# Effective Computation of Traces, Determinants, and $\zeta$-Functions for
Sturm–Liouville Operators

Fritz Gesztesy  Department of Mathematics, Baylor University, One Bear Place
#97328, Waco, TX 76798-7328, USA<EMAIL_ADDRESS>http://www.baylor.edu/math/index.php?id=935340 and  Klaus Kirsten  GCAP-
CASPER, Department of Mathematics, Baylor University, One Bear Place #97328,
Waco, TX 76798-7328, USA<EMAIL_ADDRESS>http://www.baylor.edu/math/index.php?id=54012 Dedicated with great pleasure to
Konstantin Makarov on the occasion of his 60th birthday.

###### Abstract.

The principal aim in this paper is to develop an effective and unified
approach to the computation of traces of resolvents (and resolvent
differences), Fredholm determinants, $\zeta$-functions, and $\zeta$-function
regularized determinants associated with linear operators in a Hilbert space.
In particular, we detail the connection between Fredholm and $\zeta$-function
regularized determinants.

Concrete applications of our formalism to general (i.e., three-coefficient)
regular Sturm–Liouville operators on compact intervals with various (separated
and coupled) boundary conditions, and Schrödinger operators on a half-line,
are provided and further illustrated with an array of examples.

###### Key words and phrases:

Traces, (modified) Fredholm determinants, semi-separable integral kernels,
Sturm–Liouville operators.

###### 2010 Mathematics Subject Classification:

Primary: 47A10, 47B10, 47G10. Secondary: 34B27, 34L40.

K.K. was supported by the Baylor University Summer Sabbatical Program.

##  1\. Introduction

We dedicate this paper with great pleasure to Konstantin A. Makarov on the
occasion of his 60th birthday. Happy Birthday, Konstantin, we hope our modest
contribution to spectral theory will cause some joy.

The prime motivation for writing this paper originally was to detail the
relationship between Fredholm and $\zeta$-function regularized determinants of
Hilbert space operators and providing a unified approach to these determinants
as well as traces of resolvents.

The case of traces of resolvents (and resolvent differences), Fredholm
determinants, $\zeta$-functions, and $\zeta$-function regularized determinants
associated with linear operators in a Hilbert space is described in detail in
Section 2. In particular, under an appropriate trace class hypothesis on the
resolvent of a self-adjoint operator $S$ in ${\mathcal{H}}$ we define its
$\zeta$-function and derive a formula for it in terms of the self-adjoint
functional calculus and the resolvent of $S$. In the case of trace class
resolvent differences for a pair of self-adjoint operators $(S_{1},S_{2})$ in
${\mathcal{H}}$ we also describe the underlying relative spectral
$\zeta$-function for such a pair and relate the corresponding $\zeta$-function
regularized relative determinant and a symmetrized Fredholm perturbation
determinant.

In Section 3 we provide an exhaustive discussion of regular (three-
coefficient) self-adjoint Sturm–Liouville operators, that is, operators in
$L^{2}((a,b);rdx)$ associated with self-adjoint realizations of differential
expressions of the type $\tau=-r^{-1}(d/dx)p(d/dx)+q$, with arbitrary
(separated and coupled) self-adjoint boundary conditions on compact intervals
$[a,b]$. Their traces of resolvents and associated perturbation determinants
are calculated for all self-adjoint boundary conditions in terms of concrete
expressions involving a canonical system of fundamental solutions
$\phi(z,\,\cdot\,,a)$ and $\theta(z,\,\cdot\,,a)$ of $\tau\psi=z\psi$, and
special examples such as Floquet boundary conditions and the Krein–von Neumann
extension are highlighted. The $\zeta$-function regularized determinants are
determined for all self-adjoint boundary conditions and a variety of concrete
examples complete this section.

Our final Section 4 then illustrates some of the abstract notions in Section 2
with the help of self-adjoint Schrödinger operators associated with
differential expressions of the type $-(d^{2}/dx^{2})+q$ on the half-line
${\mathbb{R}}_{+}=(0,\infty)$ with short-range potentials $q$ (i.e., we treat
the scattering theory situation). Again we study all self-adjoint boundary
conditions at $x=0$. The assumed short-range nature of $q$ then necessitates a
comparison with the case $q=0$ and thus illustrates the case of relative
perturbation determinants, relative $\zeta$-functions, and relative
$\zeta$-function regularized determinants abstractly discussed in Section 2.

Finally, we summarize some of the basic notation used in this paper
(especially, in Section 2): Let ${\mathcal{H}}$ and ${\mathcal{K}}$ be
separable complex Hilbert spaces, $(\,\cdot\,,\,\cdot\,)_{{\mathcal{H}}}$ and
$(\,\cdot\,,\,\cdot\,)_{{\mathcal{K}}}$ the scalar products in ${\mathcal{H}}$
and ${\mathcal{K}}$ (linear in the second factor), and $I_{{\mathcal{H}}}$ and
$I_{{\mathcal{K}}}$ the identity operators in ${\mathcal{H}}$ and
${\mathcal{K}}$, respectively.

Next, let $T$ be a closed linear operator from
$\text{\rm{dom}}(T)\subseteq{\mathcal{H}}$ to
$\text{\rm{ran}}(T)\subseteq{\mathcal{K}}$, with $\text{\rm{dom}}(T)$ and
$\text{\rm{ran}}(T)$ denoting the domain and range of $T$. The closure of a
closable operator $S$ is denoted by $\overline{S}$. The kernel (null space) of
$T$ is denoted by $\ker(T)$.

The spectrum, point spectrum, and resolvent set of a closed linear operator in
${\mathcal{H}}$ will be denoted by $\sigma(\cdot)$, $\sigma_{p}(\cdot)$, and
$\rho(\cdot)$; the discrete spectrum of $T$ (i.e., points in $\sigma_{p}(T)$
which are isolated from the rest of $\sigma(T)$, and which are eigenvalues of
$T$ of finite algebraic multiplicity) is abbreviated by $\sigma_{d}(T)$. The
algebraic multiplicity $m_{a}(z_{0};T)$ of an eigenvalue
$z_{0}\in\sigma_{d}(T)$ is the dimension of the range of the corresponding
Riesz projection $P(z_{0};T)$,

$m_{a}(z_{0};T)=\dim(\text{\rm{ran}}(P(z_{0};T)))=\text{\rm{tr}}_{{\mathcal{H}}}(P(z_{0};T)),$
(1.1)

where (with the symbol $\ointctrclockwise$ denoting counterclockwise oriented
contour integrals)

$P(z_{0};T)=\frac{-1}{2\pi
i}\ointctrclockwise_{C(z_{0};\varepsilon)}d\zeta\,(T-\zeta
I_{{\mathcal{H}}})^{-1},$ (1.2)

for $0<\varepsilon<\varepsilon_{0}$ and
$D(z_{0};\varepsilon_{0})\backslash\\{z_{0}\\}\subset\rho(T)$; here
$D(z_{0};r_{0})\subset{\mathbb{C}}$ is the open disk with center $z_{0}$ and
radius $r_{0}>0$, and $C(z_{0};r_{0})=\partial D(z_{0};r_{0})$ the
corresponding circle. The geometric multiplicity $m_{g}(z_{0};T)$ of an
eigenvalue $z_{0}\in\sigma_{p}(T)$ is defined by

$m_{g}(z_{0};T)=\dim(\ker((T-z_{0}I_{{\mathcal{H}}}))).$ (1.3)

The essential spectrum of $T$ is defined by
$\sigma_{ess}(T)=\sigma(T)\backslash\sigma_{d}(T)$.

The Banach spaces of bounded and compact linear operators in ${\mathcal{H}}$
are denoted by ${\mathcal{B}}({\mathcal{H}})$ and
${\mathcal{B}}_{\infty}({\mathcal{H}})$, respectively. Similarly, the
Schatten–von Neumann (trace) ideals will subsequently be denoted by
${\mathcal{B}}_{p}({\mathcal{H}})$, $p\in[1,\infty)$, and the subspace of all
finite rank operators in ${\mathcal{B}}_{1}({\mathcal{H}})$ will be
abbreviated by ${\mathcal{F}}({\mathcal{H}})$. Analogous notation
${\mathcal{B}}({\mathcal{H}}_{1},{\mathcal{H}}_{2})$,
${\mathcal{B}}_{\infty}({\mathcal{H}}_{1},{\mathcal{H}}_{2})$, etc., will be
used for bounded, compact, etc., operators between two Hilbert spaces
${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$. In addition,
$\text{\rm{tr}}_{{\mathcal{H}}}(T)$ denotes the trace of a trace class
operator $T\in{\mathcal{B}}_{1}({\mathcal{H}})$,
${\det}_{{\mathcal{H}}}(I_{{\mathcal{H}}}-T)$ the Fredholm determinant of
$I_{{\mathcal{H}}}-T$.

Finally, we find it convenient to abbreviate
${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$.

##  2\. Traces, Fredholm Determinants, and Zeta Functions of Operators

In this section we recall some well-known formulas relating traces and
Fredholm determinants and also discuss the notion of $\zeta$-functions of
self-adjoint operators. For background on the material used in this section
see, for instance, [24], [25], [26, Ch. XIII], [27, Ch. IV], [36], [47, Ch.
17], [50], [51, Ch. 3].

To set the stage we start with densely defined, closed, linear operators $A$
in ${\mathcal{H}}$ having a trace class resolvent, and hence introduce the
following assumption:

######  Hypothesis 2.1.

Suppose that $A$ is densely defined and closed in ${\mathcal{H}}$ with
$\rho(A)\neq\emptyset$, and
$(A-zI_{{\mathcal{H}}})^{-1}\in{\mathcal{B}}_{1}({\mathcal{H}})$ for some
$($and hence for all $)$ $z\in\rho(A)$.

Given Hypothesis 2.1 and $z_{0}\in\rho(A)$, consider the bounded, entire
family $A(\,\cdot\,)$ defined by

$A(z):=I_{{\mathcal{H}}}-(A-zI_{{\mathcal{H}}})(A-z_{0}I_{{\mathcal{H}}})^{-1}=(z-z_{0})(A-z_{0}I_{{\mathcal{H}}})^{-1},\quad
z\in{\mathbb{C}}.$ (2.1)

Employing the formula (cf. [27, Sect. IV.1], see also [36], [56, Sect. I.7]),

$\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(I_{{\mathcal{H}}}-T(z))^{-1}T^{\prime}(z)\big{)}=-(d/dz)\text{\rm
ln}({\det}_{{\mathcal{H}}}(I_{{\mathcal{H}}}-T(z))),$ (2.2)

valid for a trace class-valued analytic family $T(\,\cdot\,)$ on an open set
$\Omega\subset{\mathbb{C}}$ such that
$(I_{{\mathcal{H}}}-T(\,\cdot\,))^{-1}\in{\mathcal{B}}({\mathcal{H}})$, and
applying it to the entire family $A(\,\cdot\,)$ then results in

$\displaystyle\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(A-zI_{{\mathcal{H}}})^{-1}\big{)}$
$\displaystyle=-(d/dz)\text{\rm
ln}\big{(}{\det}_{{\mathcal{H}}}\big{(}I_{{\mathcal{H}}}-(z-z_{0})(A-z_{0}I_{{\mathcal{H}}})^{-1}\big{)}\big{)}$
$\displaystyle=-(d/dz)\text{\rm
ln}\big{(}{\det}_{{\mathcal{H}}}\big{(}(A-zI_{{\mathcal{H}}})(A-z_{0}I_{{\mathcal{H}}})^{-1}\big{)}\big{)},$
(2.3) $\displaystyle\hskip 163.60333ptz\in\rho(A).$

One notes that the left- and hence the right-hand side of (2.3) is independent
of the choice of $z_{0}\in\rho(A)$.

Next, following the proof of [51, Theorem 3.5 (c)] step by step, and employing
a Weierstrass-type product formula (see, e.g., [51, Theorem 3.7]), yields the
subsequent result (see also [22]).

######  Lemma 2.2.

Assume Hypothesis 2.1 and let $\lambda_{k}\in\sigma(A)$ then

${\det}_{{\mathcal{H}}}\big{(}I_{{\mathcal{H}}}-(z-z_{0})(A-z_{0}I_{{\mathcal{H}}})^{-1}\big{)}\underset{z\to\lambda_{k}}{=}(\lambda_{k}-z)^{m_{a}(\lambda_{k})}[C_{k}+O(\lambda_{k}-z)],\quad
C_{k}\neq 0,$ (2.4)

that is, the multiplicity of the zero of the Fredholm determinant
${\det}_{{\mathcal{H}}}\big{(}I_{{\mathcal{H}}}-(z-z_{0})(A-z_{0}I_{{\mathcal{H}}})^{-1}\big{)}$
at $z=\lambda_{k}$ equals the algebraic multiplicity of the eigenvalue
$\lambda_{k}$ of $A$.

In addition, denote the spectrum of $A$ by
$\sigma(A)=\\{\lambda_{k}\\}_{k\in{\mathbb{N}}}$,
$\lambda_{k}\neq\lambda_{k^{\prime}}$ for $k\neq k^{\prime}$. Then

$\displaystyle\begin{split}{\det}_{{\mathcal{H}}}(I_{{\mathcal{H}}}-(z-z_{0})(A-z_{0}I_{{\mathcal{H}}})^{-1})&=\prod_{k\in{\mathbb{N}}}\big{[}1-(z-z_{0})(\lambda_{k}-z_{0})^{-1}\big{]}^{m_{a}(\lambda_{k})}\\\
&=\prod_{k\in{\mathbb{N}}}\bigg{(}\frac{\lambda_{k}-z}{\lambda_{k}-z_{0}}\bigg{)}^{m_{a}(\lambda_{k})},\end{split}$
(2.5)

with absolutely convergent products in (2.5).

In many cases of interest not a single resolvent, but a difference of two
resolvents is trace class and hence one is naturally led to the following
generalization discussed in detail in [16], [20], [23] (see also[19]). To
avoid technicalities, we will now consider the case of self-adjoint operators
$A,B$ below, but note that [16] considers the general case of densely defined,
closed linear operators with nonempty resolvent sets:

######  Hypothesis 2.3.

Suppose $A$ and $B$ are self-adjoint operators in ${\mathcal{H}}$ with $A$
bounded from below.
$(i)$ Assume that $B$ can be written as the form sum $($denoted by $+_{q}$$)$
of $A$ and a self-adjoint operator $W$ in ${\mathcal{H}}$

$B=A+_{q}W,$ (2.6)

where $W$ can be factorized into

$W=W_{1}^{*}W_{2},$ (2.7)

such that

$\text{\rm{dom}}(W_{j})\supseteq\text{\rm{dom}}\big{(}|A|^{1/2}\big{)},\quad
j=1,2.$ (2.8)

$(ii)$ Suppose that for some $($and hence for all $)$ $z\in\rho(A)$,

$W_{2}(A-zI_{{\mathcal{H}}})^{-1/2},\,\overline{(A-zI_{{\mathcal{H}}})^{-1/2}W_{1}^{*}}\in{\mathcal{B}}_{2}({\mathcal{H}}).$
(2.9)

Given Hypothesis 2.3, one observes that

$\text{\rm{dom}}\big{(}|B|^{1/2}\big{)}=\text{\rm{dom}}\big{(}|A|^{1/2}\big{)},$
(2.10)

and that the resolvent of $B$ can be written as (cf., e.g., the detailed
discussion in [16] and the references therein)

$\displaystyle(B-zI_{{\mathcal{H}}})^{-1}$
$\displaystyle=(A-zI_{{\mathcal{H}}})^{-1}$
$\displaystyle\quad-\overline{(A-zI_{{\mathcal{H}}})^{-1}W_{1}^{*}}\big{[}I_{{\mathcal{H}}}+\overline{W_{2}(A-zI_{{\mathcal{H}}})^{-1}W_{1}^{*}}\big{]}^{-1}W_{2}(A-zI_{{\mathcal{H}}})^{-1},$
$\displaystyle\hskip 190.63338ptz\in\rho(B)\cap\rho(A).$ (2.11)

We also note the analog of Tiktopoulos’ formula (cf. [49, p. 45]),

$\displaystyle\begin{split}(B-zI_{{\mathcal{H}}})^{-1}&=(A-zI_{{\mathcal{H}}})^{-1/2}\big{[}I_{{\mathcal{H}}}+\overline{(A-zI_{{\mathcal{H}}})^{-1/2}W(A-zI_{{\mathcal{H}}})^{-1/2}}\big{]}^{-1}\\\
&\quad\times(A-zI_{{\mathcal{H}}})^{-1/2},\quad
z\in\rho(B)\cap\rho(A).\end{split}$ (2.12)

Here the closures in (2.11) and (2.12) are well-defined employing (2.9). In
addition, one observes that the resolvent formulas (2.11) and (2.12) are
symmetric with respect to $A$ and $B$ employing $A=B-_{q}W$.

