Searcharxiv⌕ Search

arXiv subjects

Christoph Fischbacher

Publications and source records attributed to Christoph Fischbacher.

At least 19 recordsLinked to original sources
arXiv2025

Complete Non-Selfadjointness of Extensions of Symmetric Operators with Bounded Dissipative Perturbations

Using boundary triples, we develop an abstract framework to investigate the complete non-selfadjointness of the maximally dissipative extensions of dissipative operators of the form $S+iV$, where $S$ is symmetric with equal finite defect indices and $V$ is a bounded non-negative operator. Our key example is the dissipative Schrödinger operator $-\tfrac{d^2}{dx^2}+\mathrm{i} V$ on the interval.

math.SP↗
arXiv2025

Maximally dissipative and self-adjoint extensions of $K$-invariant operators

We introduce the notion of $K$-invariant operators, $S$, (in a Hilbert space) with respect to a bounded and boundedly invertible operator $K$ defined via $K^*SK=S$. Conditions such that self-adjoint and maximally dissipative extensions of $K$-invariant symmetric operators are also $K$-invariant are investigated. In particular, the Friedrichs and Krein--von Neumann extensions of a nonnegative $K$-invariant symmetric operator are shown to always be $K$-invariant, while the Friedrichs extension of a $K$-invariant sectorial operator is as well. We apply our results to the case of Sturm--Liouville operators where $K$ is given by $(Kf)(x)=A(x)f(ϕ(x))$ under appropriate assumptions. Sufficient conditions on the coefficient functions for $K$-invariance to hold are shown to be related to Schröder's equation and all $K$-invariant self-adjoint extensions are characterized. Explicit examples are discussed including a Bessel-type Schrödinger operator satisfying a nontrivial $K$-invariance on the half-line.

math.SP↗
arXiv2025

Sharp Polynomial Velocity Decay Bounds for Multidimensional Periodic Schrödinger Operators

We investigate periodic Schrödinger operators in arbitrary dimensions in the large coupling regime. Our results establish that both the Lieb--Robinson velocity and the asymptotic velocity decay at an inverse polynomial rate in the coupling, with the precise exponent determined by the period of the underlying potential. In particular, we obtain sharp polynomial decay rates that capture the precise dependence on the periodic structure.

math-ph↗
arXiv2025

Some Remarks on the Product Formula for Defect Numbers of Closed Operators

This largely pedagogical paper recalls some facts on defect numbers of products of closed operators employing results from the theory of semi-Fredholm operators and then applies these facts to positive integer powers of symmetric operators and subsequently to certain minimal Sturm--Liouville and minimal higher even-order ordinary and partial differential operators. We also point out some unexpected missed opportunities when comparing the work of different groups on this subject.

math.FA↗
arXiv2025

Nonnegative extensions of Sturm-Liouville operators with an application to problems with symmetric coefficient functions

The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm-Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative self-adjoint extensions of the minimal operator in terms of generalized boundary values, as well as a parameterization of all nonnegative extensions when fixing a boundary condition at one endpoint. In addition, we investigate problems where the coefficient functions are symmetric about the midpoint of a finite interval, illustrating how every self-adjoint operator of this form is unitarily equivalent to the direct sum of two self-adjoint operators restricted to half of the interval. We also extend these result to symmetric two interval problems. We then apply our previous results to parameterize all nonnegative extensions of operators with symmetric coefficient functions. We end with an example of an operator with a symmetric Bessel-type potential (i.e., symmetric confining potential) and an application to integral inequalities.

math.SP↗
arXiv2025

An analysis of non-selfadjoint first-order differential operators with non-local point interactions

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by ${i\partial_x+V+k\langle δ,\cdot\rangle}$. We give precise estimates on the location of the eigenvalues on the complex plane and prove that the root vectors of these operators form Riesz bases of $L^2(0,2π)$. Under the additional assumption that the operator is maximally dissipative, we prove that it can have at most one real eigenvalue, and given any $λ\in\mathbb{R}$, we explicitly construct the unique operator realization such that $λ$ is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.

math.SP↗
arXiv2024

Abstract Left-Definite Theory: A Model Operator Approach, Examples, Fractional Sobolev Spaces, and Interpolation Theory

We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev spaces, and domains of (strictly) positive fractional powers of operators, employing interpolation theory. In particular, we explicitly describe the domains of positive powers of the harmonic oscillator operator in $L^2(\mathbb{R})$ $\big($and hence that of the Hermite operator in $L^2\big(\mathbb{R}; e^{-x^2}dx)\big)\big)$ in terms of fractional Sobolev spaces, certain commutation techniques, and positive powers of (the absolute value of) the operator of multiplication by the independent variable in $L^2(\mathbb{R})$.

math.SP↗
arXiv2024

Slow propagation velocities in Schrödinger operators with large periodic potential

Schrödinger operators with periodic potential have generally been shown to exhibit ballistic transport. In this work, we investigate if the propagation velocity, while positive, can be made arbitrarily small by a suitable choice of the periodic potential. We consider the discrete one-dimensional Schrödinger operator $Δ+μV$, where $Δ$ is the discrete Laplacian, $V$ is a $p$-periodic non-degenerate potential, and $μ>0$. We establish a Lieb-Robinson-type bound with a group velocity that scales like $\mathcal{O}(1/μ)$ as $μ\rightarrow\infty$. This shows the existence of a linear light cone with a maximum velocity of quantum propagation that is decaying at a rate proportional to $1/μ$. Furthermore, we prove that the asymptotic velocity, or the average velocity of the time-evolved state, exhibits a decay proportional to $\mathcal{O}(1/μ^{p-1})$ as $μ\rightarrow\infty$.

