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Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 19 recordsLinked to original sources
arXiv2026

Semiclassical resonances under local magnetic fields

We study resonances for the semiclassical magnetic Laplacian in the full plane with a compactly supported magnetic field in the framework of semiclassical complex scaling and black box scattering theory. Assuming that the magnetic field is locally constant, we prove the existence of semiclassical resonances near the Landau levels with exponentially small imaginary parts. We also prove that resonances emerge from a magnetic step discontinuity along a curved interface or a non-degenerate magnetic well, and in the vicinity of anharmonic Landau levels if the field has an isolated zero.

math-ph↗
arXiv2026

Discrete spectrum from local perturbations of leaky curves

We discuss spectrum of a class of singular Schr\"odinger operator models known as leaky curves and show that if the interaction support has a periodic shape, its local perturbations can give rise to a discrete spectrum below the continuum threshold even if they are of `zero mean'.

math.SP↗
arXiv2025

Spectral statistics of preferred orientation quantum graphs

We study the spectral statistics of quantum (metric) graphs whose vertices are equipped with preferred orientation vertex conditions. When comparing their spectral statistics to those predicted by suitable random matrix theory ensembles, one encounters some deviations. We point out these discrepancies and demonstrate that they occur in various graphs and even for Neumann-Kirchhoff vertex conditions, which was overlooked so far. Detailed explanations and computations are provided for this phenomena. To achieve this, we explore the combinatorics of periodic orbits, with a particular emphasis on counting Eulerian cycles.

math.SP↗
arXiv2025

A modified Fermi Golden Rule at threshold for 3D magnetic Schr\"odinger operators

In this paper we consider three-dimensional Schr\"odinger operators with a simple threshold eigenvalue. We show, under certain assumptions, that when a small magnetic field is introduced, this eigenvalue turns into a resonance in the time-dependent sense. We find the leading term in the asymptotic expansion of the imaginary part of the resonance and discuss the principal differences with respect to resonances induced by weak electric fields obtained previously in the literature.

math-ph↗
arXiv2025

A geometric approximation of non-local interface and boundary conditions

We analyze an approximation of a Laplacian subject to non-local interface conditions of a $\delta'$-type by Neumann Laplacians on a family of Riemannian manifolds with a sieve-like structure. We establish a (kind of) resolvent convergence for such operators, which in turn implies the convergence of spectra and eigenspaces, and demonstrate convergence of the corresponding semigroups. Moreover, we provide an explicit example of a manifold allowing to realize any prescribed integral kernel appearing in that interface conditions. Finally, we extend the discussion to similar approximations for the Laplacian with non-local Robin-type boundary conditions.

math.AP↗
arXiv2025

The Flea on the Magnetic Elephant

We investigate a two-dimensional magnetic Laplacian with two radially symmetric magnetic wells. Its spectral properties are determined by the tunneling between them. If the tunneling is weak and the wells are mirror symmetric, the two lowest eigenfunctions are localized in both wells being distributed roughly equally. In this note we show that an exponentially small symmetry violation can in this situation have a dramatic effect, making each of the eigenfunctions localized dominantly in one well only. This is reminiscent of the `flea on the elephant' effect for Schr\"odinger operators; our result shows that it has a purely magnetic counterpart.

math-ph↗
arXiv2025

Local perturbations of potential well arrays

We consider an equidistant array of disjoint potential wells in $\mathbb{R}^\nu,\: \nu\ge 2$, built over a straight line, and show that, under a restriction on the potential support aspect ratio, a perturbation consisting of longitudinal shifts of a finite number of them preserving the disjointness gives rise to a nonempty discrete spectrum below the threshold of the lowest spectral band.

math.SP↗
arXiv2025

First order operators on shrinking graph-like spaces

In this article we discuss the convergence of first order operators on a thickened graph (a graph-like space) towards a similar operator on the underlying metric graph. On the graph-like space, the first order operator is of the form exterior derivative (the gradient) on functions and its adjoint (the negative divergence) on closed 1-forms (irrotational vector fields). Under the assumption that each cross section of the tubular edge neighbourhood is convex, that each vertex neighbourhood is simply connected and under suitable uniformity assumptions (which hold in particular, if the spaces are compact) we establish generalised norm resolvent convergence of the first order operator on the graph-like space towards the one on the metric graph. The square of the first order operator is of Laplace type; on the metric graph, the function (0-form) component is the usual standard (Kirchhoff) Laplacian. A key ingredient in the proof is a uniform Gaffney estimate: such an estimate follows from an equality relating here the divergence operator with all (weak) partial derivatives and a curvature term, together with a (localised) Sobolev trace estimate.