As a consequence, $B$ is bounded from below in ${\mathcal{H}}$ and one
concludes that for some $($and hence for all $)$ $z\in\rho(B)\cap\rho(A)$,

$\big{[}(B-zI_{{\mathcal{H}}})^{-1}-(A-zI_{{\mathcal{H}}})^{-1}\big{]}\in{\mathcal{B}}_{1}({\mathcal{H}}).$
(2.13)

Moreover, one infers that (cf. [23])

$\displaystyle{\text{\rm{tr}}}_{{\mathcal{H}}}\big{(}(B-zI_{{\mathcal{H}}})^{-1}-(A-zI_{{\mathcal{H}}})^{-1}\big{)}$
$\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{{\mathcal{H}}}\Big{(}\overline{(B-zI_{{\mathcal{H}}})^{1/2}(A-zI_{{\mathcal{H}}})^{-1}(B-zI_{{\mathcal{H}}})^{1/2}}\Big{)}\Big{)}$
(2.14) $\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{{\mathcal{H}}}\Big{(}I_{{\mathcal{H}}}+\overline{W_{2}(A-zI_{{\mathcal{H}}})^{-1}W_{1}^{*}}\Big{)}\Big{)},\quad
z\in\rho(B)\cap\rho(A).$ (2.15)

Here any choice of branch cut of the normal operator
$(B-zI_{{\mathcal{H}}})^{1/2}$ (employing the spectral theorem) is
permissible. The first equality in (2.15) follows as in the proof of [23,
Theorem 2.8]. (The details are actually a bit simpler now since $A,B$ are
self-adjoint and bounded from below, and hence of positive type after some
translation, which is the case considered in [23]). For completeness we
mention that the second equality in(2.15) can be arrived at as follows:
Employing the commutation formula (cf., [9]),

$C[I_{{\mathcal{H}}}-DC]^{-1}D=-I_{{\mathcal{H}}}+[I_{{\mathcal{H}}}-CD]^{-1}$
(2.16)

for $C,D\in{\mathcal{B}}({\mathcal{H}})$ with $1\in\rho(DC)$ (and hence
$1\in\rho(CD)$ since
$\sigma(CD)\backslash\\{0\\}=\sigma(DC)\backslash\\{0\\}$), the resolvent
identity (2.12) with $A$ and $B$ interchanged yields

$\displaystyle\overline{(B-zI_{{\mathcal{H}}})^{1/2}(A-zI_{{\mathcal{H}}})^{-1}(B-zI_{{\mathcal{H}}})^{1/2}}$
$\displaystyle\quad=\big{[}I_{{\mathcal{H}}}-\overline{(B-zI_{{\mathcal{H}}})^{-1/2}W(B-zI_{{\mathcal{H}}})^{-1/2}}\big{]}^{-1}$
$\displaystyle\quad=\Big{[}I_{{\mathcal{H}}}-\overline{\big{[}(B-zI_{{\mathcal{H}}})^{-1/2}W_{1}^{*}}\big{]}\big{[}W_{2}(B-zI_{{\mathcal{H}}})^{-1/2}\big{]}\Big{]}^{-1}.$
(2.17)

On the other hand, it is well-known (cf. [16], [31]) that

$\displaystyle
I_{{\mathcal{H}}}+\overline{W_{2}(A-zI_{{\mathcal{H}}})^{-1}W_{1}^{*}}$
$\displaystyle\quad=\big{[}I_{{\mathcal{H}}}-\overline{W_{2}(B-zI_{{\mathcal{H}}})^{-1}W_{1}^{*}}\big{]}^{-1}$
$\displaystyle\quad=\Big{[}I_{{\mathcal{H}}}-\big{[}W_{2}(B-zI_{{\mathcal{H}}})^{-1/2}\big{]}\overline{\big{[}(B-zI_{{\mathcal{H}}})^{-1/2}W_{1}^{*}\big{]}}\Big{]}^{-1},$
(2.18)

and hence using the fact

${\det}_{{\mathcal{H}}}(I_{{\mathcal{H}}}-ST)={\det}_{{\mathcal{H}}}(I_{{\mathcal{H}}}-TS)$
(2.19)

for $S,T\in{\mathcal{B}}({\mathcal{H}})$ with
$ST,TS\in{\mathcal{B}}_{1}({\mathcal{H}})$ (again, since
$\sigma(ST)\backslash\\{0\\}=\sigma(TS)\backslash\\{0\\}$) then proves the
second equality in (2.15).

While this approach based on Hypothesis 2.3 is tailor-made for a discussion of
perturbations of the potential coefficient in the context of Schrödinger and,
more generally, Sturm–Liouville operators, we also mention the following
variant that is best suited for changes in the boundary conditions:

######  Hypothesis 2.4.

Suppose that $A$ and $B$ are self-adjoint operators in ${\mathcal{H}}$ bounded
from below. In addition, assume that

$\displaystyle\text{\rm{dom}}\big{(}|A|^{1/2}\big{)}\subseteq\text{\rm{dom}}\big{(}|B|^{1/2}\big{)},$
(2.20)
$\displaystyle\begin{split}&\overline{(B-tI_{{\mathcal{H}}})^{1/2}\big{[}(B-tI_{{\mathcal{H}}})^{-1}-(A-tI_{{\mathcal{H}}})^{-1}\big{]}(B-tI_{{\mathcal{H}}})^{1/2}}\in{\mathcal{B}}_{1}({\mathcal{H}})\\\
&\quad\text{for some $t<\inf(\sigma(A)\cup\sigma(B))$.}\end{split}$ (2.21)

Given Hypothesis 2.4 it has been proven in [23] (actually, in a more general
context) that for
$z\in{\mathbb{C}}\backslash[\inf(\sigma(A)\cup\sigma(B)),\infty)$,

$\displaystyle\overline{(B-zI_{{\mathcal{H}}})^{1/2}\big{[}(B-zI_{{\mathcal{H}}})^{-1}-(A-zI_{{\mathcal{H}}})^{-1}\big{]}(B-zI_{{\mathcal{H}}})^{1/2}}$
$\displaystyle\quad=I_{{\mathcal{H}}}-\overline{(B-zI_{{\mathcal{H}}})^{1/2}(A-zI_{{\mathcal{H}}})^{-1}(B-zI_{{\mathcal{H}}})^{1/2}}$
(2.22)
$\displaystyle\quad=I_{{\mathcal{H}}}-\big{[}(B-zI_{{\mathcal{H}}})^{1/2}(A-zI_{{\mathcal{H}}})^{-1/2}\big{]}\big{[}(B-{\overline{z}}I_{{\mathcal{H}}})^{1/2}(A-{\overline{z}}I_{{\mathcal{H}}})^{-1/2}\big{]}^{*}$

implying that
${\det}_{{\mathcal{H}}}\big{(}\overline{(B-zI_{{\mathcal{H}}})^{1/2}(A-zI_{{\mathcal{H}}})^{-1}(B-zI_{{\mathcal{H}}})^{1/2}}\big{)}$
is well-defined. Moreover,

$\displaystyle{\text{\rm{tr}}}_{{\mathcal{H}}}\big{(}(B-zI_{{\mathcal{H}}})^{-1}-(A-zI_{{\mathcal{H}}})^{-1}\big{)}$
$\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{{\mathcal{H}}}\Big{(}\overline{(B-zI_{{\mathcal{H}}})^{1/2}(A-zI_{{\mathcal{H}}})^{-1}(B-zI_{{\mathcal{H}}})^{1/2}}\Big{)}\Big{)},$
(2.23) $\displaystyle\hskip
142.26378ptz\in{\mathbb{C}}\backslash[\inf(\sigma(A)\cup\sigma(B)),\infty).$

Next we briefly turn to spectral $\zeta$-functions of self-adjoint operators
$S$ with a trace class resolvent (and hence purely discrete spectrum).

######  Hypothesis 2.5.

Suppose $S$ is a self-adjoint operator in ${\mathcal{H}}$, bounded from below,
satisfying

$(S-zI_{{\mathcal{H}}})^{-1}\in{\mathcal{B}}_{1}({\mathcal{H}})$ (2.24)

for some $($and hence for all $)$ $z\in\rho(S)$. We denote the spectrum of $S$
by $\sigma(S)=\\{\lambda_{j}\\}_{j\in J}$ $($with $J\subset{\mathbb{Z}}$ an
appropriate index set $)$, with every eigenvalue repeated according to its
multiplicity. Moreover, suppose that $\sum_{j\in J,\,\lambda_{j}\neq
0}|\lambda_{j}|^{-t}<\infty$ for $t>0$ sufficiently large.

Given Hypothesis 2.5, the spectral zeta function of $S$ is then defined by

$\zeta(s;S):=\sum_{\begin{subarray}{c}j\in J\\\ \lambda_{j}\neq
0\end{subarray}}\lambda_{j}^{-s}$ (2.25)

for $\text{\rm Re}(s)>0$ sufficiently large such that (2.25) converges
absolutely.

Next, let $P(0;S)$ be the spectral projection of $S$ corresponding to the
eigenvalue $0$ and denote by $m(\lambda_{0};S)$ the multiplicity of the
eigenvalue $\lambda_{0}$ of $S$, in particular,

$m(0;S)=\dim(\ker(S)).$ (2.26)

(One recalls that since $S$ is self-adjoint, the geometric and algebraic
multiplicity of each eigenvalue of $S$ coincide and hence the subscript “$g$”
or “$a$” is simply omitted from $m(\,\cdot\,;S)$.) In addition, we introduce
the simple contour $\gamma$ encircling $\sigma(S)\backslash\\{0\\}$ in a
counterclockwise manner so as to dip under (and hence avoid) the point $0$
(cf. Figure 1). In fact, following [35] (see also [34]), we will henceforth
choose as the branch cut of $z^{-s}$ the ray

$R_{\theta}=\big{\\{}z=te^{i\theta}\big{|}t\in[0,\infty)\big{\\}}\quad\theta\in(\pi/2,\pi),$
(2.27)

and note that the contour $\gamma$ avoids any contact with $R_{\theta}$ (cf.
Figure 1).

The cut $R_{\theta}$ for $z^{-s}$$z$-plane$\gamma$ Figure 1. Contour $\gamma$
in the complex $z$-plane.

We note in passing that one could also use a semigroup approach via

$\displaystyle\begin{split}\zeta(s;S)&=\Gamma(s)^{-1}\int_{0}^{\infty}dt\,t^{s-1}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}e^{-tS}[I_{{\mathcal{H}}}-P(0;S)]\big{)},\\\
&=\Gamma(s)^{-1}\int_{0}^{\infty}dt\,t^{s-1}\big{[}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}e^{-tS}\big{)}-m(0;S)\big{]},\end{split}$
(2.28)

for $\text{\rm Re}(s)>0$ sufficiently large, but we prefer to work with
resolvents in this paper.

######  Lemma 2.6.

In addition to Hypothesis 2.5 and the counterclockwise oriented contour
$\gamma$ just described $($cf. Figure 1$)$, suppose that
$\big{|}{\text{\rm{tr}}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S)]\big{)}\big{|}$
is polynomially bounded on $\gamma$. Then

$\zeta(s;S)=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\big{[}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}\big{)}+z^{-1}m(0;S)\big{]}$
(2.29)

for $\text{\rm Re}(s)>0$ sufficiently large.

###### Proof.

Assuming $\text{\rm Re}(s)>0$ sufficiently large, a contour integration
argument yields

$\displaystyle\zeta(s;S)$ $\displaystyle=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S)]\big{)}$
$\displaystyle=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}+z^{-1}P(0;S)\big{)}$
(2.30) $\displaystyle=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\big{[}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}\big{)}+z^{-1}m(0;S)\big{]},$

taking into account that
$\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S)]\big{)}$
is meromorphic with poles precisely at the nonzero eigenvalues
$\lambda_{j}\neq 0$ of $S$, with residues given by (self-adjoint) spectral
projections of $S$ of rank equal to $m(\lambda_{j};S)$. ∎

It is very tempting to continue the computation leading to (2.29) and now
deform the contour $\gamma$ so as to “hug” the branch cut $R_{\theta}$, but
this requires the right asymptotic behavior of
$\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S)]\big{)}$
as $|z|\to\infty$ as well as $|z|\to 0$, and we will investigate this in the
context of relative $\zeta$-functions next.

In cases where $(S-zI_{{\mathcal{H}}})^{-1}$ is not trace class, but one is
dealing with a pair of operators $(S_{1},S_{2})$ such that the difference of
their resolvents lies in the trace class, (2.30) naturally leads to the notion
of a relative $\zeta$-function as follows. For pertinent background
information on this circle of ideas we refer, for instance, to Forman [13],
[14], Müller [42].

######  Hypothesis 2.7.

Suppose $S_{j}$, $j=1,2$, are self-adjoint operators in ${\mathcal{H}}$,
bounded from below, satisfying

$\big{[}(S_{2}-zI_{{\mathcal{H}}})^{-1}-(S_{1}-zI_{{\mathcal{H}}})^{-1}\big{]}\in{\mathcal{B}}_{1}({\mathcal{H}})$
(2.31)

for some $($and hence for all $)$ $z\in\rho(S_{1})\cap\rho(S_{2})$. In
addition, assume that $S_{j}$, $j=1,2$, have essential spectrum contained in
$(0,\infty)$, that is, for some $\lambda_{1}>0$,

$\sigma_{ess}(S_{j})\subseteq[\lambda_{1},\infty),\quad j=1,2.$ (2.32)

We note, in particular, the essential spectrum hypothesis (2.32) includes the
case of purely discrete spectra of $S_{j}$ (i.e.,
$\sigma_{ess}(S_{j})=\emptyset$, $j=1,2$.) Since $S_{j}$ were assumed to be
bounded from below, adding a suitable constant to $S_{j}$, $j=1,2$, will shift
their (essential) spectra accordingly.

Given Hypothesis 2.7, and again choosing a counterclockwise oriented simple
contour $\gamma$ that encircles $\sigma(S_{1})\cup\sigma(S_{2})$, however,
with the stipulation that $0$ does not lie in the interior of $\gamma$, and
$0\notin\gamma$ (cf. Figure 1), the relative spectral $\zeta$-function for the
pair $(S_{1},S_{2})$ is defined by

$\displaystyle\begin{split}&\zeta(s;S_{1},S_{2})=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S_{2}-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S_{2})]\\\
&\hskip
162.18062pt-(S_{1}-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S_{1})]\big{)}\end{split}$
(2.33)

for $\text{\rm Re}(s)>0$ sufficiently large, ensuring convergence of (2.33).

Employing the contour $\gamma$ and branch cut $R_{\theta}$ as in Figure 1, and
deforming $\gamma$ so it eventually surrounds $R_{\theta}$, one arrives at the
following result.

######  Theorem 2.8.

Suppose $S_{j}$, $j=1,2$, are self-adjoint operators in ${\mathcal{H}}$
satisfying Hypothesis 2.7 and that $($cf. (2.14) and (2.23)$)$

$\displaystyle{\text{\rm{tr}}}_{{\mathcal{H}}}\big{(}(S_{2}-zI_{{\mathcal{H}}})^{-1}-(S_{1}-zI_{{\mathcal{H}}})^{-1}\big{)}$
$\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{{\mathcal{H}}}\Big{(}\overline{(S_{2}-zI_{{\mathcal{H}}})^{1/2}(S_{1}-zI_{{\mathcal{H}}})^{-1}(S_{2}-zI_{{\mathcal{H}}})^{1/2}}\Big{)}\Big{)},$
(2.34) $\displaystyle\hskip
145.10922ptz\in{\mathbb{C}}\backslash[\inf(\sigma(S_{1})\cap\sigma(S_{2})),\infty).$

In addition, assume that for some $\varepsilon>0$,

$\displaystyle\begin{split}&\big{|}{\text{\rm{tr}}}_{{\mathcal{H}}}\big{(}(S_{2}-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S_{2})]-(S_{1}-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S_{1})]\big{)}\big{|}\\\
&\quad=\begin{cases}O\big{(}|z|^{-1-\varepsilon}\big{)},&\text{as
$|z|\to\infty$,}\\\ O(1),&\text{as $|z|\to 0$.}\end{cases}\end{split}$ (2.35)

Then, for $\text{\rm Re}(s)\in(-\varepsilon,1)$,

$\displaystyle\zeta(s;S_{1},S_{2})=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)\int_{0}^{\infty}dt\,t^{-s}$ $\displaystyle\quad\times\frac{d}{dt}\text{\rm
ln}\Big{(}(e^{i\theta}t)^{[m(0;S_{1})-m(0;S_{2})]}$ (2.36)
$\displaystyle\quad\times{\det}_{{\mathcal{H}}}\big{(}(S_{2}-e^{i\theta}tI_{{\mathcal{H}}})^{1/2}(S_{1}-e^{i\theta}tI_{{\mathcal{H}}})^{-1}(S_{2}-e^{i\theta}tI_{{\mathcal{H}}})^{1/2}\big{)}\Big{)}.$

###### Proof.

Due to hypothesis (2.35) one can deform the contour $\gamma$ so that it wraps
around the branch cut $R_{\theta}$,

$\displaystyle\zeta(s;S_{1},S_{2})=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S_{2}-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S_{2})]$
$\displaystyle\hskip
165.02606pt-(S_{1}-zI_{{\mathcal{H}}})^{-1}[I_{{\mathcal{H}}}-P(0;S_{1})]\big{)}$
$\displaystyle\quad=-(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\big{\\{}\text{\rm{tr}}_{{\mathcal{H}}}\big{(}(S_{2}-zI_{{\mathcal{H}}})^{-1}-(S_{1}-zI_{{\mathcal{H}}})^{-1}\big{)}$
$\displaystyle\hskip
108.12054pt+z^{-1}\left[m(0;S_{2})-m(0;S_{1})\right]\big{\\}}$
$\displaystyle\quad=(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\Big{\\{}\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{{\mathcal{H}}}\Big{(}(S_{2}-zI_{{\mathcal{H}}})^{1/2}(S_{1}-zI_{{\mathcal{H}}})^{-1}(S_{2}-zI_{{\mathcal{H}}})^{1/2}\Big{)}\Big{)}$
$\displaystyle\hskip 105.2751pt-z^{-1}m(0;S_{2})-m(0;S_{1})]\Big{\\}}$
$\displaystyle\quad=(2\pi
i)^{-1}\ointctrclockwise_{\gamma}dz\,z^{-s}\frac{d}{dz}\text{\rm
ln}\Big{(}z^{[m(0;S_{1})-m(0;S_{2})]}$ $\displaystyle\hskip
119.50148pt\times{\det}_{{\mathcal{H}}}\Big{(}\left(S_{2}-zI_{{\mathcal{H}}}\right)^{1/2}\left(S_{1}-zI_{{\mathcal{H}}}\right)^{-1}\left(S_{2}-zI_{{\mathcal{H}}}\right)^{1/2}\Big{)}\Big{)}$
$\displaystyle\quad=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\Big{(}\left(te^{i\theta}\right)^{m(0;S_{1})-m(0;S_{2})}$
$\displaystyle\qquad\times{\det}_{{\mathcal{H}}}\Big{(}\left(S_{2}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{1/2}\left(S_{1}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{-1}\left(S_{2}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{1/2}\Big{)}\Big{)}.$
(2.37)

Here we first applied $\text{\rm{tr}}_{{\mathcal{H}}}(P(0;S_{j}))=m(0;S_{j})$
(cf. also (2.33)) and then (2.15). Carefully paying attention to the phases
when shrinking the contour to the branch cut $R_{\theta}$, one obtains (2.36).
∎

######  Theorem 2.9.

Suppose $S_{j}$, $j=1,2,$ are self-adjoint operators in ${\mathcal{H}}$
satisfying Hypothesis 2.7 and (2.35). Then

