arXiv subjects

Fritz Gesztesy

Publications and source records attributed to Fritz Gesztesy.

At least 91 records · Page 5Linked to original sources

arXiv2014

On square root domains for non-self-adjoint Sturm-Liouville operators

We determine square root domains for non-self-adjoint Sturm--Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb{R}$, the half-line $(a,\infty)$, $a \in \mathbb{R}$, or with the bounded interval $(a,b) \subset \mathbb{R}$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e.\ on $(a,b)$, we treat all separated boundary conditions at $a, b$.)

math.CA↗

arXiv2014

Stability of square root domains associated with elliptic systems of PDEs on nonsmooth domains

We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,Ω,Γ} = -\nabla\cdot a \nabla$ in $L^2(Ω)$, with mixed boundary conditions on $\partial Ω$, with respect to additive perturbations. We consider open, bounded, and connected sets $Ω\in \mathbb{R}^n$, $n \in \mathbb{N} \backslash\{1\}$, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator $L_{a,Ω,Γ}$ with respect to additive potential perturbations $V \in L^p(Ω) + L^{\infty}(Ω)$, $p>n/2$. Special emphasis is put on the case of uniformly elliptic systems with mixed boundary conditions.

math.AP↗

arXiv2014

On the Witten index in terms of spectral shift functions

We study the model operator $\mathbf{D}_{\mathbf{A}} = (d/dt) + \mathbf{A}$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(\mathbf{A} f)(t) = A(t) f(t)$ for a.e.\ $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$ (with $\mathcal{H}$ a separable, complex Hilbert space). Denoting by $A_{\pm}$ the norm resolvent limits of $A(t)$ as $t \to \pm \infty$, our setup permits $A(t)$ in $\mathcal{H}$ to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum assumptions are made on $A_{\pm}$. We introduce resolvent and semigroup regularized Witten indices of $\mathbf{D}_{\mathbf{A}}$, denoted by $W_r$ and $W_s$, and prove that these regularized indices coincide with the Fredholm index of $\mathbf{D}_{\mathbf{A}}$ whenever the latter is Fredholm. In situations where $\mathbf{D}_{\mathbf{A}}$ ceases to be a Fredholm operator in $L^2(\mathbb{R};\mathcal{H})$ we compute its resolvent (resp., semigroup) regularized Witten index in terms of the spectral shift function $ξ(\,\cdot\,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ as follows: Assuming $0$ to be a right and a left Lebesgue point of $ξ(\,\cdot\,\, ; A_+, A_-)$, denoted by $ξ_L(0_+; A_+,A_-)$ and $ξ_L(0_-; A_+, A_-)$, we prove that $0$ is also a right Lebesgue point of $ξ(\,\cdot\,\, ; \mathbf{H_2}, \mathbf{H_1})$, denoted by $ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1})$, and that \begin{align*} W_r(\mathbf{D}_{\mathbf{A}}) &= W_s(\mathbf{D}_{\mathbf{A}}) \\ & = ξ_L(0_+; \mathbf{H_2}, \mathbf{H_1}) \\ & = [ξ_L(0_+; A_+,A_-) + ξ_L(0_-; A_+, A_-)]/2, \end{align*} the principal result of this paper. In the special case where $\dim(\mathcal{H}) < \infty$, we prove that the Witten indices of $\mathbf{D}_{\mathbf{A}}$ are either integer, or half-integer-valued.

math.SP↗

arXiv2014

A Jost-Pais-type reduction of (modified) Fredholm determinants for semi-separable operators in infinite dimensions

