arXiv subjects

Sebastian Egger

Publications and source records attributed to Sebastian Egger.

12 recordsLinked to original sources

arXiv2020

Defining the spectral position of a Neumann domain

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.

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arXiv2019

Attractive conical surfaces create infinitely many bound states

In this paper we study spectral properties of a three-dimensional Schr\"odinger operator $-\Delta+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectrum which itself is identified as the ground-state energy of a certain one-dimensional operator. Most importantly, based on a result of Kirsch and Simon we are able to establish the asymptotic behavior of the eigenvalue counting function using an explicit spectral-geometric quantity associated with the cross-section. This shows a universal character of some previous results on conical layers and $\delta$-potentials created by conical surfaces.

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arXiv2018

Bound states of a pair of particles on the half-line with a general interaction potential

In this paper we study an interacting two-particle system on the positive half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues.

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arXiv2018

An asymptotic expansion of the trace of the heat kernel of a singular two-particle contact interaction in one dimension

The regularized trace of the heat kernel of a one-dimensional Schr\"odinger operator with a singular two-particle contact interaction being of Lieb-Liniger type is considered. We derive a complete small-time asymptotic expansion in (fractional) powers of the time, $t$. Most importantly, we do not invoke standard parametrix constructions for the heat kernel. Instead, we first derive the large-energy expansion of the regularized trace of the resolvent for the considered operator. Then, we exploit that the resolvent may be obtained by a Laplace transformation of the heat semi-group, and an application of a suitable inverse Watson lemma eventually yields the small-$t$ asymptotic expansion of the heat-kernel trace.

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arXiv2018

Neumann Domains on Graphs and Manifolds

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in a few recent works and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.

math.SP↗

arXiv2017

Scattering properties of two singularly interacting particles on the half-line

We analyze scattering in a system of two (distinguishable) particles moving on the half-line $\overline{\rz}_+$ under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the obtained knowledge about the kernel of the resolvent we prove a version of the limiting absorption principle. Furthermore, by an appropriate adaptation of the Lippmann-Schwinger approach we are able to construct generalized eigenfunctions which consequently allow us to establish an explicit expression for the (on-shell) scattering amplitude. An approximation of the scattering amplitude in the weak-coupling limit is also derived.

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arXiv2016

The Berry-Keating operator on a lattice

We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Keating.

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arXiv2015

A Gutzwiller trace formula for large hermitian matrices

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schr\"odinger equation. In the framework of Weyl quantisation, we construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantisation condition for eigenvalues as well as some simple examples.

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arXiv2014

Heat-kernel and resolvent asymptotics for Schroedinger operators on metric graphs

We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class regularisations of the resolvents. Our main result is a complete asymptotic expansion of the trace of the (regularised) heat-semigroup generated by the Schroedinger operator. We also determine the leading coefficients in the expansion explicitly.

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arXiv2013

Zero modes of quantum graph Laplacians and an index theorem

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.

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arXiv2011

An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics $\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$ for the eigenvalue counting function. Based on an exact trace formula, the Vorono\"i summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.

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arXiv2010

A new proof of the Vorono\"i summation formula

We present a short alternative proof of the Vorono\"i summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schr\"odinger operator on a non-compact quantum graph \mathfrak{G} [S. Egger n\'e Endres and F. Steiner, J. Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new proof of a non-trivial identity for a particular Lambert series which involves the divisor function d(n) and is identical with the trace of the Euclidean wave group of the Laplacian on the infinite graph \mathfrak{G}.

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