$\displaystyle\begin{split}&\zeta^{\prime}(0;S_{1},S_{2})=i\pi(n_{2}-n_{1})-\lim_{t\downarrow
0}\left|\text{\rm ln}\Big{(}(e^{i\theta}t)^{[m(0;S_{1})-m(0;S_{2})]}\right.\\\
&\quad\left.\times{\det}_{{\mathcal{H}}}\big{(}(S_{2}-e^{i\theta}tI_{{\mathcal{H}}})^{1/2}(S_{1}-e^{i\theta}tI_{{\mathcal{H}}})^{-1}(S_{2}-e^{i\theta}tI_{{\mathcal{H}}})^{1/2}\big{)}\Big{)}\right|,\end{split}$
(2.38)

where $n_{j}$ is the number of negative eigenvalues of $S_{j}$, $j=1,2,$. If
$n_{j}=0$, $j=1,2,$ then

$\displaystyle\zeta^{\prime}(0;S_{1},S_{2})$ $\displaystyle=-\lim_{t\downarrow
0}\text{\rm ln}\Big{(}(e^{i\theta}t)^{[m(0;S_{1})-m(0;S_{2})]}$
$\displaystyle\hskip
48.36958pt\times{\det}_{{\mathcal{H}}}\big{(}(S_{2}-e^{i\theta}tI_{{\mathcal{H}}})^{1/2}(S_{1}-e^{i\theta}tI_{{\mathcal{H}}})^{-1}(S_{2}-e^{i\theta}tI_{{\mathcal{H}}})^{1/2}\big{)}\Big{)}$
$\displaystyle=-\text{\rm ln}(C_{0}),$ (2.39)

where

$\displaystyle\begin{split}&{\det}_{{\mathcal{H}}}\big{(}(S_{2}-zI_{{\mathcal{H}}})^{1/2}(S_{1}-zI_{{\mathcal{H}}})^{-1}(S_{2}-zI_{{\mathcal{H}}})^{1/2}\big{)}\\\
&\quad\underset{z\to 0}{=}z^{[m(0;S_{2})-m(0;S_{1})]}[C_{0}+O(z)],\quad
C_{0}>0.\end{split}$ (2.40)

###### Proof.

First we note that (2.36) implies

$\displaystyle\begin{split}&\zeta^{\prime}(0;S_{1},S_{2})=\int_{0}^{\infty}dt\,\,\frac{d}{dt}\text{\rm
ln}\Big{\\{}\left(te^{i\theta}\right)^{m(0;S_{1})-m(0;S_{2})}\\\
&\quad\times{\det}_{{\mathcal{H}}}\Big{(}\left(S_{2}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{1/2}\left(S_{1}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{-1}\left(S_{2}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{1/2}\Big{)}\Big{\\}}.\end{split}$
(2.41)

In computing this quantity, one notes that for $t\in[0,\infty)$, the graph of
the function

$\displaystyle\begin{split}G(t)&=\left(te^{i\theta}\right)^{m(0;S_{1})-m(0;S_{2})}\\\
&\quad\times{\det}_{{\mathcal{H}}}\Big{(}\left(S_{2}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{1/2}\left(S_{1}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{-1}\left(S_{2}-e^{i\theta}tI_{{\mathcal{H}}}\right)^{1/2}\Big{)}\end{split}$
(2.42)

can cross the branch cut $R_{\theta}$ at several $t$-values. Therefore, the
integral has to be split at these $t$-values and pursuant real and imaginary
parts have to be summed. The real part between consecutive segments cancels
except for the contributions from zero. This explains the real part of (2.38).
The resulting imaginary part is found as follows. The sum defining the
$\zeta$-function can be split into contributions from negative and positive
eigenvalues, namely,

$\zeta(s;S_{j})=\sum_{\begin{subarray}{c}\ell=1\\\
\lambda_{-\ell}<0\end{subarray}}^{n_{j}}\lambda_{-\ell}^{-s}+\sum_{\begin{subarray}{c}k\in
J\\\ \lambda_{k}>0\end{subarray}}\lambda_{k}^{-s}.$ (2.43)

For each negative eigenvalue one computes

$\left.\frac{d}{ds}\right|_{s=0}\lambda_{-\ell}^{-s}=-\text{\rm
ln}(\lambda_{-\ell})=i\pi-\text{\rm ln}(|\lambda_{-\ell}|).$ (2.44)

This yields the imaginary part in (2.38).

Since $0$ lies outside the essential spectra of $S_{j}$, $j=1,2$, (2.40)
follows, for instance, from [56, p. 271–272]. Given the relation (2.40), the
fact (2.39) follows from (2.41) and the Lebesgue dominated convergence
theorem. ∎

In the special case where $0\in\rho(S_{1})\cap\rho(S_{2})$ (i.e.,
$m(0;S_{1})=m(0;S_{2})=0$), one thus obtains

$e^{-\zeta^{\prime}(0;S_{1},S_{2})}={\det}_{{\mathcal{H}}}\big{(}S_{2}^{1/2}S_{1}^{-1}S_{2}^{1/2}\big{)}$
(2.45)

and hence the $\zeta$-function regularized relative determinant now equals the
symmetrized (Fredholm) perturbation determinant for the pair $(S_{1},S_{2})$.
Here any choice of branch cut of the self-adjoint operator $S_{2}^{1/2}$
(employing the spectral theorem) is permissible.

##  3\. Sturm–Liouville Operators on Bounded Intervals

To illustrate the material of Section 2 we now apply it to the case of self-
adjoint Sturm–Liouville operators on bounded intervals.

We start by recalling a convenient parametrization of all self-adjoint
extensions associated with a regular, symmetric, second-order differential
expression as discussed in detail, for instance, in [55, Theorem 13.15] and
[57, Theorem 10.4.3], and recorded in [8].

Throughout this section we make the following set of assumptions:

######  Hypothesis 3.1.

Suppose $p,q,r$ satisfy the following conditions:
$(i)$ $r>0$ a.e. on $(a,b)$, $r\in L^{1}((a,b);dx)$.
$(ii)$ $p>0$ a.e. on $(a,b)$, $1/p\in L^{1}((a,b);dx)$.
$(iii)$ $q\in L^{1}((a,b);dx)$, $q$ is real-valued a.e. on $(a,b)$.

Given Hypothesis 3.1, we take $\tau$ to be the Sturm–Liouville-type
differential expression defined by

$\tau=\frac{1}{r(x)}\left[-\frac{d}{dx}p(x)\frac{d}{dx}+q(x)\right]\,\text{
for a.e.~{}$x\in(a,b)$,}\quad-\infty<a<b<\infty,$ (3.1)

and note that $\tau$ is regular on $[a,b]$. In addition, the following
convenient notation for the _first quasi-derivative_ is introduced,

$y^{[1]}(x)=p(x)y^{\prime}(x)\,\text{ for a.e.\ $x\in(a,b)$, }\,y\in
AC([a,b]).$ (3.2)

Here $AC([a,b])$ denotes the set of absolutely continuous functions on
$[a,b]$. For notational convenience we will occasionally abbreviate
$L^{2}_{r}((a,b)):=L^{2}((a,b);rdx)$.

Given that $\tau$ is regular on $[a,b]$, the _maximal operator_ $H_{max}$ in
$L^{2}((a,b);rdx)$ associated with $\tau$ is defined by

$\displaystyle H_{max}f=\tau f,$ (3.3)
$\displaystyle\,f\in\text{\rm{dom}}(H_{max})=\big{\\{}g\in
L^{2}((a,b);rdx)\,\big{|}\,g,g^{[1]}\in AC([a,b]);\,\tau g\in
L^{2}((a,b);rdx)\big{\\}},$ while the _minimal operator_ $H_{min}$ in
$L^{2}((a,b);rdx)$ associated with $\tau$ is given by $\displaystyle
H_{min}f=\tau f,$ $\displaystyle\,f\in\text{\rm{dom}}(H_{min})=\big{\\{}g\in
L^{2}((a,b);rdx)\,\big{|}\,g,g^{[1]}\in AC([a,b]);$ (3.4) $\displaystyle\hskip
88.2037ptg(a)=g^{[1]}(a)=g(b)=g^{[1]}(b)=0;\,\tau g\in
L^{2}((a,b);rdx)\big{\\}}.$

One notes that the operator $H_{min}$ is symmetric and that

$H_{min}^{*}=H_{max},\quad H_{max}^{*}=H_{min},$ (3.5)

(cf. Weidmann [54, Theorem 13.8]). Next, we summarize material found, for
instance, in [55, Ch. 13] and [57, Sects. 10.3, 10.4] in which self-adjoint
extensions of the minimal operator $H_{min}$ are characterized.

######  Theorem 3.2.

$($See, [53, Satz 2.6], [55, Theorem 13.14], [57, Theorem 10.4.2].$)$ Assume
Hypothesis 3.1 and suppose that $\widetilde{H}$ is an extension of the minimal
operator $H_{min}$ defined in (3.4). Then the following hold:
$(i)$ $\widetilde{H}$ is a self-adjoint extension of $H_{min}$ if and only if
there exist $2\times 2$ matrices $A$ and $B$ with complex-valued entries
satisfying

$\text{\rm{rank}}(A\ \ B)=2,\quad AJA^{*}=BJB^{*},\quad
J=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},$ (3.6)

with

$\displaystyle\begin{split}&\widetilde{H}f=\tau f\\\
&\text{\rm{dom}}(\widetilde{H})=\left\\{g\in\text{\rm{dom}}(H_{max})\,\bigg{|}\,A\binom{g(a)}{g^{[1]}(a)}=B\binom{g(b)}{g^{[1]}(b)}\right\\}.\end{split}$
(3.7)

Henceforth, the self-adjoint extension $\widetilde{H}$ corresponding to the
matrices $A$ and $B$ will be denoted by $H_{A,B}$.
$(ii)$ For $z\in\rho(H_{A,B})$, the resolvent $H_{A,B}$ is of the form

$\begin{split}\left((H_{A,B}-zI_{L_{r}^{2}((a,b))})^{-1}f\right)(x)=\int_{a}^{b}r(x^{\prime})dx^{\prime}\,G_{A,B}(z,x,x^{\prime})f(x^{\prime}),\\\
f\in L^{2}((a,b);rdx),\end{split}$ (3.8)

where the Green’s function $G_{A,B}(z,x,x^{\prime})$ is of the form given by

$\hskip
10.0ptG_{A,B}(z,x,x^{\prime})=\begin{cases}\sum_{j,k=1}^{2}m_{j,k}^{+}(x)u_{j}(z,x)u_{k}(z,x^{\prime}),&a\leq
x^{\prime}\leq x\leq b,\\\\[2.0pt]
\sum_{j,k=1}^{2}m_{j,k}^{-}(x)u_{j}(z,x)u_{k}(z,x^{\prime}),&a\leq
x<x^{\prime}\leq b.\end{cases}$ (3.9)

Here $\\{u_{1}(z,\,\cdot\,),u_{2}(z,\,\cdot\,)\\}$ represents a fundamental
set of solutions for $(\tau-z)u=0$ and $m_{j,k}^{\pm}(z)$, $1\leq j,k\leq 2$,
are appropriate constants.
$(iii)$ $H_{A,B}$ has purely discrete spectrum with eigenvalues of
multiplicity at most $2$. Moreover, if
$\sigma(H_{A,B})=\\{\lambda_{A,B,j}\\}_{j\in{\mathbb{N}}}$, then

$\sum_{\begin{subarray}{c}j\in{\mathbb{N}}\\\ \lambda_{A,B,j}\neq
0\end{subarray}}|\lambda_{A,B,j}|^{-1}<\infty.$ (3.10)

In particular,

$(H_{A,B}-zI_{L_{r}^{2}((a,b))})^{-1}\in{\mathcal{B}}_{1}\big{(}L^{2}((a,b);rdx)\big{)},\quad
z\in\rho(H_{A,B}).$ (3.11)

The characterization of self-adjoint extensions of $H_{\text{min}}$ in terms
of pairs of matrices $(A,B)\in{\mathbb{C}}^{2\times
2}\times{\mathbb{C}}^{2\times 2}$ satisfying (3.6) is not unique in the sense
that different pairs may lead to the same self-adjoint extension. The next
result recalls a unique characterization for all self-adjoint extensions of
$H_{min}$ and hence can be viewed as a refinement of Theorem 3.2.

######  Theorem 3.3.

$($See, e.g., [55, Theorem 13.15], [57, , Theorem 10.4.3].$)$ Assume
Hypothesis 3.1. Let $H_{min}$ be the minimal operator associated with $\tau$
and defined in (3.4) and $H_{A,B}$ a self-adjoint extension of the minimal
operator as characterized in Theorem 3.2; then, the following hold:
$(i)$ $H_{A,B}$ is a self-adjoint extension of $H_{min}$, with
$\text{\rm{rank}}(A)=\text{\rm{rank}}(B)=1$ if and only if $A$ and $B$ can be
put in the form

$A=\begin{pmatrix}\cos(\alpha)&\sin(\alpha)\\\ 0&0\end{pmatrix},\quad
B=\begin{pmatrix}0&0\\\ -\cos(\beta)&\sin(\beta)\end{pmatrix},$ (3.12)

for a unique pair $\alpha,\beta\in[0,\pi)$. Hence, upon identifying $H_{A,B}$
with $H_{\alpha,\beta}$, all self-adjoint extensions of $H_{min}$ with
separated boundary conditions are of the form

$\displaystyle H_{\alpha,\beta}f=\tau f,\quad\alpha,\beta\in[0,\pi),$
$\displaystyle
f\in\text{\rm{dom}}(H_{\alpha,\beta})=\\{g\in\text{\rm{dom}}(H_{max})\,|\,\sin(\alpha)g^{[1]}(a)+\cos(\alpha)g(a)=0,$
(3.13) $\displaystyle\hskip
152.22241pt-\sin(\beta)g^{[1]}(b)+\cos(\beta)g(b)=0\\}.$

$(ii)$ $H_{A,B}$ is a self-adjoint extension of $H_{min}$ with
$\text{\rm{rank}}(A)=\text{\rm{rank}}(B)=2$ if and only if $A$ and $B$ can be
put in the form

$A=e^{i\varphi}R,\quad B=I_{2},$ (3.14)

for a unique $\varphi\in[0,2\pi)$, and unique
$R\in\operatorname{SL}_{2}(\mathbb{R})$. Hence, upon identifying $H_{A,B}$
with $H_{\varphi,R}$, all self-adjoint extensions of $H_{min}$ with coupled
boundary conditions are of the form

$\displaystyle\begin{split}&H_{\varphi,R}f=\tau
f,\quad\varphi\in[0,2\pi),\;R\in\operatorname{SL}_{2}(\mathbb{R}),\\\
&f\in\text{\rm{dom}}(H_{\varphi,R})=\left\\{g\in\text{\rm{dom}}(H_{max})\,\bigg{|}\,\binom{g(b)}{g^{[1]}(b)}=e^{i\varphi}R\binom{g(a)}{g^{[1]}(a)}\right\\}.\end{split}$
(3.15)

$(iii)$ All self-adjoint extensions of $H_{min}$ are either of type $(i)$
$($i.e., separated $)$ or of type $(ii)$ $($i.e., coupled $)$.

For notational convenience we will adhere to the notation $H_{\alpha,\beta}$
and $H_{\varphi,R}$ in the following.

Next, we recall some results of [22]. For this purpose we introduce the
fundamental system of solutions $\theta(z,x,a)$, $\phi(z,x,a)$ of $\tau y=zy$
defined by

$\theta(z,a,a)=\phi^{[1]}(z,a,a)=1,\quad\theta^{[1]}(z,a,a)=\phi(z,a,a)=0,$
(3.16)

such that

$W(\theta(z,\,\cdot\,,a),\phi(z,\,\cdot\,,a))=1,$ (3.17)

where, for $f,g$ (locally) absolutely continuous,

$W(f,g)(\,\cdot\,)=f(\,\cdot\,)g^{[1]}(\,\cdot\,)-f^{[1]}(\,\cdot\,)g(\,\cdot\,)\big{)}.$
(3.18)

Furthermore, we introduce the boundary values for $g,\,g^{[1]}\in AC([a,b])$,
see [44, Sect. 1.2], [57, Sect. 3.2],

$\displaystyle\begin{split}U_{\alpha,\beta,1}(g)&=\sin(\alpha)g^{[1]}(a)+\cos(\alpha)g(a),\\\
U_{\alpha,\beta,2}(g)&=-\sin(\beta)g^{[1]}(b)+\cos(\beta)g(b),\end{split}$
(3.19)

in the case $(i)$ of separated boundary conditions in Theorem 3.3, and

$\displaystyle\begin{split}V_{\varphi,R,1}(g)&=g(b)-e^{i\varphi}R_{1,1}g(a)-e^{i\varphi}R_{1,2}g^{[1]}(a),\\\
V_{\varphi,R,2}(g)&=g^{[1]}(b)-e^{i\varphi}R_{2,1}g(a)-e^{i\varphi}R_{2,2}g^{[1]}(a),\end{split}$
(3.20)

in the case $(ii)$ of coupled boundary conditions in Theorem 3.3. Moreover, we
define

$\displaystyle
F_{\alpha,\beta}(z)=\det\begin{pmatrix}U_{\alpha,\beta,1}(\theta(z,\,\cdot\,,a))&U_{\alpha,\beta,1}(\phi(z,\,\cdot\,,a))\\\\[2.84526pt]
U_{\alpha,\beta,2}(\theta(z,\,\cdot\,,a))&U_{\alpha,\beta,2}(\phi(z,\,\cdot\,,a))\end{pmatrix},$
(3.21)

and

$\displaystyle
F_{\varphi,R}(z)=\det\begin{pmatrix}V_{\varphi,R,1}(\theta(z,\,\cdot\,,a))&V_{\varphi,R,1}(\phi(z,\,\cdot\,,a))\\\\[2.84526pt]
V_{\varphi,R,2}(\theta(z,\,\cdot\,,a))&V_{\varphi,R,2}(\phi(z,\,\cdot\,,a))\end{pmatrix},$
(3.22)

and note, in particular,

$\displaystyle F_{\alpha,\beta}(z)$ (3.23)
$\displaystyle\quad=\begin{cases}\phi(z,b,a),&\alpha=\beta=0,\\\\[2.84526pt]
-\sin(\beta)\phi^{[1]}(z,b,a)+\cos(\beta)\phi(z,b,a),&\alpha=0,\,\beta\in(0,\pi),\\\\[2.84526pt]
\cos(\alpha)\phi(z,b,a)-\sin(\alpha)\theta(z,b,a),&\alpha\in(0,\pi),\,\beta=0,\\\\[2.84526pt]
\cos(\alpha)[-\sin(\beta)\phi^{[1]}(z,b,a)+\cos(\beta)\phi(z,b,a)]\\\\[1.42262pt]
-\sin(\alpha)[-\sin(\beta)\theta^{[1]}(z,b,a)+\cos(\beta)\theta(z,b,a)],&\alpha,\beta\in(0,\pi).\end{cases}$

Given these preparations we can state our first result concerning the
computation of traces and determinants.