math-ph↗
arXiv2023

Entanglement Entropy Bounds for Droplet States of the XXZ Model on the Strip

The scaling behavior of the entanglement entropy of droplet states in Heisenberg spin-1/2 XXZ model defined on a strip of width $M$ under the presence of a non-negative background magnetic field is investigated. Without any assumptions on $V$, a logarithmically corrected area law is shown. Assuming that the values of $V$ are i.i.d. random variables, an area law in expectation is obtained.

math-ph↗
arXiv2022

Complete Non-Selfadjointness for Schrödinger Operators on the Semi-Axis

In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterisation of these extensions and the corresponding subspaces and present a specific example.

math.SP↗
arXiv2020

Extensions of dissipative operators with closable imaginary part

Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f\mapsto \mbox{Im}\langle f,Af\rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

math.FA↗
arXiv2020

Entanglement Entropy Bounds in the Higher Spin XXZ Chain

We consider the Heisenberg XXZ spin-$J$ chain ($J\in\mathbb{N}/2$) with anisotropy parameter $Δ$. Assuming that $Δ>2J$, and introducing threshold energies $E_{K}:=K\left(1-\frac{2J}Δ\right)$, we show that the bipartite entanglement entropy (EE) of states belonging to any spectral subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected area law with prefactor $(2\lfloor K/J\rfloor-2)$. This generalizes previous results by Beaud and Warzel as well as Abdul-Rahman, Stolz and one of the authors, who covered the spin-$1/2$ case.

math-ph↗
arXiv2020

Lower Bound to the Entanglement Entropy of the XXZ Spin Ring

We study the free XXZ quantum spin model defined on a ring of size $L$ and show that the bipartite entanglement entropy of eigenstates belonging to the first energy band above the vacuum ground state satisfies a logarithmically corrected area law. Along the way, we show a Combes-Thomas estimate for fiber operators which can also be applied to discrete many-particle Schrödinger operators on more general translation-invariant graphs.

math-ph↗
arXiv2019

Entanglement bounds in the XXZ quantum spin chain

We consider the XXZ spin chain, characterized by an anisotropy parameter $Δ>1$, and normalized such that the ground state energy is $0$ and the ground state given by the all spins up state. The energies $E_K = K(1-1/Δ)$, $K=1,2,\ldots$, can be interpreted as $K$-cluster break-up thresholds for down spin configurations. We show that, for every $K$, the bipartite entanglement of all states with energy below the $(K+1)$-cluster break-up satisfies a logarithmically corrected (or enhanced) area law. This generalizes a result by Beaud and Warzel, who considered energies in the droplet spectrum (i.e., below the 2-cluster break-up). For general $K$, we find an upper logarithmic bound with pre-factor $2K-1$. We show that this constant is optimal in the Ising limit $Δ=\infty$. Beaud and Warzel also showed that after introducing a random field and disorder averaging the enhanced area law becomes a strict area law, again for states in the droplet regime. For the Ising limit with random field, we show that this result does not extend beyond the droplet regime. Instead, we find states with energies arbitrarily close to the $(K+1)$-cluster break-up whose entanglement satisfies a logarithmically growing lower bound with pre-factor $K-1$.

math-ph↗
arXiv2019

A Schrödinger Operator Approach to Higher Spin XXZ Systems on General Graphs

We consider the spin-$J$ XXZ-Hamiltonian on general graphs $\mathcal{G}$ and show its equivalence to a direct sum of discrete many-particle Schrödinger type operators on what we call "$N$-particle graphs with maximal local occupation number $M$", where the kinetic term is described by a weighted Laplacian. Generalizing previous results for the spin-$1/2$ case, we give sufficient conditions for the existence of spectral gaps above the low-lying droplet band when the underlying graph $\mathcal{G}$ is (i) the chain and (ii) a strip of width $L$.

math-ph↗
arXiv2018

The Closed Extensions of a Closed Operator

Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed with respect to the graph norm of $A^*$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $A^*$. After this, we will express our results using the language of Gel'fand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence of densely defined operators that converge in the generalized sense to a non-densely defined operator, (ii) a non-closable extension of a symmetric operator and (iii) selfadjoint extensions of Laplacians with a generalized boundary condition.

math.FA↗
arXiv2018

A Birman-Krein-Vishik-Grubb theory for sectorial operators

We consider densely defined sectorial operators $A_\pm$ that can be written in the form $A_\pm=\pm iS+V$ with $\mathcal{D}(A_\pm)=\mathcal{D}(S)=\mathcal{D}(V)$, where both $S$ and $V\geq \varepsilon>0$ are assumed to be symmetric. We develop an analog to the Birmin-Krein-Vishik-Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $A_D$ of $A_+$ with the property that $\overline{A_+}\subset A_D\subset A_-^*$. Here, $D$ is an auxiliary operator from $\ker(A_-^*)$ to $\ker(A_+^*)$ that parametrizes the different extensions $A_D$. After this, we will give a criterion for when the quadratic form $ψ\mapsto\mbox{Re}\langleψ,A_Dψ\rangle$ is closable and show that the selfadjoint operator $\widehat{V}$ that corresponds to the closure is an extension of $V$. We will show how $\widehat{V}$ depends on $D$, which --- using the classical BKVG-theory of selfadjoint extensions --- will allow us to define a partial order on the real parts of $A_D$ depending on $D$. Applications to second order ordinary differential operators are discussed.

math.FA↗