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Joachim Kerner

Publications and source records attributed to Joachim Kerner.

At least 19 recordsLinked to original sources
arXiv2026

Ambarzumian-type theorems for Hermitian matrices with applications

A foundational result in inverse spectral theory due to Ambarzumian (1929) states that the Neumann Laplacian on an interval is not isospectral to the Neumann Laplacian with an additional non-zero potential. In this note, our aim is to investigate Ambarzumian-type theorems for certain classes of Hermitian matrices, including well-known matrices such as the discrete Laplacian on finite graphs. In addition, using different methods, we establish an Ambarzumian-type theorem for matrices with vanishing diagonal, in particular, the adjacency matrix on finite graphs. In this way, we generalize existing results on Ambarzumian-type theorems to general finite discrete graphs.

math.SP↗
arXiv2026

On the asymptotic behavior of the spectral gap for discrete Schrödinger operators

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schrödinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.

math.SP↗
arXiv2025

Spectral comparison results for Laplacians on discrete graphs

In the recent literature, various authors have studied spectral comparison results for Schrödinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note we derive such spectral comparison results in a rather general framework for general and possibly infinite discrete graphs. Along the way, we establish a discrete version of the local Weyl law whose proof does neither involve any Tauberian theorem nor the Weyl law as used in the continuous case.

math.SP↗
arXiv2024

The curious spectra and dynamics of non-locally finite crystals

This paper is devoted to the investigation of the spectral theory and dynamical properties of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour: for example, we construct a periodic graph whose Laplacian has purely singular continuous spectrum. Regarding point spectrum, and different to the locally finite case, we construct a graph with a partly flat band whose eigenvectors must have infinite support. Concerning dynamical aspects, under some assumptions we prove that motion remains ballistic along at least one layer. We also construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This provides a negative answer to an open question in this context. Furthermore, we include a discussion of the fractional Laplacian for which we prove a phase transition in its dynamical behaviour. Generally speaking, many questions still remain open, and we believe that the studied class of graphs can serve as a playground to better understand exotic spectra and dynamics.

math.SP↗
arXiv2024

On the surface area of graphs, related connectivity measures and spectral estimates

In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a special class of large graphs, called social graphs, that might prove interesting for applications. In addition, we derive spectral estimates involving the surface area and, as a main result, present an upper bound on the second eigenvalue for planar graphs which in some cases improves upon existing bounds from D. A. Spielman and S.-H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, and M. Plümer, Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs.

math.CO↗
arXiv2024

A modified local Weyl law and spectral comparison results for $δ'$-coupling conditions

We study Schrödinger operators on compact finite metric graphs subject to $δ'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different self-adjoint realisations on a given graph. Furthermore, using this spectral comparison result, we also study the limiting mean eigenvalue distance comparing $δ'$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .

math.SP↗
arXiv2023

On Bose-Einstein condensation in interacting Bose gases in the Kac-Luttinger model

We study interacting Bose gases of dimensions $2\le d \in \mathbb N$ at zero temperature in a random model known as the Kac-Luttinger model. Choosing the pair-interaction between the bosons to be of a mean-field type, we prove (complete) Bose-Einstein condensation in probability or with probability almost one into the minimizer of a Hartree-type functional. We accomplish this by building upon very recent results by Alain-Sol Sznitman on the spectral gap of the noninteracting Bose gas.

math-ph↗
arXiv2023

Some spectral comparison results on infinite quantum graphs

In this paper we establish spectral comparison results for Schrödinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite quantum graphs such as a modified local Weyl law. In this sense, we regard this paper as a starting point for a more thorough investigation of spectral comparison results on more general infinite metric graphs.

math.SP↗
arXiv2023

A note on Ambarzumian's theorem for quantum graphs

Based on the main result presented in a recent paper, we derive Ambarzumian-type theorems for Schrödinger operators defined on quantum graphs. We recover existing results such as the classical theorem by Ambarzumian and establish some seemingly new statements, too.

math.SP↗
arXiv2023

Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice

We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an ``all-or-nothing'' phenomenon in the sense that there is no perturbation which can destroy only one flat band while preserving the other.

math.SP↗
arXiv2022

Comparing the spectrum of Schrödinger operators on quantum graphs

We study Schrödinger operators on compact finite metric graphs subject to $δ$-coupling and standard boundary conditions. We compare the $n$-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. obtained for domains in $\mathbb{R}^2$ to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained in [arXiv:2212.09143] and [arXiv:2212.12531] for metric graphs. In addition, based on our main result, we introduce some notions of circumference for a (quantum) graph which might prove useful in the future.

math-ph↗
arXiv2021

On the spectral gap of higher-dimensional Schrödinger operators on large domains

We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.

math.SP↗
arXiv2021

Bose--Einstein condensation for particles with repulsive short-range pair interactions in a Poisson random external potential in $\mathbb R^d$

We study Bose gases in $d$ dimensions, $d \ge 2$, with short-range repulsive pair interactions, at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose--Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the eigenstates of the underlying one-particle Hamiltonian.

math-ph↗
arXiv2021

On the spectral gap of the path graph in the limit of large volume

In this paper we study the spectral gap of the path graph and illustrate an interesting effect which has been described recently in the continuous setting. More explicitly, in the large-volume limit and in the presence of a certain external potential, it is shown that the spectral gap converges to zero strictly faster than it does for the free Laplacian. The underlying mechanism is a combination of the increase in volume and an effective degeneracy of the ground state in the limiting regime.

math.SP↗
arXiv2020

On the effect of repulsive interactions on Bose-Einstein condensation in the Luttinger-Sy model

In this paper we investigate the effect of repulsive pair interactions on Bose-Einstein condensation in a well-established random one-dimensional system known as the Luttinger-Sy model at positive temperature. We study separately hard core interactions as well as a class of more general repulsive interactions, also allowing for a scaling of certain interaction parameters in the thermodynamic limit. As a main result, we prove in both cases that for sufficiently strong interactions all eigenstates of the non-interacting one-particle Luttinger-Sy Hamiltonian as well as any sufficiently localized one-particle state are almost surely not macroscopically occupied.

math-ph↗
arXiv2019

On a condition for type-I Bose-Einstein condensation in random potentials in $d$ dimensions

In this paper we discuss Bose-Einstein condensation (BEC) in systems of pairwise non-interacting bosons in random potentials in $d$ dimensions. Working in a rather general framework, we provide a "gap condition" which is sufficient to conclude existence of type-I BEC in probability and in the $r$th mean. We illustrate our results in the context of the well-known (one-dimensional) Luttinger-Sy model. Here, whenever the particle density exceeds a critical value, we show in addition that only the ground state is macroscopically occupied.

math-ph↗