######  Theorem 3.4.

Assume Hypothesis 3.1 and denote by $H_{\alpha,\beta}$ and $H_{\varphi,R}$ the
self-adjoint extensions of $H_{min}$ as described in cases $(i)$ and $(ii)$ of
Theorem 3.3, respectively.
$(i)$ Suppose $z_{0}\in\rho(H_{\alpha,\beta})$, then

$\displaystyle\begin{split}&{\det}_{L^{2}_{r}((a,b))}\big{(}I_{L^{2}_{r}((a,b))}-(z-z_{0})(H_{\alpha,\beta}-z_{0}I_{L^{2}_{r}((a,b))})^{-1}\big{)}\\\
&\quad=F_{\alpha,\beta}(z)/F_{\alpha,\beta}(z_{0}),\quad
z\in{\mathbb{C}}.\end{split}$ (3.24)

In particular,

$\text{\rm{tr}}_{L^{2}_{r}((a,b))}\big{(}(H_{\alpha,\beta}-zI_{L^{2}_{r}((a,b))})^{-1}\big{)}=-(d/dz)\text{\rm
ln}(F_{\alpha,\beta}(z)),\quad z\in\rho(H_{\alpha,\beta}).$ (3.25)

$(ii)$ Suppose $z_{0}\in\rho(H_{\varphi,R})$, then

$\displaystyle\begin{split}&{\det}_{L^{2}_{r}((a,b))}\big{(}I_{L^{2}_{r}((a,b))}-(z-z_{0})(H_{\varphi,R}-z_{0}I_{L^{2}_{r}((a,b))})^{-1}\big{)}\\\
&\quad=F_{\varphi,R}(z)/F_{\varphi,R}(z_{0}),\quad
z\in{\mathbb{C}}.\end{split}$ (3.26)

In particular,

$\text{\rm{tr}}_{L^{2}_{r}((a,b))}\big{(}(H_{\varphi,R}-zI_{L^{2}_{r}((a,b))})^{-1}\big{)}=-(d/dz)\text{\rm
ln}(F_{\varphi,R}(z)),\quad z\in\rho(H_{\varphi,R}).$ (3.27)

###### Proof.

In the special case $p=r=1$ and given separated (but generally, non-self-
adjoint) boundary conditions, the fact (3.24) was proved in [22] upon
combining the eigenvalue results in [44, Sect. 1.2] and Lemma 2.2. The proof
in [22] extends to the present situation with $p,r$ satisfying Hypothesis 3.1
in the cases $(i)$ and $(ii)$ since actually the eigenvalues $\lambda_{A,B,j}$
of $H_{A,B}$ all have the universal leading Weyl-type asymptotics [28, Sect.
VI.7] (see also [2], [57, Theorem 4.3.1])

$\lim_{j\to\infty}j^{-2}\lambda_{A,B,j}=\pi^{2}\bigg{(}\int_{a}^{b}dx\,[r(x)/p(x)]^{1/2}\bigg{)}^{-2},$
(3.28)

independently of the chosen boundary conditions, improving upon relation
(3.10). More precisely, [28, Sect. VI.7] determines the leading asymptotic
behavior of the eigenvalue counting function (independently of the underlying
choice of boundary conditions) which is known to yield (3.28).

Relations (3.25) and (3.27) are then clear from combining formula (2.3) with
(3.24) and (3.26), respectively. ∎

######  Remark 3.5.

Considering traces of resolvent differences for various boundary conditions
(separated and/or coupled) permits one to make a direct connection with the
boundary data maps studied in [7], [8], and [23]. More precisely, suppose the
pairs $A,B\in{\mathbb{C}}^{2\times 2}$ and
$A^{\prime},B^{\prime}\in{\mathbb{C}}^{2\times 2}$ satisfy (3.6) and define
$H_{A,B}$ and $H_{A^{\prime},B^{\prime}}$ according to (3.7). Then

$\displaystyle\text{\rm{tr}}_{L^{2}((a,b);rdx)}\big{(}(H_{A^{\prime},B^{\prime}}-zI_{L_{r}^{2}((a,b))})^{-1}-(H_{A,B}-zI_{L_{r}^{2}((a,b))})^{-1}\big{)}$
$\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{{\mathbb{C}}^{2}}\Big{(}\Lambda_{A,B}^{A^{\prime},B^{\prime}}(z)\Big{)}\Big{)},\quad
z\in\rho(H_{A,B})\cap\rho(H_{A^{\prime},B^{\prime}}),$ (3.29)

where the $2\times 2$ matrix $\Lambda_{A,B}^{A^{\prime},B^{\prime}}(z)$
represents the boundary data map associated to the pair
$(H_{A,B},H_{A^{\prime},B^{\prime}})$. A comparison of (3.29) with (3.25) and
(3.27) yields

${\det}_{{\mathbb{C}}^{2}}\Big{(}\Lambda_{A,B}^{A^{\prime},B^{\prime}}(z)\Big{)}=C_{0}\,F_{A^{\prime},B^{\prime}}(z)/F_{A,B}(z),\quad
z\in\rho(H_{A,B})\cap\rho(H_{A^{\prime},B^{\prime}}),$ (3.30)

where, in obvious notation, $F_{A^{\prime},B^{\prime}}(z),F_{A,B}(z)$
represent either $F_{\alpha,\beta}(z)$ or $F_{\varphi,R}(z)$, depending on
which boundary conditions (separated or coupled) are represented by the pairs
$(A^{\prime},B^{\prime})$, $(A,B)$, and
$C_{0}=C_{0}(A^{\prime},B^{\prime},A,B)$ is a $z$-independent constant.
Indeed, $C_{0}=1$ for separated as well as coupled boundary conditions. The
separated case was shown in [23], Lemma 3.4, using the explicit matrix
representation of $\Lambda_{AB}^{A^{\prime}B^{\prime}}(z)$ for that case, and
the result for coupled boundary conditions follows along the same lines.
$\diamond$

For the case of determinants of general higher-order differential operators
(with matrix-valued coefficients) and general boundary conditions on compact
intervals we refer to Burghelea, Friedlander, and Kappeler [4], Dreyfus and
Dym [11], Falco, Fedorenko, and Gruzberg [12], and Forman [13, Sect. 3]. We
also refer to [40] for a closed form of an infinite product of ratios of
eigenvalues of Sturm–Liouville operators on compact intervals.

We briefly look at two prominent examples associated with coupled boundary
conditions next:

######  Example 3.6.

$($Floquet boundary conditions. $)$ Consider the family of operators
$H_{\varphi,I_{2}}$, familiar from Floquet theory, defined by taking
$R=B=I_{2}$ in (3.15). In this case

$F_{\varphi,I_{2}}(z)=-2e^{i\varphi}[\Delta(z)-\cos(\varphi)],\quad
z\in{\mathbb{C}},$ (3.31)

where

$\Delta(z)=\big{[}\theta(z,b,a)+\phi^{[1]}(z,b,a)\big{]}\big{/}2,\quad
z\in{\mathbb{C}},$ (3.32)

represents the well-known Floquet discriminant and hence

$\displaystyle{\det}_{L^{2}_{r}((a,b))}\big{(}I_{L^{2}_{r}((a,b))}-(z-z_{0})(H_{\varphi,I_{2}}-z_{0}I_{L^{2}_{r}((a,b))})^{-1}\big{)}$
$\displaystyle\quad=[\Delta(z)-\cos(\varphi)]/[\Delta(z_{0})-\cos(\varphi)],\quad
z\in{\mathbb{C}},$ (3.33)
$\displaystyle\text{\rm{tr}}_{L^{2}_{r}((a,b))}\big{(}(H_{\varphi,I_{2}}-zI_{L^{2}_{r}((a,b))})^{-1}\big{)}$
$\displaystyle\quad=-(d/dz)\text{\rm ln}([\Delta(z)-\cos(\varphi)]),\quad
z\in\rho(H_{\varphi,I_{2}}).$ (3.34)

######  Example 3.7.

$($The Krein–von Neumann extension. $)$ Consider the case $\varphi=0$,
$A=R_{K}$, with

$R_{K}=\begin{pmatrix}\theta(0,b,a)&\phi(0,b,a)\\\\[2.84526pt]
\theta^{[1]}(0,b,a)&\phi^{[1]}(0,b,a)\end{pmatrix}.$ (3.35)

As shown in [8, Example 3.3], the resulting operator $H_{0,R_{K}}$ represents
the Krein–von Neumann extension of $H_{min}$. Thus,

$F_{0,R_{K}}(z)=-2[D_{K}(z)-1],\quad z\in{\mathbb{C}},$ (3.36)

where

$\displaystyle\begin{split}D_{K}(z)&=\big{[}\phi^{[1]}(0,b,a)\theta(z,b,a)+\theta(0,b,a)\phi^{[1]}(z,b,a)-\phi(0,b,a)\theta^{[1]}(z,b,a)\\\
&\quad-\theta^{[1]}(0,b,a)\phi(z,b,a)\big{]}\big{/}2,\quad
z\in{\mathbb{C}}.\end{split}$ (3.37)

Hence,

$\displaystyle{\det}_{L^{2}_{r}((a,b))}\big{(}I_{L^{2}_{r}((a,b))}-(z-z_{0})(H_{0,R_{K}}-z_{0}I_{L^{2}_{r}((a,b))})^{-1}\big{)}$
$\displaystyle\quad=[D_{K}(z)-1]/[D_{K}(z_{0})-1],\quad z\in{\mathbb{C}},$
(3.38)
$\displaystyle\text{\rm{tr}}_{L^{2}_{r}((a,b))}\big{(}(H_{0,R_{K}}-zI_{L^{2}_{r}((a,b))})^{-1}\big{)}$
$\displaystyle\quad=-(d/dz)\text{\rm ln}(D_{K}(z)-1),\quad
z\in\rho(H_{0,R_{K}}).$ (3.39)

Because of the Wronskian relation (3.17), $D_{K}(0)=1$, furthermore

$D_{K}(z)-1\underset{z\to 0}{=}z^{2}[c+O(z)],$ (3.40)

where

$\displaystyle\begin{split}c&=\frac{1}{2}\Big{(}\overset{\textbf{\Large.}}{\phi}(0,b,a)\,\big{[}\overset{\textbf{\Large.}}{\theta}\big{]}^{[1]}(0,b,a)-\big{[}\overset{\textbf{\Large.}}{\phi}\big{]}^{[1]}(0,b,a)\,\overset{\textbf{\Large.}}{\theta}(0,b,a)\Big{)}\\\
&=\frac{1}{2}W\Big{(}\overset{\textbf{\Large.}}{\phi}(0,\,\cdot\,,a),\overset{\textbf{\Large.}}{\theta}(0,\,\cdot\,,a)\Big{)}(b),\end{split}$
(3.41)

abbreviating $\overset{\textbf{\Large.}}{}=d/dz$.

The small-$z$ behavior (3.40) is clear from general results in [44, Sect. 1.2]
or [57, Sect. 3.2] and is in accordance with a two-dimensional nullspace of
the Krein–von Neumann extension on a compact interval. To actually show that
$c\neq 0$ $($in fact, $c<0$$)$ in (3.40) one recalls that
$\overset{\textbf{\Large.}}{\phi}(z,x,a)$ and
$\overset{\textbf{\Large.}}{\theta}(z,x,a)$ satisfy an inhomogeneous
Sturm–Liouville equation and as a result one obtains $($ for
$z\in{\mathbb{C}}$, $x\in[a,b]$$)$

$\displaystyle\begin{split}\overset{\textbf{\Large.}}{\theta}(z,x,a)&=\theta(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(z,x^{\prime},a)\theta(z,x^{\prime},a)\\\
&\quad-\phi(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\theta(z,x^{\prime},a)^{2},\\\
\big{[}\overset{\textbf{\Large.}}{\theta}\big{]}^{[1]}(z,x,a)&=\theta^{[1]}(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(z,x^{\prime},a)\theta(z,x^{\prime},a)\\\
&\quad-\phi^{[1]}(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\theta(z,x^{\prime},a)^{2},\\\
\overset{\textbf{\Large.}}{\phi}(z,x,a)&=\theta(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(z,x^{\prime},a)^{2}\\\
&\quad-\phi(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(z,x^{\prime},a)\theta(z,x^{\prime},a),\\\
\big{[}\overset{\textbf{\Large.}}{\phi}\big{]}^{[1]}(z,x,a)&=\theta^{[1]}(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(z,x^{\prime},a)^{2}\\\
&\quad-\phi^{[1]}(z,x,a)\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(z,x^{\prime},a)\theta(z,x^{\prime},a).\end{split}$
(3.42)

Equations (3.42) and Cauchy’s inequality then imply

$\displaystyle
W\Big{(}\overset{\textbf{\Large.}}{\phi}(\lambda,\,\cdot\,,a),\overset{\textbf{\Large.}}{\theta}(\lambda,\,\cdot\,,a)\Big{)}(x)=\bigg{[}\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(\lambda,x^{\prime},a)\theta(\lambda,x^{\prime},a)\bigg{]}^{2}$
(3.43)
$\displaystyle\quad-\bigg{[}\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\theta(\lambda,x^{\prime},a)^{2}\bigg{]}\bigg{[}\int_{a}^{x}r(x^{\prime})dx^{\prime}\,\phi(\lambda,x^{\prime},a)^{2}\bigg{]}\leq
0,\quad\lambda\in{\mathbb{R}},\;x\in[a,b],$

using the fact that $\phi(\lambda,x,a)$ and $\theta(\lambda,x,a)$ are real-
valued for $\lambda\in{\mathbb{R}}$, $x\in[a,b]$. Equality in Cauchy’s
inequality for $x>a$ would imply that for some $\alpha,\beta\in[0,\infty)$,
$(\alpha,\beta)\neq(0,0)$,

$\alpha\,\phi(\lambda,x^{\prime},a)=\beta\,\theta(\lambda,x^{\prime},a),\quad
x^{\prime}\in(a,x],$ (3.44)

a contradiction. Thus,
$W\Big{(}\overset{\textbf{\Large.}}{\phi}(\lambda,\,\cdot\,,a),\overset{\textbf{\Large.}}{\theta}(\lambda,\,\cdot\,,a)\Big{)}(x)<0$
for $\lambda\in{\mathbb{R}}$, $x\in(a,b]$ and hence $c<0$ in (3.40), (3.41).

In the case of separated boundary conditions, that is, case $(i)$ in Theorem
3.3, one can shed more light on $F_{\alpha,\beta}(z)$ in terms of appropriate
Weyl solutions $\psi_{-}(z,\,\cdot\,,a,\alpha)$ and
$\psi_{+}(z,\,\cdot\,,a,\beta)$ that satisfy the boundary conditions in
$\text{\rm{dom}}(H_{\alpha,\beta})$ in (3.3) at $a$ and $b$, respectively. Up
to normalizations, $\psi_{\pm}(z,\,\cdot\,,a,\alpha)$ are given by

$\displaystyle\psi_{-}(z,\,\cdot\,,a,\alpha)=c_{-}\begin{cases}\cos(\alpha)\phi(z,\,\cdot\,,a)-\sin(\alpha)\theta(z,\,\cdot\,,a),\quad\alpha\in(0,\pi),\\\
\phi(z,\,\cdot\,,a),\quad\alpha=0,\end{cases}$ (3.45)
$\displaystyle\psi_{+}(z,\,\cdot\,,a,\beta)=c_{+}\begin{cases}\big{\\{}\big{[}-\sin(\beta)\theta^{[1]}(z,b,a)+\cos(\beta)\theta(z,b,a)\big{]}\phi(z,\,\cdot\,,a)\\\
+\big{[}\sin(\beta)\phi^{[1]}(z,b,a)-\cos(\beta)\phi(z,b,a)\big{]}\theta(z,\,\cdot\,,a)\big{\\}},\\\
\hskip 169.29381pt\beta\in(0,\pi),\\\
\theta(z,b,a)\phi(z,\,\cdot\,,a)-\phi(z,b,a)\theta(z,\,\cdot\,,a),\quad\beta=0,\end{cases}$
(3.46)

and hence the Green’s function of $H_{\alpha,\beta}$ is of the semi-separable
form,

$\displaystyle G_{\alpha,\beta}(z,x,x^{\prime})$
$\displaystyle=(H_{\alpha,\beta}-zI_{L_{r}^{2}((a,b))})^{-1}(x,x^{\prime})$
(3.47)
$\displaystyle=\frac{1}{W(\psi_{+}(z,\,\cdot\,,a,\beta),\psi_{-}(z,\,\cdot\,,a,\alpha))}$
$\displaystyle\quad\times\begin{cases}\psi_{-}(z,x,a,\alpha)\psi_{+}(z,x^{\prime},a,\beta),&a\leq
x\leq x^{\prime}\leq b,\\\
\psi_{-}(z,x^{\prime},a,\alpha)\psi_{+}(z,x,a,\beta),&a\leq x^{\prime}\leq
x\leq b.\end{cases}$

A direct computation then reveals the following connection between
$F_{\alpha,\beta}(z)$, $\psi_{-}(z,\,\cdot\,,a,\alpha)$, and
$\psi_{+}(z,\,\cdot\,,a,\beta)$,

