}\leq
b_{0}<b$ the sesquilinear form associated with $\ell$ by

$[f,g]\bigg{|}_{a_{0}}^{b_{0}}:=\int_{a_{0}}^{b_{0}}\left\\{\ell[f(x)]\overline{g(x)}-\ell[\overline{g(x)}]f(x)\right\\}w(x)dx.$
(15)

The notation for the sesquilinear form does not involve $\ell$ explicitly,
rather the differential expression will be clear from context.

######  Theorem 6.2 (see, e.g. (N, , Section 17.4))

The limits $[f,g](b):=\lim_{x\to b^{-}}[f,g](x)$ and $[f,g](a):=\lim_{x\to
a^{+}}[f,g](x)$ exist and are finite for
$f,g\in\mathcal{D}_{\scriptstyle\text{\rm max}}(\ell)$.

The equation (15) is Green’s formula for $\ell[\,\cdot\,]$, and in the case of
Sturm–Liouville operators (7) it can be explicitly computed using integration
by parts to be the modified Wronskian

$\displaystyle[f,g]\bigg{|}_{a}^{b}:=p(x)[f^{\prime}(x)g(x)-f(x)g^{\prime}(x)]\bigg{|}_{a}^{b}.$

######  Definition 5 (see, e.g. (N, , Section 17.4))

The minimal domain of $\ell[\,\cdot\,]$ is given by

$\displaystyle\mathcal{D}_{\scriptstyle\text{\rm
min}}=\mathcal{D}_{\scriptstyle\text{\rm
min}}(\ell):=\left\\{f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}(\ell)~{}:~{}[f,g]\big{|}_{a}^{b}=0~{}~{}\forall
g\in\mathcal{D}_{\scriptstyle\text{\rm max}}(\ell)\right\\}.$

The maximal and minimal operators associated with the expression
$\ell[\,\cdot\,]$ are then defined as $L_{\scriptstyle\text{\rm
min}}=\\{\ell,\mathcal{D}_{\scriptstyle\text{\rm min}}\\}$ and
$L_{\scriptstyle\text{\rm max}}=\\{\ell,\mathcal{D}_{\scriptstyle\text{\rm
max}}\\}$, respectively. By (N, , Section 17.2), these operators are adjoints
of one another, i.e. $(L_{\scriptstyle\text{\rm
min}})^{*}=L_{\scriptstyle\text{\rm max}}$ and $(L_{\scriptstyle\text{\rm
max}})^{*}=L_{\scriptstyle\text{\rm min}}$. The operator
$L_{\scriptstyle\text{\rm min}}$ is thus symmetric.

Note that the self-adjoint extensions of a symmetric operator coincide with
those of the closure of the symmetric operator (DS, , Theorem XII.4.8), so
without loss of generality we assume that $L_{\scriptstyle\text{\rm min}}$ is
closed.

######  Definition 6 (variation of (N, , Section 14.2))

Define the positive defect space and the negative defect space, respectively,
by

$\mathcal{D}_{+}:=\left\\{f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}~{}:~{}L_{\scriptstyle\text{\rm
max}}f=if\right\\}\quad\text{and}\quad\mathcal{D}_{-}:=\left\\{f\in\mathcal{D}_{\scriptstyle\text{\rm
max}}~{}:~{}L_{\scriptstyle\text{\rm max}}f=-if\right\\}.$

The dimensions dim$(\mathcal{D}_{+})=m_{+}$ and dim$(\mathcal{D}_{-})=m_{-}$,
called the positive and negative deficiency indices of
$L_{\scriptstyle\text{\rm min}}$ respectively, will play an important role.
They are usually conveyed as the pair $(m_{+},m_{-})$. The symmetric operator
$L_{\scriptstyle\text{\rm min}}$ has self-adjoint extensions if and only if
its deficiency indices are equal (N, , Section 14.8.8).

The classical von Neumann formula then says that

$\displaystyle\mathcal{D}_{\scriptstyle\text{\rm
max}}=\mathcal{D}_{\scriptstyle\text{\rm
min}}\dotplus\mathcal{D}_{+}\dotplus\mathcal{D}_{-}.$

The decomposition can be made into an orthogonal direct sum by using the graph
norm, see FFL . If the operator $L_{\scriptstyle\text{\rm min}}$ has any self-
adjoint extensions, then the deficiency indices of $L_{\scriptstyle\text{\rm
min}}$ have the form $(m,m)$, where $0\leq m\leq 2$ (N, , Section 14.8.8).
Hence, Sturm–Liouville expressions that generate self-adjoint operators must
have deficiency indices $(0,0)$, $(1,1)$ or $(2,2)$. If a differential
expression is either in the limit-circle case or regular at the endpoint $a$,
it requires a boundary condition at $a$. If it is in the limit-point case at
the endpoint $a$, it does not require a boundary condition. The analogous
statements are true at the endpoint $b$.

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