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Fritz Gesztesy

Publications and source records attributed to Fritz Gesztesy.

At least 55 records · Page 3Linked to original sources
arXiv2020

On Spectral Theory for Schrödinger Operators with Strongly Singular Potentials

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and $2\times 2$ matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on $(a,\infty)$ with a potential strongly singular at the end point $a$. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is self-adjoint (equivalently, the associated minimally defined Schrödinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point $a$. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators.

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arXiv2019

On Birman's sequence of Hardy-Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_{n}\},$ for $n\in\mathbb{N},$ valid for functions in $C_0^{n}((0,\infty))\subset L^{2}((0,\infty)).$ In particular, $I_{1}$ is the classical (integral) Hardy inequality and $I_{2}$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_{n}([0,\infty))$ of functions defined on $[0,\infty).$ Moreover, $f\in H_{n}([0,\infty))$ implies $f^{\prime}\in H_{n-1}([0,\infty));$ as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0,$ these inequalities hold on the standard Sobolev space $H_0^{n}((0,b))$. Furthermore, in all cases, the Birman constants $[(2n-1)!!]^{2}/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^{2}((0,\infty))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.

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arXiv2019

On Weighted Hardy-Type Inequalities

We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.

math.CA↗
arXiv2019

The inverse approach to Dirac-type systems based on the $A$-function concept

The principal objective in this paper is a new inverse approach to general Dirac-type systems modeled after B. Simon's 1999 inverse approach to half-line Schrödinger operators. In particular, we derive the so-called A-equation associated to Dirac-type systems and, given a fundamental positivity condition, we prove that this integro-differential equation for A is uniquely solvable. We show how to recover the matrix-valued potential coefficient from A. This approach is also applicable to matrix-valued Schrödinger operators on a half-line.

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arXiv2018

Trace formulas applied to the Riemann $ζ$-function

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

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arXiv2018

On traces and modified Fredholm determinants for half-line Schrödinger operators with purely discrete spectra

After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger operators $(- d^2/dx^2) + q$ on $(0,\infty)$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials $q$ that, for some fixed $C_0 > 0$, $\varepsilon > 0$, $x_0 \in (0, \infty)$, diverge at infinity in the manner that $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treat all self-adjoint boundary conditions at the left endpoint $0$.

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arXiv2018

Some Remarks on the Operator T^*T

This note deals with the operator $T^*T$, where $T$ is a densely defined operator on a complex Hilbert space. We reprove a recent result of Z. Sebestyén and Zs. Tarcsay [13]: If $T^*T$ and $TT^*$ are self-adjoint, then $T$ is closed. In addition, we describe the Friedrichs extension of $S^2$, where $S$ is a symmetric operator, recovering results due to Yu. Arlinskiĭ and Yu. Kovalev [1], [2].

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arXiv2017

On the Global Limiting Absorption Principle for Massless Dirac Operators

We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = α\cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.

math.SP↗
arXiv2017

A spectral shift function for Schrödinger operators with singular interactions

For the pair $\{-Δ, -Δ-αδ_\mathcal{C}\}$ of self-adjoint Schrödinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here $δ_{\cal{C}}$ denotes a singular $δ$-potential which is supported on a smooth compact hypersurface $\mathcal{C}\subset\mathbb{R}^n$ and $α$ is a real-valued function on $\mathcal{C}$.

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arXiv2017

Renormalized oscillation theory for Hamiltonian systems

We extend a result on renormalized oscillation theory, originally derived for Sturm-Liouville and Dirac-type operators on arbitrary intervals in the context of scalar coefficients, to the case of general Hamiltonian systems with block matrix coefficients. In particular, this contains the cases of general Sturm-Liouville and Dirac-type operators with block matrix-valued coefficients as special cases. The principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate Wronskians of solutions, oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails.