$\displaystyle
F_{\alpha,\beta}(z)=\frac{1}{W(\psi_{+}(z,\,\cdot\,,a,\beta),\psi_{-}(z,\,\cdot\,,a,\alpha))}$
(3.48)
$\displaystyle\quad\times\begin{cases}\psi_{-}(z,b,a,0)\psi_{+}(z,a,a,0),\quad\alpha=\beta=0,\\\\[2.84526pt]
\big{[}-\sin(\beta)\psi^{[1]}_{-}(z,b,a,0)+\cos(\beta)\psi_{-}(z,b,a,0)\big{]}\psi_{+}(z,a,a,\beta),\\\\[1.42262pt]
\hskip 180.67517pt\alpha=0,\,\beta\in(0,\pi),\\\\[2.84526pt]
\psi_{-}(z,b,a,\alpha)\big{[}\sin(\alpha)\psi^{[1]}_{+}(z,a,a,0)+\cos(\alpha)\psi_{+}(z,a,a,0)\big{]},\\\\[1.42262pt]
\hskip 172.13925pt\alpha\in(0,\pi),\,\beta=0,\\\\[2.84526pt]
\big{[}-\sin(\beta)\psi^{[1]}_{-}(z,b,a,\alpha)+\cos(\beta)\psi_{-}(z,b,a,\alpha)\big{]}\\\\[1.42262pt]
\times\big{[}\sin(\alpha)\psi^{[1]}_{+}(z,a,a,\beta)+\cos(\alpha)\psi_{+}(z,a,a,\beta)\big{]},\quad\alpha,\beta\in(0,\pi).\end{cases}$

Combining (3.48) and (3.25) thus yields

$\displaystyle\text{\rm{tr}}_{L^{2}_{r}((a,b))}\big{(}(H_{\alpha,\beta}-zI_{L^{2}_{r}((a,b))})^{-1}\big{)}$
$\displaystyle\quad=\begin{cases}-(d/dz)\text{\rm
ln}\big{(}\psi_{+}(z,a,a,0)\big{/}\psi^{[1]}_{+}(z,b,a,0)\big{)}\\\\[1.42262pt]
\quad=-(d/dz)\text{\rm
ln}\big{(}\psi_{-}(z,b,a,0)\big{/}\psi^{[1]}_{-}(z,a,a,0)\big{)},&\alpha=\beta=0,\\\\[2.84526pt]
-(d/dz)\text{\rm
ln}(\psi_{+}(z,a,a,\beta)/\psi_{+}(z,b,a,\beta)),&\alpha=0,\,\beta\in(0,\pi),\\\\[2.84526pt]
-(d/dz)\text{\rm
ln}(\psi_{-}(z,b,a,\alpha)/\psi_{-}(z,a,a,\alpha)),&\alpha\in(0,\pi),\,\beta=0,\\\\[2.84526pt]
-(d/dz)\text{\rm
ln}\Big{(}\frac{W(\psi_{+}(z,\,\cdot\,,a,\beta),\psi_{-}(z,\,\cdot\,,a,\alpha))}{\psi_{+}(z,b,a,\beta)\psi_{-}(z,a,a,\alpha)}\Big{)},&\alpha,\beta\in(0,\pi),\end{cases}$
(3.49) $\displaystyle\hskip 238.14941ptz\in\rho(H_{\alpha,\beta}).$

Next, applying Theorem 3.4 in the context of Theorem 2.9 immediately yields
results about $\zeta$-regularized determinants (see also [15], [32, Chs. 2,3],
[38]).

######  Remark 3.8.

In Example 3.19 we will consider a simple case with negative eigenvalues
present. Otherwise, in the examples of this section we will always assume that
eigenvalues are non-negative. If that is not the case, an appropriate
imaginary part according to Theorem 2.9 has to be included. $\diamond$

To deal with $\zeta$-regularized determinants we now slightly strengthen
Hypothesis 3.1 and hence introduce the following assumptions on $p,q,r$:

######  Hypothesis 3.9.

Suppose $p,q,r$ satisfy the following conditions:
$(i)$ $r>0$ a.e. on $(a,b)$, $r\in L^{1}((a,b);dx)$, $1/r\in
L^{\infty}((a,b))$.
$(ii)$ $p>0$ a.e. on $(a,b)$, $1/p\in L^{1}((a,b);dx)$.
$(iii)$ $q\in L^{1}((a,b);dx)$, $q$ is real-valued a.e. on $(a,b)$.
$(iv)$ $p\,r$ and $(p\,r)^{\prime}/r$ are absolutely continuous on $[a,b]$.

The substitutions (cf. [41, p. 2])

$v(x)=\frac{1}{c}\int_{a}^{x}dt\,[r(t)/p(t)]^{1/2},\quad
u(x)=[r(x)p(x)]^{1/4}y(x),$ (3.50)

where $c$ is given by

$c=\int_{a}^{b}dt\,[r(t)/p(t)]^{1/2},$ (3.51)

such that $v\in[0,1]$, transforms the Sturm–Liouville problem $(\tau
y)(x)=zy(x)$ into

$-u^{\prime\prime}(v)+V(v)u(v)=c^{2}zu(v),$ (3.52)

with ($\mu(v)=[r(x)p(x)]^{1/4}$)

$\displaystyle V(v)$ $\displaystyle=$
$\displaystyle\frac{\mu^{\prime\prime}(v)}{\mu(v)}+c^{2}\frac{q(x)}{r(x)}$
(3.53)
$\displaystyle=-\frac{c^{2}}{16}\,\,\frac{1}{r(x)p(x)}\,\,\frac{(r(x)p(x))^{\prime}}{r(x)}+\frac{c^{2}}{4}\,\,\frac{1}{r(x)}\frac{d}{dx}\frac{(r(x)p(x))^{\prime}}{r(x)}+c^{2}\frac{q(x)}{r(x)}.$

Hypothesis 3.9 guarantees111We are indebted to C. Bennewitz [3] for a very
helpful discussion of this issue. that $V\in L^{1}((0,1);dx)$, and as a
consequence one has asymptotically,

$\displaystyle\begin{split}\phi(z,x,a)\underset{|z|\to\infty}{=}&2^{-1}z^{-1/2}[p(x)p(a)r(x)r(a)]^{-1/4}\\\
&\times\exp\bigg{(}z^{1/2}\int_{a}^{x}dt\,[r(t)/p(t)]^{1/2}\bigg{)}\big{(}1+O\big{(}|z|^{-1/2}\big{)}\big{)},\\\
\phi^{[1]}(z,x,a)\underset{|z|\to\infty}{=}&2^{-1}[p(a)r(a)]^{-1/4}[p(x)r(x)]^{1/4}\\\
&\times\exp\bigg{(}z^{1/2}\int_{a}^{x}dt\,[r(t)/p(t)]^{1/2}\bigg{)}\big{(}1+O\big{(}|z|^{-1/2}\big{)}\big{)},\\\
\theta(z,x,a)\underset{|z|\to\infty}{=}&2^{-1}[r(x)/p(x)]^{1/2}[p(x)p(a)r(x)r(a)]^{-1/4}\\\
&\times\exp\bigg{(}z^{1/2}\int_{a}^{x}dt\,[r(t)/p(t)]^{1/2}\bigg{)}\big{(}1+O\big{(}|z|^{-1/2}\big{)}\big{)},\\\
\theta^{[1]}(z,x,a)\underset{|z|\to\infty}{=}&2^{-1}z^{1/2}r(x)[p(x)p(a)r(x)r(a)]^{-1/4}\\\
&\times\exp\bigg{(}z^{1/2}\int_{a}^{x}dt\,[r(t)/p(t)]^{1/2}\bigg{)}\big{(}1+O\big{(}|z|^{-1/2}\big{)}\big{)}.\end{split}$
(3.54)

This asymptotic behavior is used to guarantee that assumption (2.35) is
satisfied in several of the following examples.

######  Theorem 3.10.

Assume Hypothesis 3.9 and denote by $H_{\alpha,\beta,j}$ and
$H_{\varphi,R,j}$, $j=1,2$, the self-adjoint extensions of $H_{min}$ as
described in Theorem 3.3 $(i)$ and $(ii)$, respectively. Here the index $j$
refers to a potential $q_{j}$ in (3.1), $j=1,2$. Then the following items
$(i)$ and $(ii)$ hold:
$(i)$

$\zeta^{\prime}\left(0;H_{\alpha,\beta,1},H_{\alpha,\beta,2}\right)=-\lim_{t\downarrow
0}\text{\rm
ln}\bigg{(}\left|\frac{F_{\alpha,\beta,2}(te^{i\theta})}{F_{\alpha,\beta,1}(te^{i\theta})}\,\,(te^{i\theta})^{m(0,H_{\alpha,\beta,1})-m(0,H_{\alpha,\beta,2})}\right|\bigg{)}.$
(3.55)

$(ii)$

$\zeta^{\prime}\left(0;H_{\varphi,R,1},H_{\varphi,R,2}\right)=-\lim_{t\downarrow
0}\text{\rm
ln}\bigg{(}\left|\frac{F_{\varphi,R,2}(te^{i\theta})}{F_{\varphi,R,1}(te^{i\theta})}\,\,(te^{i\theta})^{m(0,H_{\varphi,R,1})-m(0,H_{\varphi,R,2})}\right|\bigg{)}.$
(3.56)

###### Proof.

This follows immediately from Theorem 2.8 and Theorem 2.9 applied to
$S_{j}=H_{\alpha,\beta,j}$, respectively, $S_{j}=H_{\varphi,R,j}$, $j=1,2$. ∎

######  Remark 3.11.

In the absence of zero eigenvalues of $H_{\alpha,\beta,j}$, respectively,
$H_{\varphi,R,j}$, $j=1,2$, these results simplify to

$\zeta^{\prime}\left(0;H_{\alpha,\beta,1},H_{\alpha,\beta,2}\right)=\text{\rm
ln}\bigg{(}\left|\frac{F_{\alpha,\beta,1}(0)}{F_{\alpha,\beta,2}(0)}\right|\bigg{)},$
(3.57)

respectively,

$\zeta^{\prime}\left(0;H_{\varphi,R,1},H_{\varphi,R,2}\right)=\text{\rm
ln}\bigg{(}\left|\frac{F_{\varphi,R,1}(0)}{F_{\varphi,R,2}(0)}\right|\bigg{)}.$
(3.58)

In case there are zero modes, a suitable energy shift will again lead to this
case. $\diamond$

######  Example 3.12.

If one of the potentials vanishes, say $q_{1}=0$, more explicit results can be
obtained. We consider separated boundary conditions with no zero modes present
for $j=1,2$. Then

$\displaystyle\begin{split}F_{\alpha,\beta,j}(0)&=\cos(\alpha)\left[-\sin(\beta)\phi_{j}^{[1]}(0,b,a)+\cos(\beta)\phi_{j}(0,b,a)\right]\\\
&\quad-\sin(\alpha)\left[-\sin(\beta)\theta_{j}^{[1]}(0,b,a)+\cos(\beta)\theta_{j}(0,b,a)\right],\quad
j=1,2.\end{split}$ (3.59)

For $q_{1}=0$, $\theta_{1}(0,x,a)$ and $\phi_{1}(0,x,a)$ satisfy the initial
value problems

$\displaystyle\begin{split}-\bigg{(}\frac{d}{dx}p(x)\frac{d}{dx}\bigg{)}\theta_{1}(0,x,a)=0,\quad\quad\theta_{1}(0,a,a)=1,\quad\theta_{1}^{[1]}(0,a,a)=0,\\\
\
-\bigg{(}\frac{d}{dx}p(x)\frac{d}{dx}\bigg{)}\phi_{1}(0,x,a)=0,\quad\quad\phi_{1}(0,a,a)=0,\quad\phi_{1}^{[1]}(0,a,a)=1,\end{split}$
(3.60)

solutions of which are

$\phi_{1}(0,x,a)=\int_{a}^{x}dt\,p(t)^{-1},\quad\theta_{1}(0,x,a)=1.$ (3.61)

Then one can show that

$F_{\alpha,\beta,1}(0)=-\sin(\alpha+\beta)+\cos(\alpha)\cos(\beta)\int_{a}^{b}dt\,p(t)^{-1}.$
(3.62)

The relative $\zeta$-regularized determinant in this case then reads

$\zeta^{\prime}\left(0;H_{\alpha,\beta,1},H_{\alpha,\beta,2}\right)=\text{\rm
ln}\Bigg{(}\left|\frac{\cos(\alpha)\cos(\beta)\int_{a}^{b}dt\,p(t)^{-1}-\sin(\alpha+\beta)}{F_{\alpha,\beta,2}(0)}\right|\Bigg{)}.$
(3.63)

######  Example 3.13.

Restricting Example 3.12 to the case $p(x)=r(x)=1$, the $\zeta$-determinant
$\zeta^{\prime}(0;H_{\alpha,\beta,2})$ can be computed explicitly. First, one
notes that under Hypothesis 3.9 the $\zeta$-function can be analytically
continued to a neighborhood of $s=0$ and
$\zeta^{\prime}(0;H_{\alpha,\beta,2})$ is a well-defined quantity; similarly,
$\zeta^{\prime}(0;H_{\alpha,\beta,1})$ is well-defined for $q_{1}=0$.
Computing $\zeta^{\prime}(0;H_{\alpha,\beta,1})$, employing (3.63), one
obtains $\zeta^{\prime}(0;H_{\alpha,\beta,2})$.

First we assume there are no zero modes for $j=1$, which is the case if

$(b-a)\cos(\alpha)\cos(\beta)-\sin(\alpha+\beta)\neq 0.$ (3.64)

Next, one notes that

$\phi_{1}(z,x,a)=z^{-1/2}\sin\big{(}z^{1/2}(x-a)\big{)},\quad\theta_{1}(z,x,a)=\cos\big{(}z^{1/2}(x-a)\big{)},$
(3.65)

and thus from (3.23),

$\displaystyle F_{\alpha,\beta,1}(z)$
$\displaystyle=\cos(\alpha)\big{[}-\sin(\beta)\cos\big{(}z^{1/2}(b-a)\big{)}+z^{-1/2}\cos(\beta)\sin\big{(}z^{1/2}(b-a)\big{)}\big{]}$
$\displaystyle\quad-\sin(\alpha)\big{[}z^{1/2}\sin(\beta)\sin\big{(}z^{1/2}(b-a)\big{)}+\cos(\beta)\cos\big{(}z^{1/2}(b-a)\big{)}\big{]}.$
(3.66)

The details of what follows depend on the boundary conditions imposed. In this
example, we consider $\alpha,\beta\in(0,\pi)$. Along the relevant contour, as
$|z|\to\infty$, one has $\text{\rm Im}\big{(}z^{1/2}\big{)}>0$ and the
asymptotics for the boundary conditions considered reads

$F_{\alpha,\beta,1}(z)=F_{\alpha,\beta,1}^{asym}(z)\big{[}1+O\big{(}z^{-1/2}\big{)}\big{]},$
(3.67)

where

$F_{\alpha,\beta,1}^{asym}(z)=-(i/2)z^{1/2}\sin(\alpha)\sin(\beta)e^{-iz^{1/2}(b-a)}.$
(3.68)

Adding and subtracting this asymptotics where applicable, we rewrite the
$\zeta$-function for $H_{\alpha,\beta,1}$ in the form

$\displaystyle\zeta(s;H_{\alpha,\beta,1})$
$\displaystyle=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{0}^{1}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\big{(}F_{\alpha,\beta,1}\left(te^{i\theta}\right)\big{)}$
$\displaystyle\quad+e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{1}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\bigg{(}\frac{F_{\alpha,\beta,1}\left(te^{i\theta}\right)}{F_{\alpha,\beta,1}^{asym}\left(te^{i\theta}\right)}\bigg{)}$
$\displaystyle\quad+\zeta^{asym}(s;H_{\alpha,\beta,1}),$ (3.69)

where

$\zeta^{asym}(s;H_{\alpha,\beta,1})=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{1}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\big{(}F_{\alpha,\beta,1}^{asym}\left(te^{i\theta}\right)\big{)}.$ (3.70)

This representation is valid for $\frac{1}{2}<\text{\rm Re}(s)<1$. The last
term is easily computed and yields

$\zeta^{asym}(s;H_{\alpha,\beta,1})=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\left[\frac{1}{2s}-\frac{i}{2}e^{i\theta/2}(b-a)\frac{1}{s-(1/2)}\right],$
(3.71)

yielding its analytic continuation to $-\frac{1}{2}<\text{\rm Re}(s)<1$. The
$\zeta$-determinant for $H_{\alpha,\beta,1}$ then follows from

$\displaystyle\zeta^{\prime}(0;H_{\alpha,\beta,1})$ $\displaystyle=\text{\rm
Re}\left(\text{\rm
ln}F_{\alpha,\beta,1}^{asym}\left(e^{i\theta}\right)-\text{\rm
ln}F_{\alpha,\beta,1}(0)+\zeta^{asym\,\,\prime}(0;H_{\alpha,\beta,1})\right)$
$\displaystyle=-\text{\rm
ln}\bigg{(}\left|\frac{2\left[\cos(\alpha)\,\cos(\beta)\,\,(b-a)-\sin(\alpha+\beta)\right]}{\sin(\alpha)\sin(\beta)}\right|\bigg{)}.$
(3.72)

Using expression (3.72) in (3.63) yields

$\zeta^{\prime}(0;H_{\alpha,\beta,2})=\text{\rm
ln}\bigg{(}\left|\frac{\sin(\alpha)\sin(\beta)}{2F_{\alpha,\beta,2}(0)}\right|\bigg{)}.$
(3.73)

If there is a zero mode for $j=1$, namely, if

$(b-a)\cos(\alpha)\cos(\beta)-\sin(\alpha+\beta)=0,$ (3.74)

the relevant formula for the relative $\zeta$-determinant is

$\zeta^{\prime}(0;H_{\alpha,\beta,1},H_{\alpha,\beta,2})=\text{\rm
ln}\bigg{(}\left|\frac{F_{\alpha,\beta,1}^{\prime}(0)}{F_{\alpha,\beta,2}(0)}\right|\bigg{)}.$
(3.75)

Employing (3.66), this can be cast in the form

$\zeta^{\prime}(0;H_{\alpha,\beta,1},H_{\alpha,\beta,2})=\text{\rm
ln}\bigg{(}\left|\frac{(b-a)\left[(b-a)\sin(\alpha+\beta)-3\sin(\alpha)\sin\beta\right]}{3F_{\alpha,\beta,2}(0)}\right|\bigg{)}.$
(3.76)

In order to compute $\zeta^{\prime}(0;H_{\alpha,\beta,2})$, the part
$\zeta^{\prime}(0;H_{\alpha,\beta,1})$ can be computed as before, the only
difference being that $F_{\alpha,\beta,1}(z)$ is replaced by
$F_{\alpha,\beta,1}(z)/z$. The final answer then reads