math-ph↗
arXiv2024

Quantum lattice transport along an infinitely extended perturbation

We consider a periodic quantum graph in the form of a rectangular lattice with the $\delta$-coupling of strength $\gamma$ in the vertices perturbed by changing the latter at an infinite straight array of vertices to a $\widetilde\gamma\ne\gamma$. We analyze the band spectrum of the system and show that it remains preserved as a set provided $\widetilde\gamma>\gamma>0$ while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals $\frac12$.

math.SP↗
arXiv2024

Optimization of quantum graph eigenvalues with preferred orientation vertex conditions

We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which the ground state eigenvalue is maximized. Furthermore, for general finite metric graphs we provide upper bounds for all eigenvalues.

math-ph↗
arXiv2024

Magnetic transport due to a translationally invariant potential obstacle

We consider a two-dimensional system in which a charged particle is exposed to a homogeneous magnetic field perpendicular to the plane and a potential that is translationally invariant in one dimension. We derive several conditions on such a perturbation under which the Landau levels change into an absolutely continuous spectrum.

math-ph↗
arXiv2024

Spectral properties of hexagonal lattices with the -R coupling

We analyze the spectrum of the hexagonal lattice graph with a vertex coupling which manifestly violates the time reversal invariance and at high energies it asymptotically decouples edges at even degree vertices; a comparison is made to the case when such a decoupling occurs at odd degree vertices. We also show that the spectral character does not change if the equilateral elementary cell of the lattice is dilated to have three different edge lengths, except that flat bands are absent if those are incommensurate.

math-ph↗
arXiv2024

Optimizing the ground of a Robin Laplacian: asymptotic behavior

In this note we consider achieving the largest principle eigenvalue of a Robin Laplacian on a bounded domain $\Omega$ by optimizing the Robin parameter function under an integral constraint. The main novelty of our approach lies in establishing a close relation between the problem under consideration and the asymptotic behavior of the Dirichlet heat content of $\Omega$. By using this relation we deduce a two-term asymptotic expansion of the principle eigenvalue and discuss several applications.

math.SP↗
arXiv2024

Vertex coupling interpolation in quantum chain graphs

We analyze band spectrum of the periodic quantum graph in the form of a chain of rings connected by line segments with the vertex coupling which violates the time reversal invariance, interpolating between the $\delta$ coupling and the one determined by a simple circulant matrix. We find that flat bands are generically absent and that the negative spectrum is nonempty even for interpolation with a non-attractive $\delta$ coupling; we also determine the high-energy asymptotic behavior of the bands.

math-ph↗
arXiv2023

Cairo lattice with time-reversal non-invariant vertex couplings

We analyze the spectrum of a periodic quantum graph of the Cairo lattice form. The used vertex coupling violates the time reversal invariance and its high-energy behavior depends on the vertex degree parity; in the considered example both odd and even parities are involved. The presence of the former implies that the spectrum is dominated by gaps. In addition, we discuss two modifications of the model in which this is not the case, the zero limit of the length parameter in the coupling, and the sign switch of the coupling matrix at the vertices of degree three; while different they both yield the same probability that a randomly chosen positive energy lies in the spectrum.

math.SP↗
arXiv2023

On two-dimensional Dirac operators with $\delta$-shell interactions supported on unbounded curves with straight ends

In this paper we study the self-adjointness and spectral properties of two-dimensional Dirac operators with electrostatic, Lorentz scalar, and anomalous magnetic $\delta$-shell interactions with constant weights that are supported on a smooth unbounded curve that is straight outside a compact set and whose ends are rays that are not parallel to each other. For all possible combinations of interaction strengths we describe the self-adjoint realizations and compute their essential spectra. Moreover, we prove in different situations the existence of geometrically induced discrete eigenvalues.

math.SP↗
arXiv2023

Tunneling in soft waveguides:closing a book

We investigate the spectrum of a soft quantum waveguide in two dimensions of the generalized `bookcover' shape, that is, Schr\"odinger operator with the potential in the form of a ditch consisting of a finite curved part and straight asymptotes which are parallel or almost parallel pointing in the same direction. We show how the eigenvalues accumulate when the angle between the asymptotes tends to zero. In case of parallel asymptotes the existence of a discrete spectrum depends on the ditch profile. We prove that it is absent in the weak-coupling case, on the other hand, it exists provided the transverse potential is strong enough. We also present a numerical example in which the critical strength can be assessed.

math.SP↗