Searcharxiv⌕ Search

arXiv subjects

David Krejcirik

Publications and source records attributed to David Krejcirik.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative uniform resolvent estimates

We derive quantitative uniform resolvent estimates for Schrödinger operators on the half-line with inverse-square potentials, which provide a sharp behaviour in the limit of large coupling. Our approach is based on a matrix representation of the boundary value of a weighted resolvent. The partial wave decomposition then turns these one-dimensional channel estimates into explicit weighted resolvent estimates for the Laplacian, its inverse-square potential perturbations and for the magnetic Laplacian with an Aharonov--Bohm potential. We also obtain exact Simon-type identities for the imaginary parts of the weighted resolvents of these operators.

math.AP↗
arXiv2026

A reverse Faber--Krahn inequality for the Robin Laplacian with negative boundary parameter: small coupling in all dimensions

We establish Bareket's conjecture from 1977 for convex domains in all dimensions in the regime of weak boundary coupling. In other words, we consider the Laplace operator, subject to negative boundary conditions, and show that the ball maximises the first eigenvalue among all bounded convex domains of fixed volume, provided that the boundary parameter is sufficiently close to zero. The smallness depends on the volume and dimension only. The proof relies on a comparison with spherical shells with combined Neumann--Robin boundary conditions obtained via the method of parallel coordinates, which we manage to extend to all dimensions, and on a careful analysis of the corresponding radial problem.

math.SP↗
arXiv2026

Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane

We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the domain, there exist discrete eigenvalues below $\frac14$ if, and only if, the Robin parameter is below a non-positive critical constant, which depends on the shape of the domain. As the main result, we prove that the lowest Robin eigenvalue for the exterior of a bounded geodesically convex domain $Ω$ in the hyperbolic plane does not exceed such an eigenvalue for the exterior of the geodesic disk, whose geodesic curvature of the boundary is not smaller than the averaged geodesic curvature of the boundary of $Ω$. This result implies as a consequence that under fixed area or fixed perimeter constraints the exterior of the geodesic disk maximises the lowest Robin eigenvalue among exteriors of bounded geodesically convex domains. Moreover, we obtain under the same geometric constraints a reverse inequality between the critical constants.

math.AP↗
arXiv2026

Sharp estimates for the Laplacian torsional rigidity with negative Robin boundary conditions

Motivated by pioneering works of Bandle and Wagner, given a bounded Lipschitz domain $Ω\subset \mathbb R^d$ with $d\ge3$, we consider the Robin-Laplacian torsional rigidity $τ_α(Ω)$ with negative boundary parameter $α$ and we show that sharp inequalities for $τ_α(Ω)$ hold if $|α|$ is small enough. In particular, we prove that, if $|α|$ is smaller than the first non-trivial Steklov-Laplacian eigenvalue, then the ball maximises $τ_α(Ω)$ among all convex domains under perimeter or volume constraints.This solves an open problem raised by Bandle and Wagner. We also prove the result in the planar case among simply connected sets and under perimeter constraint.

math.OC↗
arXiv2025

The Laplacian with complex magnetic fields

We study the two-dimensional magnetic Laplacian when the magnetic field is allowed to be complex-valued. Under the assumption that the imaginary part of the magnetic potential is relatively form-bounded with respect to the real part of the magnetic Laplacian, we introduce the operator as an m-sectorial operator. In two dimensions, sufficient conditions are established to guarantee that the resolvent is compact. In the case of non-critical complex magnetic fields, a WKB approach is used to construct semiclassical pseudomodes, which do not exist when the magnetic field is real-valued.

math-ph↗
arXiv2025

Is the optimal magnetic rectangle a square?

We are concerned with the dependence of the lowest eigenvalue of the magnetic Dirichlet Laplacian on the geometry of rectangles, subject to homogeneous fields. We conjecture that the square is a global minimiser both under the area or perimeter constraints. Contrary to the well-known magnetic-free analogue, the present spectral problem does not admit explicit solutions. By establishing lower and upper bound to the eigenvalue, we establish the conjecture for weak magnetic fields. Moreover, we relate the validity of the conjecture to the simplicity of the eigenvalue and symmetries of minimisers of a non-convex minimisation problem.

math.SP↗
arXiv2025

Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump

In this paper we present a complete spectral analysis of Dirac operators with non-Hermitian matrix potentials of the form $i\operatorname{sgn}(x)+V(x)$ where $V\in L^1$. For $V=0$ we compute explicitly the matrix Green function. This allows us to determine the spectrum, which is purely essential, and its different types. It also allows us to find sharp enclosures for the pseudospectrum and its complement, in all parts of the complex plane. Notably, this includes the instability region, corresponding to the interior of the band that forms the numerical range. Then, with the help of a Birman-Schwinger principle, we establish in precise manner how the spectrum and pseudospectrum change when $V\not=0$, assuming the hypotheses $\|V\|_{L^1}<1$ or $V\in L^1\cap L^p$ where $p>1$. We show that the essential spectra remain unchanged and that the $\varepsilon$-pseudospectrum stays close to the instability region for small $\varepsilon$. We determine sharp asymptotic for the discrete spectrum, whenever $V$ satisfies further conditions of decay at infinity. Finally, in one of our main findings, we give a complete description of the weakly-coupled model.