$\zeta^{\prime}(0;H_{\alpha,\beta,1})=-\text{\rm
ln}\bigg{(}\left|2(b-a)\left[1-\frac{(b-a)\sin(\alpha+\beta)}{3\sin(\alpha)\sin\beta}\right]\right|\bigg{)}.$
(3.77)

$($This confirms the known result in the case of Neumann boundary conditions
at $x=a$ and $x=b$, that is, for $\alpha=\beta=\pi/2$.$)$ From (3.77) it is
immediate that

$\zeta^{\prime}(0;H_{\alpha,\beta,2})=\text{\rm
ln}\bigg{(}\left|\frac{\sin(\alpha)\sin(\beta)}{2F_{\alpha,\beta,2}(0)}\right|\bigg{)}.$
(3.78)

######  Example 3.14.

Next, consider the Dirichlet boundary condition at $x=a$ and a Robin boundary
condition at $x=b$, that is, $\alpha=0$ and $\beta\in(0,\pi)$. In the absence
of zero modes one then has

$\zeta^{\prime}(0;H_{0,\beta,1},H_{0,\beta,2})=\text{\rm
ln}\bigg{(}\left|\frac{F_{0,\beta,1}(0)}{F_{0,\beta,2}(0)}\right|\bigg{)}.$
(3.79)

In order to find the $\zeta$-determinant $\zeta^{\prime}(0;H_{0,\beta,2})$ we
consider as before $q_{1}=0$. Then,

$F_{0,\beta,1}(z)=-\sin(\beta)\cos\big{(}z^{1/2}(b-a)\big{)}+z^{-1/2}\cos\big{(}\beta)\sin(z^{1/2}(b-a)\big{)}.$
(3.80)

The relevant asymptotic large-$|z|$ behavior is

$F_{0,\beta,1}^{asym}(z)=-\sin(\beta)e^{-iz^{1/2}(b-a)}/2,$ (3.81)

and proceeding along the lines of previous computations, one finds

$\zeta^{\prime}(0;H_{0,\beta,1})=\text{\rm
ln}\bigg{(}\left|\frac{\sin(\beta)}{2\left[\sin(\beta)-(b-a)\cos(\beta)\right]}\right|\bigg{)}.$
(3.82)

$($Again, this confirms the known case where $\beta=\pi/2$$)$. From (3.82) it
is immediate that

$\zeta^{\prime}(0;H_{0,\beta,2})=\text{\rm
ln}\bigg{(}\left|\frac{\sin(\beta)}{2F_{0,\beta,2}(0)}\right|\bigg{)}.$ (3.83)

######  Example 3.15.

For Dirichlet boundary conditions at both endpoints, that is, for
$\alpha=\beta=0$, the relative zeta-determinant follows from

$\zeta^{\prime}(0;H_{0,0,1},H_{0,0,2})=\text{\rm
ln}\bigg{(}\left|\frac{F_{0,0,1}(0)}{F_{0,0,2}(0)}\right|\bigg{)}.$ (3.84)

For $q_{1}=0$, from

$F_{0,0,1}(z)=z^{-1/2}\sin\big{(}z^{1/2}(b-a)\big{)},$ (3.85)

the relevant asymptotics is

$F_{0,0,1}^{asym}(z)=(i/2)z^{-1/2}e^{-iz^{1/2}(b-a)}.$ (3.86)

Thus, one finds

$\zeta^{\prime}(0;H_{0,0,1})=-\text{\rm ln}(2(b-a)),$ (3.87)

and hence,

$\zeta^{\prime}(0;H_{0,0,2})=-\text{\rm ln}(|2F_{0,0,2}(0)|).$ (3.88)

######  Remark 3.16.

Under the assumptions of Example 3.13 no additional computations are needed
when considering certain relative determinants for different boundary
conditions. Indeed, for $\alpha_{j},\beta_{j}\in(0,\pi)$, $j=1,2$, eq. (3.73)
is valid replacing $\alpha,\beta\to\alpha_{1},\beta_{1}$ and
$F_{\alpha,\beta,2}\to F_{\alpha_{2},\beta_{2},2}$, and eq. (3.83) is valid
replacing $\beta\to\beta_{1}$, and $F_{0,\beta,2}\to F_{0,\beta_{2},2}$.
$\diamond$

######  Example 3.17.

As an example for coupled boundary conditions we reconsider the Krein–von
Neumann extension, Example 3.7. We first note that different potentials
$q_{1}\neq q_{2}$ lead to different Krein–von Neumann extensions
$R_{K_{1}}\neq R_{K_{2}}$; see, for instance, (3.35). Nevertheless, under the
assumptions made, Theorem 3.10 $(ii)$ remains valid and

$\zeta^{\prime}(0;H_{0,R_{K_{1}},1},H_{0,R_{K_{2}},2})=\lim_{z\to 0}\text{\rm
ln}\bigg{(}\left|\frac{F_{0,R_{K_{1}},1}(z)}{F_{0,R_{K_{2}},2}(z)}\right|\bigg{)},$
(3.89)

and from (3.40) one finds

$\zeta^{\prime}(0;H_{0,R_{K_{1}},1},H_{0,R_{K_{2}},2})=\text{\rm
ln}(|c_{1}/c_{2}|),$ (3.90)

with

$c_{j}=\frac{1}{2}\left(\overset{\textbf{\Large.}}{\theta}_{j}^{[1]}(0,b,a)\,\,\overset{\textbf{\Large.}}{\phi}_{j}(0,b,a)-\overset{\textbf{\Large.}}{\theta}_{j}(0,b,a)\overset{\textbf{\Large.}}{\phi}_{j}^{[1]}(0,b,a)\right),\quad
j=1,2.$ (3.91)

For the case of vanishing potential, $q_{1}=0$, the constant $c_{1}$ can be
determined explicitly. To this end we need the small-$z$ expansion of the
solutions of

$-\bigg{(}\frac{1}{r(x)}\frac{d}{dx}p(x)\frac{d}{dx}\bigg{)}\theta_{1}(z,x,a)=z\theta_{1}(z,x,a),\quad\theta_{1}(z,a,a)=1,\,\,\theta_{1}^{[1]}(z,a,a)=0,$
(3.92)

and

$-\bigg{(}\frac{1}{r(x)}\frac{d}{dx}p(x)\frac{d}{dx}\bigg{)}\phi_{1}(z,x,a)=z\phi_{1}(z,x,a),\quad\phi_{1}(z,a,a)=0,\,\,\phi_{1}^{[1]}(z,a,a)=1.$
(3.93)

Expanding

$\theta_{1}(z,x,a)=\theta_{1}(0,x,a)+z\overset{\textbf{\Large.}}{\theta}_{1}(0,x,a)+O\big{(}z^{2}\big{)},$
(3.94)

one compares $O(z)$-terms in (3.92) to find with (3.61)

$\bigg{(}\frac{d}{dx}p(x)\frac{d}{dx}\bigg{)}\overset{\textbf{\Large.}}{\theta}_{1}(0,x,a)=-r(x).$
(3.95)

Integrating, this yields

$\overset{\textbf{\Large.}}{\theta}_{1}^{[1]}(0,x,a)-\overset{\textbf{\Large.}}{\theta}_{1}^{[1]}(0,a,a)=-\int_{a}^{x}du\,r(u),$
(3.96)

but given $\theta_{1}^{[1]}(z,a,a)=0$, one concludes that

$\overset{\textbf{\Large.}}{\theta}_{1}^{[1]}(0,x,a)=-\int_{a}^{x}du\,r(u).$
(3.97)

Similarly, integrating (3.97),

$\overset{\textbf{\Large.}}{\theta}_{1}(0,b,a)=-\int_{a}^{b}dv\,p(v)^{-1}\int_{a}^{v}du\,r(u).$
(3.98)

Proceeding in the same way for $\phi_{1}(z,x,a)$, one first shows

$\bigg{(}\frac{d}{dx}p(x)\frac{d}{dx}\bigg{)}\overset{\textbf{\Large.}}{\phi}_{1}(0,x,a)=-r(x)\int_{a}^{x}du\,p(u)^{-1},$
(3.99)

and thus

$\overset{\textbf{\Large.}}{\phi}_{1}^{[1]}(0,x,a)=-\int_{a}^{x}dv\,r(v)\int_{a}^{v}du\,p(u)^{-1}.$
(3.100)

Furthermore,

$\overset{\textbf{\Large.}}{\phi}_{1}(0,x,a)=-\int_{a}^{x}dw\,p(w)^{-1}\int_{a}^{w}dv\,r(v)\int_{a}^{v}du\,p(u)^{-1}.$
(3.101)

Altogether this yields

$\displaystyle\begin{split}c_{1}&=\frac{1}{2}\left[\left(\int_{a}^{b}dt\,r(t)\right)\left(\int_{a}^{b}dw\int_{a}^{w}dv\int_{a}^{v}du\,\,\frac{r(v)}{p(w)p(u)}\right)\right.\\\
&\quad\left.-\left(\int_{a}^{b}dv\int_{a}^{v}du\,\,\frac{r(u)}{p(v)}\right)\left(\int_{a}^{b}dw\int_{a}^{w}dt\,\frac{r(w)}{p(t)}\right)\right].\end{split}$
(3.102)

######  Example 3.18.

For the particular case $r(x)=p(x)=1$, we now recompute the
$\zeta$-determinant $\zeta^{\prime}(0;H_{0,R_{K_{2}},2})$ by choosing
$q_{1}=0$. First, from (3.102) one finds

$c_{1}=-(b-a)^{4}/24.$ (3.103)

From Example 3.7 one determines

$F_{0,R_{K_{1}},1}(z)=2\big{[}1-\cos\big{(}z^{1/2}(b-a)\big{)}\big{]}-(b-a)z^{1/2}\sin\big{(}z^{1/2}(b-a)\big{)}.$
(3.104)

The relevant leading asymptotics reads

$F_{0,R_{K_{1}},1}^{asym}(z)=-i(b-a)z^{1/2}e^{-iz^{1/2}(b-a)}\big{/}2.$
(3.105)

The $zeta$-function for the Krein–von Neumann extension is therefore analyzed
using

$\displaystyle\zeta(s;H_{0,R_{K_{1}},1})$
$\displaystyle=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\bigg{(}\frac{F_{0,R_{K_{1}},1}(te^{i\theta})}{t^{2}e^{2i\theta}}\bigg{)}$
$\displaystyle=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{0}^{1}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\left(\frac{F_{0,R_{K_{1}},1}(te^{i\theta})}{t^{2}e^{2i\theta}}\right)$
$\displaystyle\quad+e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{1}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\left(\frac{F_{0,R_{K_{1}},1}(te^{i\theta})}{F_{0,R_{K_{1}},1}^{asym}(te^{i\theta})}\right)$
$\displaystyle\quad+\zeta^{asym}(s;H_{0,R_{K_{1}},1}),$ (3.106)

where

$\displaystyle\zeta^{asym}(s;H_{0,R_{K_{1}},1})$
$\displaystyle=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{1}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\left(\frac{F_{0,R_{K_{1}},1}^{asym}(te^{i\theta})}{t^{2}e^{2i\theta}}\right)$
$\displaystyle=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\left(-\frac{3}{2s}-\frac{i}{2}(b-a)e^{i\theta/2}\frac{1}{s-(1/2)}\right),$
(3.107)

such that

${\zeta^{asym}}^{\prime}(0;H_{0,R_{K_{1}},1})=i(b-a)e^{i\theta/2}+[3i(\theta-\pi)/2].$
(3.108)

From (3.106) we then find

$\zeta^{\prime}(0;H_{0,R_{K_{1}},1})=-\text{\rm ln}\big{(}(b-a)^{3}/6\big{)},$
(3.109)

in agreement with [43]. Finally, this proves

$\zeta^{\prime}(0;H_{0,R_{K_{2}},2})=\zeta^{\prime}(0;H_{0,R_{K_{1}},1})+\text{\rm
ln}(|c_{1}/c_{2}|)=\text{\rm
ln}\bigg{(}\left|\frac{b-a}{4c_{2}}\right|\bigg{)}.$ (3.110)

######  Example 3.19.

As our final example we consider a case where negative eigenvalues are
present. Let $x\in(0,\pi)$ and $\tau_{j}=-(d^{2}/dx^{2})-m_{j}^{2}$ with
$m_{j}\in(n_{j},n_{j}+1)$, $n_{j}\in{\mathbb{N}}$, $j=1,2$. Imposing Dirichlet
boundary conditions at both endpoints, the eigenvalues are
$\lambda_{\ell}^{(j)}=\ell^{2}-m_{j}^{2}$, $\ell\in{\mathbb{N}}$, such that
there are $n_{j}$ negative eigenvalues for $H_{0,0,j}$. The $\zeta$-function
representation for each $j=1,2$, can be found following Examples 3.13 and
3.15. We note that

$\displaystyle F_{0,0,j}(z)$
$\displaystyle=\big{(}z+m_{j}^{2}\big{)}^{-1/2}\sin\Big{(}\big{(}z+m_{j}^{2}\big{)}^{1/2}\pi\Big{)}$
$\displaystyle=\big{(}z+m_{j}^{2}\big{)}^{-1/2}(i/2)e^{-i(z+m_{j}^{2})^{1/2}\pi}\left(1-e^{2\pi
i(z+m_{j}^{2})^{1/2}}\right)$
$\displaystyle=:F_{0,0,j}^{asym}(z)\left(1-e^{2\pi
i(z+m_{j}^{2})^{1/2}}\right),$ (3.111)

and hence,

$\zeta(s;H_{0,0,j})=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\bigg{(}\frac{F_{0,0,j}\left(te^{i\theta}\right)}{F_{0,0,j}^{asym}\left(te^{i\theta}\right)}\bigg{)}+\zeta^{asym}(s;H_{0,0,j}),$
(3.112)

where

$\zeta^{asym}(s;H_{0,0,j})=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\big{(}F_{0,0,j}^{asym}\big{(}te^{i\theta}\big{)}\big{)},$ (3.113)

this representation being valid for $(1/2)<\text{\rm Re}(s)<1$. From [29,
3.193, 3.194] one infers

$\displaystyle\int_{0}^{\infty}dt\,\,\frac{t^{-s}}{(t+M)^{1/2}}=\frac{M^{(1/2)-s}\Gamma(1-s)\Gamma\left(s-\frac{1}{2}\right)}{\pi^{1/2}},$
(3.114)
$\displaystyle\int_{0}^{\infty}dt\,\,\frac{t^{-s}}{t+M}=\frac{\pi}{M^{s}\sin(\pi
s)},$ (3.115)

and hence one obtains

$\zeta^{asym}(s;H_{0,0,j})=-\frac{e^{is\pi}}{2m_{j}^{2s}}\left[\frac{i\pi^{1/2}m_{j}\Gamma\left(s-(1/2)\right)}{\Gamma(s)}+1\right],$
(3.116)

implying

$\zeta^{asym\prime}(0;H_{0,0,j})=-i(\pi/2)+im_{j}\pi+\text{\rm ln}(m_{j}).$
(3.117)

It then follows that

$\zeta^{\prime}(0;H_{0,0,j})=-\text{\rm ln}\left(1-e^{2\pi
im_{j}}\right)-i(\pi/2)+im_{j}\pi+\text{\rm ln}(m_{j}).$ (3.118)

In order to obtain the final answer explicitly, showing the relation between
the imaginary part and the number of negative eigenvalues, we first note that

$1-e^{2\pi im_{j}}=-2ie^{\pi im_{j}}\sin(\pi m_{j}).$ (3.119)

A careful analysis of the argument of $1-e^{2\pi im_{j}}$ then shows that it
is $\pi[m_{j}-n_{j}-(1/2)]$, such that

$\zeta^{\prime}(0;H_{0,0,j})=i\pi n_{j}-\text{\rm
ln}\bigg{(}\bigg{|}\frac{2\sin(\pi m_{j})}{m_{j}}\bigg{|}\bigg{)}.$ (3.120)

Considering instead

$\zeta(s;H_{0,0,1},H_{0,0,2})=e^{is(\pi-\theta)}\frac{\sin(\pi
s)}{\pi}\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\bigg{(}\frac{F_{0,0,2}(te^{i\theta})}{F_{0,0,1}(te^{i\theta})}\bigg{)},$
(3.121)

in (3.120), one obtains

$\zeta^{\prime}(0;H_{0,0,1},H_{0,0,2})=i\pi(n_{2}-n_{1})+\text{\rm
ln}\bigg{(}\left|\frac{\sin(\pi m_{1})}{\sin(\pi
m_{2})}\,\,\frac{m_{2}}{m_{1}}\right|\bigg{)}.$ (3.122)

The real part of this answer is readily reproduced from (2.38) in Theorem 2.9.
However, even for this simple example, the behavior of
$F_{0,0,1}(te^{i\theta})/F_{0,0,2}(te^{i\theta})$ along the integration range
$t\in[0,\infty)$ is quite intricate so that finding the correct imaginary part
from (3.121), namely, from

$\zeta^{\prime}(0;H_{0,0,1},H_{0,0,2})=\int_{0}^{\infty}dt\,\frac{d}{dt}\text{\rm
ln}\bigg{(}\frac{F_{0,0,1}(te^{i\theta})}{F_{0,0,2}(te^{i\theta})}\bigg{)},$
(3.123)

is rather involved, and appears to be next to impossible for more general
cases.

For the case of Schrödinger operators with strongly singular potentials at one
or both endpoints of a compact intervals, see [33], [37], [39], [52].

##  4\. Schrödinger Operators on a Half-Line

In our final section we illustrate some of the abstract notions in Section 2
with the help of self-adjoint Schrödinger operators on the half-line
${\mathbb{R}}_{+}=(0,\infty)$. We will focus on the case of short-range
potentials $q$ (cf. (4.1)) and hence the scattering theory situation which
necessitates a comparison with the case $q=0$ and thus illustrates the case of
relative perturbation determinants, relative $\zeta$-functions, and relative
$\zeta$-function regularized determinants.

We assume that the potential coefficient $q$ satisfies the following
conditions.