We study the analog of semi-separable integral kernels in $\mathcal{H}$ of the type $$ K(x,x')=\begin{cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, \end{cases} $$ where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \mathcal{B}_2(\mathcal{H}_j,\mathcal{H})$ and $G_j(x) \in \mathcal{B}_2(\mathcal{H},\mathcal{H}_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and $$ \|F_j(\cdot)\|_{\mathcal{B}_2(\mathcal{H}_j,\mathcal{H})} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\mathcal{B}_2(\mathcal{H},\mathcal{H}_j)} \in L^2((a,b)), \quad j=1,2, $$ with $\mathcal{H}$ and $\mathcal{H}_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a Hilbert-Schmidt operator $\mathbf{K}$ in $L^2((a,b);\mathcal{H})$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the modified Fredholm determinant ${\det}_{2, L^2((a,b);\mathcal{H})}(\mathbf{I} - α\mathbf{K})$, $α\in \mathbb{C}$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\mathcal{H}$ (and $\mathcal{H} \oplus \mathcal{H}$). Some applications to Schrödinger operators with operator-valued potentials are provided.

math.FA↗

arXiv2014

A Description of All Self-Adjoint Extensions of the Laplacian and Krein-Type Resolvent Formulas on Nonsmooth Domains

This paper has two main goals. First, we are concerned with the classification of self-adjoint extensions of the Laplacian $-Δ\big|_{C^\infty_0(Ω)}$ in $L^2(Ω; d^n x)$. Here, the domain $Ω$ belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), which contain all convex domains, as well as all domains of class $C^{1,r}$, for $r\in(1/2,1)$. Second, we establish Krein-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasi-convex domains and study the properties of the corresponding Weyl--Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant technical innovation in this paper is an extension of the classical boundary trace theory for functions in spaces which lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

math.AP↗

arXiv2014

A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets

For an arbitrary nonempty, open set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- Δ)^m\big|_{C_0^{\infty}(Ω)}$, $m \in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,Ω,m}$ in $L^2(Ω)$. With $N(λ,A_{K,Ω,m})$, $λ> 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,Ω,m}$, we derive the bound $$ N(λ,A_{K,Ω,m}) \leq (2 π)^{-n} v_n |Ω| \{1 + [2m/(2m+n)]\}^{n/(2m)} λ^{n/(2m)}, \quad λ> 0, $$ where $v_n := π^{n/2}/Γ((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. The proof relies on variational considerations and exploits the fundamental link between the Krein--von Neumann extension and an underlying (abstract) buckling problem.

math.SP↗

arXiv2014

Some Operator Bounds Employing Complex Interpolation Revisited

We revisit and extend known bounds on operator-valued functions of the type $$ T_1^{-z} S T_2^{-1+z}, \quad z \in \ol Σ= \{z\in\bbC\,|\, \Re(z) \in [0,1]\}, $$ under various hypotheses on the linear operators $S$ and $T_j$, $j=1,2$. We particularly single out the case of self-adjoint and sectorial operators $T_j$ in some separable complex Hilbert space $\cH_j$, $j=1,2$, and suppose that $S$ (resp., $S^*$) is a densely defined closed operator mapping $\dom(S) \subseteq \cH_1$ into $\cH_2$ (resp., $\dom(S^*) \subseteq \cH_2$ into $\cH_1$), relatively bounded with respect to $T_1$ (resp., $T_2^*$). Using complex interpolation methods, a generalized polar decomposition for $S$, and Heinz's inequality, the bounds we establish lead to inequalities of the following type, \begin{align*} & \big\|\ol{T_2^{-x}ST_1^{-1+x}}\big\|_{\cB(\cH_1,\cH_2)} \leq N_1 N_2 e^{(θ_1 + θ_2) [x(1-x)]^{1/2}} \\ & \quad \times \big\|ST_1^{-1}\big\|_{\cB(\cH_1,\cH_2)}^{1-x} \, \big\|S^*(T_2^*)^{-1}\big\|_{\cB(\cH_2,\cH_1)}^{x}, \quad x \in [0,1], \end{align*} assuming that $T_j$ have bounded imaginary powers, that is, for some $N_j\ge 1$ and $θ_j \ge 0,$ $$ \big\|T_j^{is}\big\|_{\cB(\cH)} \leq N_j e^{θ_j |s|}, \quad s \in \bbR, \; j=1,2. $$ We also derive analogous bounds with $\cB(\cH_1,\cH_2)$ replaced by trace ideals, $\cB_p(\cH_1, \cH_2)$, $p \in [1,\infty)$. The methods employed are elementary, predominantly relying on Hadamard's three-lines theorem and Heinz's inequality.

