arXiv subjects

Jonathan Rohleder

Publications and source records attributed to Jonathan Rohleder.

At least 37 records · Page 2Linked to original sources

arXiv2019

Laplacians on bipartite metric graphs

We study spectral properties of the standard (also called Kirchhoff) Laplacian and the anti-standard (or anti-Kirchhoff) Laplacian on a finite, compact metric graph. We show that the positive eigenvalues of these two operators coincide whenever the graph is bipartite; this leads to a precise relation between their eigenvalues enumerated with multiplicities and including the possible eigenvalue zero. Several spectral inequalities for, e.g., trees are among the consequences of this. In the second part we study inequalities between standard and Dirichlet eigenvalues in more detail and expose another connection to bipartiteness.

math.SP↗

arXiv2018

Spectral monotonicity for Schrödinger operators on metric graphs

We study the influence of certain geometric perturbations on the spectra of self-adjoint Schrödinger operators on compact metric graphs. Results are obtained for permutation invariant vertex conditions, which, amongst others, include $δ$ and $δ'$-type conditions. We show that adding edges to the graph or joining vertices changes the eigenvalues monotonically. However, the monotonicity properties may differ from what is known for the previously studied cases of Kirchhoff (or standard) and $δ$-conditions and may depend on the signs of the coefficients in the vertex conditions.

math.SP↗

arXiv2018

On the hot spots of quantum trees

We show that any second eigenfunction of the Laplacian with standard vertex conditions on a metric tree graph attains its extremal values only at degree one vertices, and give an example where these vertices do not realise the diameter of the graph.

math.SP↗

arXiv2018

Spectral enclosures for non-self-adjoint extensions of symmetric operators

The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator $B$. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for $A_{[B]}$ to have a non-empty resolvent set are provided in terms of the parameter $B$ and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for $A_{[B]}$ are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with $δ$-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.

math.SP↗

arXiv2018

Visibility of quantum graph spectrum from the vertices

We investigate the relation between the eigenvalues of the Laplacian with Kirchhoff vertex conditions on a finite metric graph and a corresponding Titchmarsh-Weyl function (a parameter-dependent Neumann-to-Dirichlet map). We give a complete description of all real resonances, including multiplicities, in terms of the edge lengths and the connectivity of the graph, and apply it to characterize all eigenvalues which are visible for the Titchmarsh-Weyl function.

math.SP↗

arXiv2017

A remark on the order of mixed Dirichlet-Neumann eigenvalues of polygons

Given the Laplacian on a planar, convex domain with piecewise linear boundary subject to mixed Dirichlet-Neumann boundary conditions, we provide a sufficient condition for its lowest eigenvalue to dominate the lowest eigenvalue of the Laplacian with the complementary boundary conditions (i.e. with Dirichlet replaced by Neumann and vice versa). The application of this result to triangles gives an affirmative partial answer to a recent conjecture. Moreover, we prove a further observation of similar flavor for right triangles.

math.SP↗

arXiv2017

Absolutely continuous spectrum for Laplacians on radial metric trees and periodicity

On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $δ$- and weighted $δ'$-couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolutely continuous spectrum implies that the sequence of geometric data of the tree as well as the coupling conditions are eventually periodic. On the other hand, we provide examples of self-adjoint, non-periodic couplings which admit absolutely continuous spectrum.

math.SP↗

arXiv2017

Quasi boundary triples and semibounded self-adjoint extensions

In this note semibounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second order PDEs on domains with non-compact boundaries.

math.SP↗

arXiv2016

Spectral theory for Schrödinger operators with $δ$-interactions supported on curves in $\mathbb R^3$

The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schrödinger operators with $δ$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten--von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.

math.SP↗

arXiv2016

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.

math.SP↗

arXiv2016

Eigenvalue estimates for the Laplacian on a metric tree

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular, we establish a sharp upper bound for the spectral gap, i.e. the smallest positive eigenvalue, and show that equilateral star graphs are the unique maximizers of the spectral gap among all trees of a given average length.

math.SP↗

arXiv2016

Eigenvalue inequalities for Schrödinger operators on unbounded Lipschitz domains

Given a Schrödinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary conditions, respectively. Moreover, we prove a strict inequality between the eigenvalues of two different elliptic differential operators on the same domain with Dirichlet boundary conditions.

math.SP↗

arXiv2015

Generalized interactions supported on hypersurfaces

We analyze a family of singular Schrödinger operators with local singular interactions supported by a hypersurface $Σ\subset \mathbb{R}^n, n \ge 2$, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of $Σ$ the interaction is characterized by four real parameters, the earlier studied case of $δ$- and $δ'$-interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

math-ph↗

arXiv2015

Recovering quantum graph spectrum from vertex data

We study the question to what extent spectral information of a Schrödinger operator on a finite, compact metric graph subject to standard or $δ$-type matching conditions can be recovered from a corresponding Titchmarsh-Weyl function on the boundary of the graph. In contrast to the case of ordinary or partial differential operators, the knowledge of the Titchmarsh-Weyl function is in general not sufficient for recovering the complete spectrum of the operator (or the potentials on the edges). However, it is shown that those eigenvalues with sufficiently high (depending on the cyclomatic number of the graph) multiplicities can be recovered. Moreover, we prove that under certain additional conditions the Titchmarsh-Weyl function contains even the full spectral information.

math.SP↗

arXiv2014

Titchmarsh-Weyl theory for Schrödinger operators on unbounded domains

In this note it is proved that the complete spectral data of selfadjoint Schrödinger operators on unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogues of classical facts from singular Sturm-Liouville theory.

math.SP↗

arXiv2014

An eigenvalue inequality for Schrödinger operators with $δ$ and $δ'$-interactions supported on hypersurfaces

We consider self-adjoint Schrödinger operators in $L^2 (\mathbb{R}^d)$ with a $δ$-interaction of strength $α$ and a $δ'$-interaction of strength $β$, respectively, supported on a hypersurface, where $α$ and $β^{-1}$ are bounded, real-valued functions. It is known that the inequality $0 < β\leq 4/α$ implies inequality of the eigenvalues of these two operators below the bottoms of the essential spectra. We show that this eigenvalue inequality is strict whenever $β< 4 / α$ on a nonempty, open subset of the hypersurface. Moreover, we point out special geometries of the interaction support, such as broken lines or infinite cones, for which strict inequality of the eigenvalues even holds in the borderline case $β= 4 / α$.

math.SP↗

arXiv2014

Spectral analysis of selfadjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions

The spectrum of a selfadjoint second order elliptic differential operator in $L^2(\mathbb{R}^n)$ is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and correspond to an interior and an exterior boundary value problem. This leads to PDE analogs of renowned facts in spectral theory of ODEs. The main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding Weyl functions, and are applied to the PDE setting afterwards.

math.SP↗

arXiv2014

Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains

On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.

math.AP↗