######  Hypothesis 4.1.

Suppose $q$ satisfies the short-range assumption

$q\in L^{1}({\mathbb{R}}_{+};(1+|x|)dx),\,\text{ $q$ is real-valued a.e.\ on
${\mathbb{R}}_{+}$.}$ (4.1)

Given Hypothesis 4.1, we take $\tau_{+}$ to be the Schrödinger differential
expression

$\tau_{+}=-\frac{d^{2}}{dx^{2}}+q(x)\,\text{ for
a.e.~{}$x\in{\mathbb{R}}_{+}$,}$ (4.2)

and note that $\tau_{+}$ is regular at $0$ and in the limit point case at
$+\infty$. The _maximal operator_ $H_{+,max}$ in $L^{2}({\mathbb{R}}_{+};dx)$
associated with $\tau_{+}$ is defined by

$\displaystyle H_{+,max}f=\tau_{+}f,$
$\displaystyle\,f\in\text{\rm{dom}}(H_{+,max})=\big{\\{}g\in
L^{2}({\mathbb{R}}_{+};dx)\,\big{|}\,g,g^{\prime}\in AC([0,b])\,\text{for all
$b>0$};$ (4.3) $\displaystyle\hskip 223.3543pt\tau_{+}g\in
L^{2}({\mathbb{R}}_{+};dx)\big{\\}},$ while the _minimal operator_ $H_{+,min}$
in $L^{2}({\mathbb{R}}_{+};dx)$ associated with $\tau_{+}$ is given by
$\displaystyle H_{+,min}f=\tau_{+}f,$
$\displaystyle\,f\in\text{\rm{dom}}(H_{+,min})=\big{\\{}g\in
L^{2}({\mathbb{R}}_{+};dx)\,\big{|}\,g,g^{\prime}\in AC([0,b])\,\text{for all
$b>0$};$ (4.4) $\displaystyle\hskip
146.53151ptg(0)=g^{\prime}(0)=0;\,\tau_{+}g\in
L^{2}({\mathbb{R}}_{+};dx)\big{\\}}.$

Again, one notes that the operator $H_{+,min}$ is symmetric and that

$H_{+,min}^{*}=H_{+,max},\quad H_{+,max}^{*}=H_{+,min}.$ (4.5)

Moreover, all self-adjoint extensions of $H_{+,min}$ are given by the one-
parameter family $H_{+,\alpha}$ in $L^{2}({\mathbb{R}}_{+};dx)$,

$\displaystyle H_{+,\alpha}f=\tau_{+}f,$
$\displaystyle\,f\in\text{\rm{dom}}(H_{+,\alpha})=\big{\\{}g\in
L^{2}({\mathbb{R}}_{+};dx)\,\big{|}\,g,g^{\prime}\in AC([0,b])\,\text{for all
$b>0$};$ (4.6) $\displaystyle\hskip
86.78099pt\sin(\alpha)g^{\prime}(0)+\cos(\alpha)g(0)=0;\,\tau_{+}g\in
L^{2}({\mathbb{R}}_{+};dx)\big{\\}},$ $\displaystyle\hskip
260.34253pt\alpha\in[0,\pi).$

The corresponding comparison operator with vanishing potential coefficient
$q\equiv 0$ will be denoted by $H_{+,\alpha}^{(0)}$, $\alpha\in[0,\pi)$.

Next, introducing the Jost solutions

$\displaystyle
f_{+}(z,x)=f_{+}^{(0)}(z,x)-\int_{x}^{\infty}dx^{\prime}\,z^{-1/2}\sin(z^{1/2}(x-x^{\prime}))q(x^{\prime})f_{+}(z,x^{\prime}),$
(4.7) $\displaystyle f_{+}^{(0)}(z,x)=e^{iz^{1/2}x},\quad
z\in{\mathbb{C}},\;\text{\rm Im}\big{(}z^{1/2}\big{)}\geq 0,\;x\geq 0,$ (4.8)

satisfying $\tau_{+}y=zy$, $z\in{\mathbb{C}}$, on ${\mathbb{R}}_{+}$, and
abbreviating $I_{L^{2}({\mathbb{R}}_{+};dx)}=I_{+}$, and

$v=|q|^{1/2},\;u=v\,\text{\rm{sign}}(q),\,\text{ such that }\,q=uv=vu,$ (4.9)

one infers the following facts:

$\displaystyle{\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}\overline{(H_{+,\alpha}-zI_{+})^{1/2}\big{(}H_{+,\alpha}^{(0)}-zI_{+}\big{)}^{-1}(H_{+,\alpha}-zI_{+})^{1/2}}\Big{)}$
$\displaystyle\quad={\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}I_{+}+\overline{u\big{(}H_{+,\alpha}^{(0)}-zI_{+}\big{)}^{-1}v}\Big{)}$
(4.10)
$\displaystyle\quad=\frac{\sin(\alpha)f_{+}^{\prime}(z,0)+\cos(\alpha)f_{+}(z,0)}{\sin(\alpha)iz^{1/2}+\cos(\alpha)},\quad\alpha\in[0,\pi),\;z\in\rho(H_{+,\alpha}^{(0)})\cap\rho(H_{+,\alpha}).$

Here the first equality in (4.10) is shown as in the abstract context
(2.15)–(2.19), and the second equality in (4.10) for the Dirichlet and Neumann
cases $\alpha=0$, $\alpha=\pi/2$ has been discussed in [16], [17], [18]; the
general case $\alpha\in[0,\pi)$ is proved in [19, Theorem 2.6].

Since

$\sigma_{ess}(H_{+,\alpha})=[0,\infty),\quad\alpha\in[0,\pi),$ (4.11)

we now shift all operators $H_{+,\alpha}$ by $\lambda_{1}I_{+}$, with a fixed
$\lambda_{1}>0$, and consider

$H_{+,\alpha}(\lambda_{1})=H_{+,\alpha}+\lambda_{1}I_{+},\quad\alpha\in[0,\pi),$
(4.12)

from this point on and hence obtain

$\displaystyle{\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}\overline{(H_{+,\alpha}(\lambda_{1})-zI_{+})^{1/2}\big{(}H_{+,\alpha}^{(0)}(\lambda_{1})-zI_{+}\big{)}^{-1}(H_{+,\alpha}(\lambda_{1})-zI_{+})^{1/2}}\Big{)}$
$\displaystyle\quad={\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}I_{+}+\overline{u\big{(}H_{+,\alpha}^{(0)}(\lambda_{1})-zI_{+}\big{)}^{-1}v}\Big{)}$
$\displaystyle\quad=\frac{\sin(\alpha)f_{+}^{\prime}(z-\lambda_{1},0)+\cos(\alpha)f_{+}(z-\lambda_{1},0)}{\sin(\alpha)i(z-\lambda_{1})^{1/2}+\cos(\alpha)},$
(4.13) $\displaystyle\hskip
22.76219pt\alpha\in[0,\pi),\;z\in\rho(H_{+,\alpha}^{(0)}(\lambda_{1}))\cap\rho(H_{+,\alpha}(\lambda_{1})).$

In this half-line context all discrete eigenvalues $H_{+,\alpha}(\lambda_{1})$
(i.e., all eigenvalues of $H_{+,\alpha}(\lambda_{1})$ below $\lambda_{1}$) are
simple and hence

$m(0;H_{+,\alpha}(\lambda_{1}))\in\\{0,1\\},\quad\alpha\in[0,\pi).$ (4.14)

In addition, it is known that under Hypothesis 4.1, the threshold of the
essential spectrum of $H_{+,\alpha}(\lambda_{1})$, $\lambda_{1}$, is never an
eigenvalue of $H_{+,\alpha}(\lambda_{1})$.

Iterating the Volterra integral equation (4.7) for $f_{+}(z-\lambda_{1},0)$,
and analogously for its $x$-derivative yields uniform asymptotic expansions
near $z=0$ and as $z\to\infty$ (in terms of powers of $|z|^{-1/2}$). The same
applies to their $z$-derivatives (cf., e.g., [6, Ch. I]) and explicit
computations yield the following. For fixed $0<\varepsilon_{0}$ sufficiently
small, and using the abbreviation
${\mathbb{C}}_{\varepsilon_{0}}={\mathbb{C}}\backslash
B(\lambda_{1};\varepsilon_{0})$, with $B(z_{0};r_{0})$ the open ball in
${\mathbb{C}}$ of radius $r_{0}>0$ centered at $z_{0}\in{\mathbb{C}}$, one
obtains

$\displaystyle
f_{+}(z-\lambda_{1},0)\underset{\begin{subarray}{c}|z|\to\infty\\\
z\in{\mathbb{C}}_{\varepsilon_{0}}\end{subarray}}{=}1-\frac{i}{2z^{1/2}}\int_{0}^{\infty}dx_{1}\,\big{[}e^{2i(z-\lambda_{1})^{1/2}x_{1}}-1\big{]}q(x_{1})+O\big{(}|z|^{-1}\big{)},$
$\displaystyle
f_{+}^{\prime}(z-\lambda_{1},0)\underset{\begin{subarray}{c}|z|\to\infty\\\
z\in{\mathbb{C}}_{\varepsilon_{0}}\end{subarray}}{=}iz^{1/2}-\frac{1}{2}\int_{0}^{\infty}dx_{1}\,\big{[}e^{2i(z-\lambda_{1})^{1/2}x_{1}}+1\big{]}q(x_{1})+O\big{(}|z|^{-1/2}\big{)},$
$\displaystyle\overset{\textbf{\Large.}}{f}_{+}(z-\lambda_{1},0)\underset{\begin{subarray}{c}|z|\to\infty\\\
z\in{\mathbb{C}}_{\varepsilon_{0}}\end{subarray}}{=}\frac{1}{2z}\int_{0}^{\infty}dx_{1}\,e^{2i(z-\lambda_{1})^{1/2}x_{1}}x_{1}q(x_{1})+O\big{(}|z|^{-3/2}\big{)},$
$\displaystyle\overset{\textbf{\Large.}}{f}_{+}\phantom{f}^{\prime}\,\,(z-\lambda_{1},0)\underset{\begin{subarray}{c}|z|\to\infty\\\
z\in{\mathbb{C}}_{\varepsilon_{0}}\end{subarray}}{=}\frac{i}{2z^{1/2}}-\frac{i}{2z^{1/2}}\int_{0}^{\infty}dx_{1}\,e^{2i(z-\lambda_{1})^{1/2}x_{1}}x_{1}q(x_{1})+O\big{(}|z|^{-1}\big{)}$
(4.15)

(abbreviating again $\overset{\textbf{\Large.}}{}=d/dz$).

Given the asymptotic expansions (4.15) as $|z|\to\infty$, and employing the
fact that
$f_{+}(\,\cdot-\lambda_{1},0),f_{+}^{\prime}(\,\cdot-\lambda_{1},0),\overset{\textbf{\Large.}}{f}_{+}(\,\cdot-\lambda_{1},0),{\overset{\textbf{\Large.}}{f}}_{+}\phantom{f}^{\prime}\,\,(\,\cdot-\lambda_{1},0)$
are all analytic with respect to $z$ around $z=0$, investigating the case
distinctions $\alpha\in[0,\pi)\backslash\\{0,\pi/2\\}$, $\alpha=0$,
$\alpha=\pi/2$, $f_{+}(z-\lambda_{1},0)\neq 0$, $f_{+}(z-\lambda_{1},0)=0$,
$f_{+}^{\prime}(z-\lambda_{1},0)\neq 0$, $f_{+}^{\prime}(z-\lambda_{1},0)=0$,
etc., one verifies in each case that the logarithmic $z$-derivative of (4.10)
satisfies the hypotheses in Theorem 2.8, hence the latter applies with
$\varepsilon=1/2$ to the pairs
$\big{(}H_{+,\alpha}^{(0)}(\lambda_{1}),H_{+,\alpha}(\lambda_{1})\big{)}$,
$\alpha\in[0,\pi)$.

More generally, we now replace the pair $(0,q)$ by $(q_{1},q_{2})$, where
$q_{j}$, $j=1,2$, satisfy Hypothesis 4.1, and denote the corresponding
Schrödinger operators in $L^{2}({\mathbb{R}}_{+};dx)$ with $q$ (resp., $u$,
$v$) replaced by $q_{j}$ (resp., $u_{j}$, $v_{j}$) by $H_{+,\alpha,j}$ and
similarly, after the shift with $\lambda_{1}$, by
$H_{+,\alpha,j}(\lambda_{1})$, $j=1,2$. Analogously, we denote the
corresponding Jost solutions by $f_{+,j}(z,\,\cdot\,)$, $j=1,2$. This then
yields the following results:

######  Theorem 4.2.

Suppose $q_{j}$, $j=1,2$, satisfy Hypothesis 4.1. Then,

$\displaystyle{\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}\overline{(H_{+,\alpha,2}(\lambda_{1})-zI_{+})^{1/2}\big{(}H_{+,\alpha,1}(\lambda_{1})-zI_{+}\big{)}^{-1}(H_{+,\alpha,2}(\lambda_{1})-zI_{+})^{1/2}}\Big{)}$
$\displaystyle\quad={\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}I_{+}+\overline{u_{1,2}\big{(}H_{+,\alpha,1}(\lambda_{1})-zI_{+}\big{)}^{-1}v_{1,2}}\Big{)}$
$\displaystyle\quad=\frac{\sin(\alpha)f_{+,2}^{\prime}(z-\lambda_{1},0)+\cos(\alpha)f_{+,2}(z-\lambda_{1},0)}{\sin(\alpha)f_{+,1}^{\prime}(z-\lambda_{1},0)+\cos(\alpha)f_{+,1}(z-\lambda_{1},0)},$
(4.16) $\displaystyle\hskip
20.48596pt\alpha\in[0,\pi),\;z\in\rho\big{(}H_{+,\alpha,1}(\lambda_{1})\big{)}\cap\rho(H_{+,\alpha,2}(\lambda_{1})),$

where

$\displaystyle\begin{split}&v_{1,2}=|q_{2}-q_{1}|^{1/2},\;u_{1,2}=v_{1,2}\,\text{\rm{sign}}(q_{2}-q_{1}),\\\
&\quad\text{such that
}\,q_{2}-q_{1}=u_{1,2}v_{1,2}=v_{1,2}u_{1,2}.\end{split}$ (4.17)

In addition,

$\displaystyle\zeta(s;H_{+,\alpha,1}(\lambda_{1}),H_{+,\alpha,2}(\lambda_{1}))=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)$ $\displaystyle\quad\times\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\bigg{(}(e^{i\theta}t)^{[m(0;H_{+,\alpha,1}(\lambda_{1}))-m(0;H_{+,\alpha,2}(\lambda_{1}))]}$
(4.18) $\displaystyle\hskip
91.04872pt\times\frac{\sin(\alpha)f_{+,2}^{\prime}(e^{i\theta}t-\lambda_{1},0)+\cos(\alpha)f_{+,2}(e^{i\theta}t-\lambda_{1},0)}{\sin(\alpha)f_{+,1}^{\prime}(e^{i\theta}t-\lambda_{1},0)+\cos(\alpha)f_{+,1}(e^{i\theta}t-\lambda_{1},0)}\bigg{)},$
$\displaystyle\hskip 202.01474pt\alpha\in[0,\pi),\;\text{\rm
Re}(s)\in(-1/2,1).$

###### Proof.

Relations (4.16), (4.17) follow as summarized in (4.7)–(4.15) and the two
paragraphs preceding Theorem 4.2. Thus, Theorem 2.8 applies to the pairs of
self-adjoint operators
$(H_{+,\alpha,1}(\lambda_{1}),H_{+,\alpha,2}(\lambda_{1}))$,
$\alpha\in[0,\pi)$. ∎

The $\zeta$-function regularized determinant now follows immediately from
Theorem 2.9.

Special cases of (4.16) (pertaining to the Dirichlet boundary conditions
$\alpha_{j}=0$, $j=1,2$) appeared in the celebrated work by Jost and Pais [30]
and Buslaev and Faddeev [5] (see also [10], [17], [45], [46], [48]).