math.FA↗

arXiv2014

A Jost-Pais-type reduction of Fredholm determinants and some applications

We study the analog of semi-separable integral kernels in $\cH$ of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \cB_2(\cH_j,\cH)$ and $G_j(x) \in \cB_2(\cH,\cH_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and {equation*} \|F_j(\cdot)\|_{\cB_2(\cH_j,\cH)} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\cB_2(\cH,\cH_j)} \in L^2((a,b)), \quad j=1,2, {equation*} with $\cH$ and $\cH_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a trace class operator $\bsK$ in $L^2((a,b);\cH)$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant ${\det}_{L^2((a,b);\cH)}(\bsI - α\bsK)$, $α\in \bbC$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\cH$ (and $\cH_1 \oplus \cH_2$). Explicit applications of this reduction theory are made to Schrödinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.

math.FA↗

arXiv2014

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,Ω}$, the Krein--von Neumann extension of the perturbed Laplacian $-Δ+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(Ω)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Ω\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.

math.SP↗

arXiv2013

Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[τf = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right), \] where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$ and real-valued with $p\not=0$ and $r>0$ a.e.\ on $(a,b)$. In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg-Marchenko-type inverse spectral results. The special cases of Schrödinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular.

math.SP↗

arXiv2013

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions II

The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A\nabla u + Θ\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partialΩ, $$ where $Θ$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, and $ν$ denotes the outward pointing normal unit vector on $\partial\Om$.

math.AP↗

arXiv2013

Heat Kernel Bounds for Elliptic Partial Differential Operators in Divergence Form with Robin-Type Boundary Conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{Θ, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$ Lu = - \biggl(\sum_{j,k=1}^n\partial_j\bigg(\sum_{β= 1}^m a^{α,β}_{j,k}\partial_k u_β\bigg) \bigg)_{1\leqα\leq m}, \quad u=(u_1,...,u_m). $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A D u + Θ\big[u\big|_{\partial \Om}\big] = 0 \, \text{on $\partial \Om$}, $$ where $Θ$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, $ν$ denotes the outward pointing normal unit vector on $\partial\Om$, and $Du:=\bigl(\partial_j u_α\bigr)_{\substack{1\leqα\leq m 1\leq j\leq n}}$. Assuming $Θ\geq 0$ in the scalar case $m=1$, we prove Gaussian heat kernel bounds for $L_{Θ, \Om}$ by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on $\partial \Om$. We also discuss additional zero-order potential coefficients $V$ and hence operators corresponding to the form sum $L_{Θ, \Om} + V$.

math.AP↗

arXiv2013

Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ τf = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $τ$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. Finally, in the special case where $τ$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

math.SP↗

arXiv2013

Some Remarks on the Spectral Problem Underlying the Camassa-Holm Hierarchy

We consider left-definite eigenvalue problems $A ψ= λB ψ$, with $A \geq \varepsilon I$ for some $\varepsilon > 0$ and $B$ self-adjoint, but $B$ not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa-Holm hierarchy. In fact, we will treat a more general version where $A$ represents a positive definite Schrödinger or Sturm-Liouville operator $T$ in $L^2(\bbR; dx)$ associated with a differential expression of the form $τ= - (d/dx) p(x) (d/dx) + q(x)$, $x \in \bbR$, and $B$ represents an operator of multiplication by $r(x)$ in $L^2(\bbR; dx)$, which, in general, is not a weight, that is, it is not nonnegative a.e.\ on $\bbR$. Our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients $q$ and $r$ and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space $L^2(\bbR; dx)$. Our approach relies on rewriting the eigenvalue problem $A ψ= λB ψ$ in the form $A^{-1/2} B A^{-1/2} χ= λ^{-1} χ$, $χ= A^{1/2} ψ$, and a careful study of (appropriate realizations of) the operator $A^{-1/2} B A^{-1/2}$ in $L^2(\bbR; dx)$. In the course of our treatment we employ a supersymmetric formalism which permits us to factor the second-order operator $T$ into a product of two first-order operators familiar from (and inspired by) Miura's transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients $q$ and $r$, where $q$ may be a distribution and $r$ generates a measure and hence no smoothness is assumed for $q$ and $r$.