math.SP↗
arXiv2025

Critical quasilinear Schroedinger equations with electromagnetic fields

The p-Laplace operator in the entire N-dimensional Euclidean space, subject to external electromagnetic potentials, is investigated. In the general case 1<p<N, the existence of at least one solution of mountain pass type to a weighted critical equation is proved. Our technique relies on variational methods and faces a twofold difficulty: double lack of compactness, which requires concentration compactness arguments; and a complex quasilinear framework, which entails appropriate inequalities.

math.AP↗
arXiv2025

Virtual bound states of the Pauli operator with an Aharonov-Bohm potential

A maximal realisation of the two-dimensional Pauli operator, subject to Aharonov--Bohm magnetic field, is investigated. Contrary to the case of the Pauli operator with regular magnetic potentials, it is shown that both components of the Pauli operator are critical. Asymptotics of the weakly coupled eigenvalues, generated by electric (not necessarily self-adjoint) perturbations, are derived.

math-ph↗
arXiv2024

Abrupt changes in the spectra of the Laplacian with constant complex magnetic field

We analyze the spectrum of the Laplace operator, subject to homogeneous complex magnetic fields in the plane. For real magnetic fields, it is well-known that the spectrum consists of isolated eigenvalues of infinite multiplicities (Landau levels). We demonstrate that when the magnetic field has a nonzero imaginary component, the spectrum expands to cover the entire complex plane. Additionally, we show that the Landau levels (appropriately rotated and now embedded in the complex plane) persists, unless the magnetic field is purely imaginary in which case they disappear and the spectrum becomes purely continuous.

math.SP↗
arXiv2024

The virial theorem and the method of multipliers in spectral theory

We provide a link between the virial theorem in functional analysis and the method of multipliers in theory of partial differential equations. After giving a physical insight into the techniques, we show how to use them to deduce the absence of eigenvalues and other spectral properties of electromagnetic quantum Hamiltonians. We focus on our recent developments in non-self-adjoint settings, namely on Schroedinger operators with matrix-valued potentials, relativistic operators of Pauli and Dirac types, and complex Robin boundary conditions.

math.SP↗
arXiv2024

Quasi-conical domains with embedded eigenvalues

The spectrum of the Dirichlet Laplacian on any quasi-conical open set coincides with the non-negative semi-axis. We show that there is a connected quasi-conical open set such that the respective Dirichlet Laplacian has a positive (embedded) eigenvalue. This open set is constructed as the tower of cubes of growing size connected by windows of vanishing size. Moreover, we show that the sizes of the windows in this construction can be chosen so that the absolutely continuous spectrum of the Dirichlet Laplacian is empty.

math.SP↗
arXiv2024

Dirac operators on the half-line: stability of spectrum and non-relativistic limit

We consider Dirac operators on the half-line, subject to generalised infinite-mass boundary conditions. We derive sufficient conditions which guarantee the stability of the spectrum against possibly non-self-adjoint potential perturbations and study the optimality of the obtained results. Finally, we establish a non-relativistic limit which makes a relationship of the present model to the Robin Laplacian on the half-line.

math.SP↗
arXiv2024

Spectral determinant for the wave equation on an interval with Dirac damping

A closed formula for the spectral determinant for the wave equation on a bounded interval, subject to Dirichlet boundary conditions and an $α$-multiple of the Dirac $δ$-type damping, is derived. Depending on the choice of the branch cut of the logarithm used in its definition, the spectral determinant diverges either for $α=2$ or $α=-2$.

math.SP↗
arXiv2024

Numerical optimisation of Dirac eigenvalues

Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the optimality of the regular polygon (respectively, disk) among all polygons of a given number of sides (respectively, arbitrary sets), subject to area or perimeter constraints. We consider the three lowest positive eigenvalues and their ratios. Roughly, we find results analogous to known or expected for the Dirichlet Laplacian, except for the third eigenvalue which does not need to be minimised by the regular polygon (respectively, the disk) for all masses. In addition to the numerical results, a new, mass-dependent upper bound to the lowest eigenvalue in rectangles is proved and its extension to arbitrary quadrilaterals is conjectured.

math.OC↗
arXiv2024

Semiclassical asymptotics of the Bloch--Torrey operator in two dimensions

The Bloch--Torrey operator $-h^2Δ+e^{iα}x_1$ on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming $α\in\left[0,\frac{3π}{5}\right)$ and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of the eigenvalues with the smallest real part in the limit $h \to 0$ are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.

math.SP↗
arXiv2023

Curved nonlinear waveguides

The Dirichlet p-Laplacian in tubes of arbitrary cross-section along infinite curves in Euclidean spaces of arbitrary dimension is investigated. First, it is shown that the gap between the lowest point of the generalised spectrum and the essential spectrum is positive whenever the cross-section is circular and the tube is asymptotically straight, untwisted and non-trivially bent. Second, a Hardy-type inequality is derived for unbent and non-trivially twisted tubes.

math.AP↗