Up to this point we kept the boundary condition, that is, $\alpha$, fixed and
varied the potential coefficient $q$. Next, we keep $q$ fixed, but vary
$\alpha$. Returning to the operator $H_{+,\alpha}$, $\alpha\in[0,\pi)$, we
turn to its underlying quadratic form $Q_{H_{+,\alpha}}$ next,

$\displaystyle
Q_{H_{+,\alpha}}(f,g)=\int_{0}^{\infty}dx\big{[}\overline{f^{\prime}(x)}g^{\prime}(x)+q(x)\overline{f(x)}g(x)\big{]}-\cot(\alpha)\overline{f(0)}g(0),$
$\displaystyle
f,g\in\text{\rm{dom}}(Q_{H_{+,\alpha}})=\text{\rm{dom}}\big{(}|H_{+,\alpha}|^{1/2}\big{)}=H^{1}({\mathbb{R}}_{+})$
(4.19) $\displaystyle\quad=\big{\\{}h\in L^{2}({\mathbb{R}}_{+};dx)\,|\,h\in
AC([0,b])\,\text{for all $b>0$};\,h^{\prime}\in
L^{2}({\mathbb{R}}_{+};dx)\big{\\}},$ $\displaystyle\hskip
254.65207pt\alpha\in(0,\pi),$ $\displaystyle
Q_{H_{+,0}}(f,g)=\int_{0}^{\infty}dx\big{[}\overline{f^{\prime}(x)}g^{\prime}(x)+q(x)\overline{f(x)}g(x)\big{]},$
$\displaystyle
f,g\in\text{\rm{dom}}(Q_{H_{+,0}})=\text{\rm{dom}}\big{(}|H_{+,0}|^{1/2}\big{)}=H^{1}_{0}({\mathbb{R}}_{+})$
(4.20) $\displaystyle\quad=\big{\\{}h\in L^{2}({\mathbb{R}}_{+};dx)\,|\,h\in
AC([0,b])\,\text{for all $b>0$};\,h(0)=0;\,h^{\prime}\in
L^{2}({\mathbb{R}}_{+};dx)\big{\\}}.$

Moreover, introducing the regular solution $\phi_{\alpha}(z,\,\cdot\,)$
associated with $H_{+,\alpha}$ satisfying $\tau_{+}y=zy$, $z\in{\mathbb{C}}$,
on ${\mathbb{R}}_{+}$, and

$\sin(\alpha)\phi_{\alpha}^{\prime}(z,0)+\cos(\alpha)\phi_{\alpha}(z,0)=0,\quad\alpha\in[0,\pi),\;z\in{\mathbb{C}},$
(4.21)

one infers

$\displaystyle\begin{split}\phi_{\alpha}(z,x)=\phi_{\alpha}^{(0)}(z,x)+\int_{0}^{x}dx^{\prime}\,z^{-1/2}\sin(z^{1/2}(x-x^{\prime}))q(x^{\prime})\phi_{\alpha}(z,x^{\prime}),&\\\
z\in{\mathbb{C}},\;\text{\rm Im}(z^{1/2})\geq 0,\;x\geq 0,&\end{split}$ (4.22)

with

$\displaystyle\begin{split}\phi_{\alpha}^{(0)}(z,x)=\cos(\alpha)z^{-1/2}\sin(z^{1/2}x)-\sin(\alpha)\cos(z^{1/2}x),&\\\
z\in{\mathbb{C}},\;\text{\rm Im}(z^{1/2})\geq 0,\;x\geq 0.&\end{split}$ (4.23)

Given the solutions $\phi_{\alpha}(z,\,\cdot\,)$, $f_{+}(z,\,\cdot\,)$ of
$\tau_{+}y=z$, $z\in{\mathbb{C}}$, the resolvent of $H_{+,\alpha}$ is given by

$\displaystyle\begin{split}\big{(}(H_{+,\alpha}-zI_{+})^{-1}f\big{)}(x)=\int_{0}^{\infty}dx^{\prime}\,G_{+,\alpha}(z,x,x^{\prime})f(x^{\prime}),&\\\
z\in\rho(H_{+,\alpha}),\;x\geq 0,\;f\in
L^{2}({\mathbb{R}}_{+};dx),&\end{split}$ (4.24)

with the Green’s function $G_{+,\alpha}$ of $H_{+,\alpha}$ expressed in terms
of $\phi_{\alpha}$ and $f_{+}$ by

$\displaystyle\begin{split}&G_{+,\alpha}(z,x,x^{\prime})=(H_{+,\alpha}-zI_{+})^{-1}(x,x^{\prime})\\\
&\quad=\frac{1}{W(f_{+}(z,\cdot),\phi_{\alpha}(z,\cdot))}\begin{cases}\phi_{\alpha}(z,x)\,f_{+}(z,x^{\prime}),&0\leq
x\leq x^{\prime}<\infty,\\\ \phi_{\alpha}(z,x^{\prime})\,f_{+}(z,x),&0\leq
x^{\prime}\leq x<\infty,\end{cases}\\\ &\hskip
225.63014ptz\in\rho(H_{+,\alpha}).\end{split}$ (4.25)

In the special case $\alpha=0$ one verifies

$W(f_{+}(z,\cdot),\phi_{0}(z,\cdot))=f_{+}(z,0),\quad z\in\rho(H_{+,0}),$
(4.26)

with $f_{+}(z,0)$ the well-known Jost function.

Next, we compare the half-line Green’s functions $G_{+,\alpha_{1}}$ and
$G_{+,\alpha_{2}}$, that is, we investigate the integral kernels associated
with a special case of Krein’s formula for resolvents (cf. [1, & 106]): Assume
$\alpha_{1},\alpha_{2}\in[0,\pi)$, with $\alpha_{1}\neq\alpha_{2}$. Then,

$\displaystyle\begin{split}&G_{+,\alpha_{2}}(z,x,x^{\prime})=G_{+,\alpha_{1}}(z,x,x^{\prime})-\frac{\psi_{+,\alpha_{1}}(z,x)\psi_{+,\alpha_{1}}(z,x^{\prime})}{\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(z)},\\\
&\hskip
88.2037ptz\in\rho(H_{+,\alpha_{1}})\cap\rho(H_{+,\alpha_{2}}),\;x,x^{\prime}\in[0,\infty),\end{split}$
(4.27)

and

$\int_{0}^{\infty}dx\,\psi_{+,\alpha}(z,x)^{2}=\overset{\textbf{\Large.}}{m}_{+,\alpha}(z),$
(4.28)

implying

$\displaystyle\begin{split}&{\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{-1}-(H_{+,\alpha_{1}}(\lambda_{1})-zI_{+})^{-1}\big{)}\\\
&\quad=-\frac{d}{dz}\text{\rm
ln}[\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(z)],\quad
z\in\rho(H_{+,\alpha_{1}})\cap\rho(H_{+,\alpha_{2}}),\end{split}$ (4.29)

according to Lemma 2.2 and (A.44) in [21]. Here $\psi_{+,\alpha}(z,\,\cdot\,)$
and $m_{+,\alpha}(z)$ are the Weyl–Titchmarsh solution and $m$-function
corresponding to $H_{+,\alpha}$. More precisely,

$\psi_{+,\alpha}(z,\,\cdot\,)=\theta_{\alpha}(z,\,\cdot\,)+m_{+,\alpha}(z)\phi_{\alpha}(z,\,\cdot\,),\quad
z\in\rho(H_{+,\alpha}),$ (4.30)

where

$\displaystyle\begin{split}\theta_{\alpha}(z,x)=\theta_{\alpha}^{(0)}(z,x)+\int_{0}^{x}dx^{\prime}\,z^{-1/2}\sin(z^{1/2}(x-x^{\prime}))q(x^{\prime})\theta_{\alpha}(z,x^{\prime}),&\\\
z\in{\mathbb{C}},\;\text{\rm Im}(z^{1/2})\geq 0,\;x\geq 0,&\end{split}$ (4.31)

with

$\displaystyle\begin{split}\theta_{\alpha}^{(0)}(z,x)=\cos(\alpha)\cos(z^{1/2}x)+\sin(\alpha)z^{-1/2}\sin(z^{1/2}x),&\\\
z\in{\mathbb{C}},\;\text{\rm Im}(z^{1/2})\geq 0,\;x\geq 0.&\end{split}$ (4.32)

Due to the limit point property of $\tau_{+}$ at $+\infty$, one actually has

$\psi_{+,\alpha}(z,\,\cdot\,)=C_{\alpha}(z)f_{+}(z,\,\cdot\,),\quad
z\in\rho(H_{+,\alpha}),$ (4.33)

for some $z$-dependent complex-valued constant $C_{\alpha}(z)$. Actually,
since $\psi_{+,\alpha}(z,0)=C_{\alpha}f_{+}(z,0)$, one can show (using (A.18)
in [21]) that

$C_{\alpha}(z)=\frac{\cos(\alpha)-\sin(\alpha)m_{+,\alpha}(z)}{f_{+}(z,0)}=\frac{1}{\cos(\alpha)f_{+}(z,0)+\sin(\alpha)f_{+}^{\prime}(z,0)}.$
(4.34)

Similarly,

$m_{+,0}(z)=\psi^{\prime}_{+,0}(z,0)/\psi_{+,0}(z,0)=f^{\prime}_{+}(z,0)/f_{+}(z,0),\quad
z\in\rho(H_{+,\alpha}),$ (4.35)

and

$\displaystyle\begin{split}m_{+,\alpha}(z)=\frac{-\sin(\alpha)+\cos(\alpha)m_{+,0}(z)}{\cos(\alpha)+\sin(\alpha)m_{+,0}(z)}=\frac{\cos(\alpha)f_{+}^{\prime}(z,0)-\sin(\alpha)f_{+}(z,0)}{\sin(\alpha)f_{+}^{\prime}(z,0)+\cos(\alpha)f_{+}(z,0)},&\\\
z\in\rho(H_{+,\alpha}).&\end{split}$ (4.36)

Moreover, one verifies that

$\displaystyle\cos(\alpha_{2}-\alpha_{1})+\sin(\alpha_{2}-\alpha_{1})m_{+,\alpha_{1}}(z)=\frac{\sin(\alpha_{2})f_{+}^{\prime}(z,0)+\cos(\alpha_{2})f_{+}(z,0)}{\sin(\alpha_{1})f_{+}^{\prime}(z,0)+\cos(\alpha_{1})f_{+}(z,0)}.$
(4.37)

Combining (4.29) and (4.37) yields

$\displaystyle{\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{-1}-(H_{+,\alpha_{1}}(\lambda_{1})-zI_{+})^{-1}\big{)}$
$\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\bigg{(}\frac{\sin(\alpha_{2})f_{+}^{\prime}(z-\lambda_{1},0)+\cos(\alpha_{2})f_{+}(z-\lambda_{1},0)}{\sin(\alpha_{1})f_{+}^{\prime}(z-\lambda_{1},0)+\cos(\alpha_{1})f_{+}(z-\lambda_{1},0)}\bigg{)},$
(4.38) $\displaystyle\hskip
45.52458ptz\in\rho(H_{+,\alpha_{1}}(\lambda_{1}))\cap\rho(H_{+,\alpha_{2}}(\lambda_{1})),\;\alpha_{1},\alpha_{2}\in[0,\pi).$

In particular,
$\big{[}(H_{+,\alpha_{2}}-zI_{+})^{-1}-(H_{+,\alpha_{1}}-zI_{+})^{-1}\big{]}$
is rank-one and hence trace class. Moreover, since by (4.19)

$f_{+}(z,\,\cdot\,)\in\text{\rm{dom}}\big{(}|H_{+,\alpha}|^{1/2}\big{)}=H^{1}({\mathbb{R}}_{+}),\quad\alpha\in(0,\pi),$
(4.39)

Hypothesis 2.4 and hence relations (2.22), (2.23) are now satisfied for the
pair $(H_{+,\alpha_{1}}(\lambda_{1}),H_{+,\alpha_{2}}(\lambda_{1}))$ for
$\alpha_{1}\in[0,\pi)$, $\alpha_{2}\in(0,\pi)$, implying the following result:

######  Theorem 4.3.

Suppose $q_{j}$, $j=1,2$, satisfy Hypothesis 4.1. Then,

$\displaystyle{\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{-1}-(H_{+,\alpha_{1}}(\lambda_{1})-zI_{+})^{-1}\big{)}$
$\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\Big{(}{\det}_{L^{2}({\mathbb{R}}_{+};dx)}\Big{(}\big{\\{}(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{1/2}\big{(}H_{+,\alpha_{1}}(\lambda_{1})-zI_{+}\big{)}^{-1}$
$\displaystyle\hskip
118.07875pt\times(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{1/2}\big{\\}}^{c}\Big{)}\Big{)}$
(4.40) $\displaystyle\quad=-\frac{d}{dz}\text{\rm
ln}\bigg{(}\frac{\sin(\alpha_{2})f_{+}^{\prime}(z-\lambda_{1},0)+\cos(\alpha_{2})f_{+}(z-\lambda_{1},0)}{\sin(\alpha_{1})f_{+}^{\prime}(z-\lambda_{1},0)+\cos(\alpha_{1})f_{+}(z-\lambda_{1},0)}\bigg{)},$
(4.41)
$\displaystyle\quad\;z\in\rho(H_{+,\alpha_{1}}(\lambda_{1}))\cap\rho(H_{+,\alpha_{2}}(\lambda_{1})),\;\alpha_{1}\in[0,\pi),\;\alpha_{2}\in(0,\pi),\;\alpha_{1}\neq\alpha_{2},$

temporarily abbreviating the operator closure symbol by $\\{\cdots\\}^{c}$ due
to lack of space in (4.40). In addition,

$\displaystyle\zeta(s;H_{+,\alpha_{1}}(\lambda_{1}),H_{+,\alpha_{2}}(\lambda_{1}))=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)$ $\displaystyle\quad\times\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\left((te^{i\theta})^{[m(0,H_{+,\alpha_{1}}(\lambda_{1}))-m(0,H_{+,\alpha_{2}}(\lambda_{1}))]}\right.$
(4.42)
$\displaystyle\quad\times\left.\frac{\sin(\alpha_{2})f_{+}^{\prime}(te^{i\theta}-\lambda_{1},0)+\cos(\alpha_{2})f_{+}(te^{i\theta}-\lambda_{1},0)}{\sin(\alpha_{1})f_{+}^{\prime}(te^{i\theta}-\lambda_{1},0)+\cos(\alpha_{1})f_{+}(te^{i\theta}-\lambda_{1},0)}\right),$
$\displaystyle\hskip 73.97733pt\alpha_{j}\in(0,\pi),\;j=1,2,\,\,\text{\rm
Re}(s)\in(-1/2,1).$

###### Proof.

Relations (4.40), (4.41) summarize the discussion in (4.21)–(4.39). Applying
Theorem 2.8 in the case $\alpha_{1},\alpha_{2}\in(0,\pi)$ then yields (4.42).
∎

The $\zeta$-function regularized determinant again follows immediately from
Theorem 2.9,

$\displaystyle\zeta^{\prime}\Big{(}0;H_{+,0}^{(0)}(\lambda_{1}),H_{+,\alpha_{2}}^{(0)}(\lambda_{1})\Big{)}=-\lim_{t\downarrow
0}\frac{d}{dt}\text{\rm
ln}\bigg{(}\big{(}te^{i\theta}\big{)}^{m\big{(}0,H_{+,0}^{(0)}(\lambda_{1})\big{)}-m\big{(}0,H_{+,\alpha_{2}}^{(0)}(\lambda_{1})\big{)}}$
$\displaystyle\quad\times\frac{\sin(\alpha_{2})f_{+}^{\prime}(te^{i\theta}-\lambda_{1},0)+\cos(\alpha_{2})f_{+}(te^{i\theta}-\lambda_{1},0)}{\sin(\alpha_{1})f_{+}^{\prime}(te^{i\theta}-\lambda_{1},0)+\cos(\alpha_{1})f_{+}(te^{i\theta}-\lambda_{1},0)}\bigg{)}.$
(4.43)

######  Remark 4.4.

The case $\alpha_{1}=0$, $\alpha_{2}\in(0,\pi)$, is more involved in that the
representation (4.42) is only valid for $\text{\rm Re}(s)\in(0,1)$. This is
due to the fact that

$\frac{\sin(\alpha_{2})f_{+}^{\prime}(z-\lambda_{1},0)+\cos(\alpha_{2})f_{+}(z-\lambda_{1},0)}{f_{+}(z-\lambda_{1},0)}\underset{|z|\to\infty}{=}O\big{(}|z|^{1/2}\big{)},$
(4.44)

and hence assumption (2.35) is not satisfied. One then proceeds as follows.
Let $H_{+,\alpha}^{(0)}(\lambda_{1})$ denote the case with vanishing
potential. We rewrite

$\displaystyle{\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{-1}-(H_{+,0}(\lambda_{1})-zI_{+})^{-1}\big{)}$
$\displaystyle\quad={\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,\alpha_{2}}(\lambda_{1})-zI_{+})^{-1}-(H_{+,\alpha_{2}}^{(0)}(\lambda_{1})-zI_{+})^{-1}\big{)}$
$\displaystyle\qquad+{\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,0}^{(0)}(\lambda_{1})-zI_{+})^{-1}-(H_{+,0}(\lambda_{1})-zI_{+})^{-1}\big{)}$
$\displaystyle\qquad+{\text{\rm{tr}}}_{L^{2}({\mathbb{R}}_{+};dx)}\big{(}(H_{+,\alpha_{2}}^{(0)}(\lambda_{1})-zI_{+})^{-1}-(H_{+,0}^{(0)}(\lambda_{1})-zI_{+})^{-1}\big{)}.$
(4.45)

For the first two of the three contributions on the right-hand side of (4.45)
the relative $\zeta$-determinants can be computed from Theorem 4.2. For the
third contribution more care is needed as the subleading large-$|z|$ behavior
differs due to one boundary condition being Dirichlet and the other being
Robin. The starting point is the representation

$\displaystyle\begin{split}&\zeta\Big{(}s;H_{+,0}^{(0)}(\lambda_{1}),H_{+,\alpha_{2}}^{(0)}(\lambda_{1})\Big{)}=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)\\\ &\quad\times\int_{0}^{\infty}dt\,t^{-s}\frac{d}{dt}\text{\rm
ln}\left(\sin(\alpha_{2})i\big{(}te^{i\theta}-\lambda_{1}\big{)}^{1/2}+\cos(\alpha_{2})\right).\end{split}$
(4.46)

Along the lines of (3.69) one rewrites this as

$\displaystyle\begin{split}&\zeta\Big{(}s;H_{+,0}^{(0)}(\lambda_{1}),H_{+,\alpha_{2}}^{(0)}(\lambda_{1})\Big{)}=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)\int_{0}^{\infty}dt\,t^{-s}\\\ &\quad\times\frac{d}{dt}\text{\rm
ln}\left(\frac{\sin(\alpha_{2})i\big{(}te^{i\theta}-\lambda_{1}\big{)}^{1/2}+\cos(\alpha_{2})}{\sin(\alpha_{2})i\big{(}te^{i\theta}-\lambda_{1}\big{)}^{1/2}}\right)+\zeta^{asym}(s),\end{split}$
(4.47)

where, using (3.115),

$\displaystyle\zeta^{asym}(s)$
$\displaystyle=e^{is(\pi-\theta)}\pi^{-1}\sin(\pi
s)\int_{0}^{\infty}dtt^{-s}\frac{d}{dt}\text{\rm
ln}\left(i\sin(\alpha_{2})\big{(}te^{i\theta}-\lambda_{1}\big{)}^{1/2}\right)$
$\displaystyle=\lambda_{1}^{-s}/2.$ (4.48)

The relative $\zeta$-determinant for the last term in (4.45) then follows from

$\displaystyle\zeta^{\prime}\Big{(}0;H_{+,0}^{(0)}(\lambda_{1}),H_{+,\alpha_{2}}^{(0)}(\lambda_{1})\Big{)}$
$\displaystyle=\text{\rm Re}\left(-\text{\rm
ln}\bigg{(}\frac{\sin(\alpha_{2})i(-\lambda_{1})^{1/2}+\cos(\alpha_{2})}{\sin(\alpha_{2})i(-\lambda_{1})^{1/2}}\bigg{)}-\text{\rm
ln}\big{(}\lambda_{1}^{1/2}\big{)}\right)$ $\displaystyle=-\text{\rm
ln}\big{(}\big{|}\lambda_{1}^{1/2}-\cot(\alpha_{2})\big{|}\big{)}.$ (4.49)

$\diamond$

One notes that formally, (4.41) extends to the trivial case
$\alpha_{1}=\alpha_{2}$.

For the case of a strongly singular potential on the half-line with
$x^{-2}$-type singularity at $x=0$ we refer to [33].

Acknowledgments. We are indebted to Christer Bennewitz for very helpful
discussions.

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