math.CA↗
arXiv2017

Factorizations and Hardy-Rellich-Type Inequalities

The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{α,β} := - Δ+ α|x|^{-2} x \cdot \nabla + β|x|^{-2}$, $α, β\in \mathbb{R}$, $x \in \mathbb{R}^n \backslash \{0\}$, $n \in \mathbb{N}$, $n \geq 2$, and its formal adjoint, denoted by $T_{α,β}^+$, we show that nonnegativity of $T_{α,β}^+ T_{α,β}$ on $C_0^{\infty}(\mathbb{R}^n \backslash \{0\})$ implies the fundamental inequality, \begin{align} \int_{\mathbb{R}^n} [(Δf)(x)]^2 \, d^n x &\geq [(n - 4) α- 2 β] \int_{\mathbb{R}^n} |x|^{-2} |(\nabla f)(x)|^2 \, d^n x \notag \\ & \quad - α(α- 4) \int_{\mathbb{R}^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \notag \\ & \quad + β[(n - 4) (α- 2) - β] \int_{\mathbb{R}^n} |x|^{-4} |f(x)|^2 \, d^n x, \notag \end{align} for $f \in C^{\infty}_0(\mathbb{R}^n \backslash \{0\})$. A particular choice of values for $α$ and $β$ yields known Hardy-Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where $\mathbb{R}^n$ is replaced by an arbitrary open set $Ω\subseteq \mathbb{R}^n$ for functions $f \in C^{\infty}_0(Ω\backslash \{0\})$. Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order situations.

math.AP↗
arXiv2017

On (conditional) positive semidefiniteness in a matrix-valued context

In a nutshell, we intend to extend Schoenberg's classical theorem connecting conditionally positive semidefinite functions $F\colon \mathbb{R}^n \to \mathbb{C}$, $n \in \mathbb{N}$, and their positive semidefinite exponentials $\exp(tF)$, $t > 0$, to the case of matrix-valued functions $F \colon \mathbb{R}^n \to \mathbb{C}^{m \times m}$, $m \in \mathbb{N}$. Moreover, we study the closely associated property that $\exp(t F(- i \nabla))$, $t>0$, is positivity preserving and its failure to extend directly in the matrix-valued context.

math.CA↗
arXiv2016

Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves

Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which latter possibility appears to offer applications in the multi-dimensional case. A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [23] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.

math.AP↗
arXiv2016

On the index of meromorphic operator-valued functions and some applications

We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided.

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arXiv2016

Trace Formulas for a Class of non-Fredholm Operators: A Review

We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.

math.AP↗
arXiv2016

Spectral shift functions and Dirichlet-to-Neumann maps

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

math.SP↗
arXiv2016

Decoupling of Deficiency Indices and Applications to Schrödinger-Type Operators with Possibly Strongly Singular Potentials

We investigate closed, symmetric $L^2(\mathbb{R}^n)$-realizations $H$ of Schrödinger-type operators $(- Δ+V)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus Σ)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $Σ_j$, $j \in J$, of $n$-dimensional Lebesgue measure zero, with $J \subseteq \mathbb{N}$ an index set and $Σ= \bigcup_{j \in J} Σ_j$. We show that the defect, $\mathrm{def}(H)$, of $H$ can be computed in terms of the individual defects, $\mathrm{def}(H_j)$, of closed, symmetric $L^2(\mathbb{R}^n)$-realizations of $(- Δ+ V_j)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus Σ_j)}$ with potential coefficient $V_j$ localized around the singularity $Σ_j$, $j \in J$, where $V = \sum_{j \in J} V_j$. In particular, we prove \[ \mathrm{def}(H) = \sum_{j \in J} \mathrm{def}(H_j), \] including the possibility that one, and hence both sides equal $\infty$. We first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of Schrödinger-type operators in $L^2(\mathbb{R}^n)$. Moreover, we also show how operator (and form) bounds for $V$ relative to $H_0= - Δ\upharpoonright_{H^2(\mathbb{R}^n)}$ can be estimated in terms of the operator (and form) bounds of $V_j$, $j \in J$, relative to $H_0$. Again, we first prove an abstract result and then show its applicability to Schrödinger-type operators in $L^2(\mathbb{R}^n)$. Extensions to second-order (locally uniformly) elliptic differential operators on $\mathbb{R}^n$ with a possibly strongly singular potential coefficient are treated as well.

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