math.SP↗

arXiv2013

On Spectral Theory for Schrödinger Operators with Operator-Valued Potentials

Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $τ= - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$ or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in \cB(\cH)$ such that $V(\cdot)$ is weakly measurable and the operator norm $\|V(\cdot)\|_{\cB(\cH)}$ is locally integrable. We consider self-adjoint half-line $L^2$-realizations $H_α$ in $L^2((a,\infty); dx; \cH)$ associated with $τ$, assuming $a$ to be a regular endpoint necessitating a boundary condition of the type $\sin(α)u'(a) + \cos(α)u(a)=0$, indexed by the self-adjoint operator $α= α^* \in \cB(\cH)$. In addition, we study self-adjoint full-line $L^2$-realizations $H$ of $τ$ in $L^2(\bbR; dx; \cH)$. In either case we treat in detail basic spectral theory associated with $H_α$ and $H$, including Weyl--Titchmarsh theory, Green's function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.

math.SP↗

arXiv2012

Abstract Wave Equations and Associated Dirac-Type Operators

We discuss the unitary equivalence of generators $G_{A,R}$ associated with abstract damped wave equations of the type $\ddot{u} + R \dot{u} + A^*A u = 0$ in some Hilbert space $\mathcal{H}_1$ and certain non-self-adjoint Dirac-type operators $Q_{A,R}$ (away from the nullspace of the latter) in $\mathcal{H}_1 \oplus \mathcal{H}_2$. The operator $Q_{A,R}$ represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of $A^*A$. In addition to the unitary equivalence results concerning $G_{A,R}$ and $Q_{A,R}$, we provide a detailed study of the domain of the generator $G_{A,R}$, consider spectral properties of the underlying quadratic operator pencil $M(z) = |A|^2 - iz R - z^2 I_{\mathcal{H}_1}$, $z\in\mathbb{C}$, derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The special example where $R$ represents an appropriate function of $|A|$ is treated in depth and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator. The cases of undamped (R=0) and damped ($R \neq 0$) abstract wave equations as well as the cases $A^* A \geq εI_{\mathcal{H}_1}$ for some $ε> 0$ and $0 \in σ(A^* A)$ (but 0 not an eigenvalue of $A^*A$) are separately studied in detail.

math.AP↗

arXiv2012

Boundary Data Maps and Krein's Resolvent Formula for Sturm-Liouville Operators on a Finite Interval

We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm-Liouville operators on the finite interval $(a,b)$, to more general boundary conditions. While earlier studies of boundary data maps focused on the case of general separated boundary conditions at $a$ and $b$, the present work develops a unified treatment for all possible self-adjoint boundary conditions (i.e., separated as well as non-separated ones). In the course of this paper we describe the connections with Krein's resolvent formula for self-adjoint extensions of the underlying minimal Sturm-Liouville operator (parametrized in terms of boundary conditions), with some emphasis on the Krein extension, develop the basic trace formulas for resolvent differences of self-adjoint extensions, especially, in terms of the associated spectral shift functions, and describe the connections between various parametrizations of all self-adjoint extensions, including the precise relation to von Neumann's basic parametrization in terms of unitary maps between deficiency subspaces.

math.SP↗

arXiv2011

Weak Convergence of Spectral Shift Functions for One-Dimensional Schrödinger Operators

We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on the finite interval $(0,R)$ converge in the infinite volume limit $R\to\infty$ to the half-line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at $x=0$ and $x=R$.

math.SP↗