arXiv subjects

Vladimir Lotoreichik

Publications and source records attributed to Vladimir Lotoreichik.

At least 55 records · Page 3Linked to original sources

arXiv2018

The Landau Hamiltonian with $δ$-potentials supported on curves

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian $A_α=(i \nabla + A)^2 + αδ$ in $L^2(R^2)$ with a $δ$-potential supported on a finite $C^{1,1}$-smooth curve $Σ$ are studied. Here $A = \frac{1}{2} B (-x_2, x_1)^\top$ is the vector potential, $B>0$ is the strength of the homogeneous magnetic field, and $α\in L^\infty(Σ)$ is a position-dependent real coefficient modeling the strength of the singular interaction on the curve $Σ$. After a general discussion of the qualitative spectral properties of $A_α$ and its resolvent, one of the main objectives in the present paper is a local spectral analysis of $A_α$ near the Landau levels $B(2q+1)$. Under various conditions on $α$ it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of $α$. Furthermore, the use of Landau Hamiltonians with $δ$-perturbations as model operators for more realistic quantum systems is justified by showing that $A_α$ can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.

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arXiv2018

Spectral isoperimetric inequality for the $δ'$-interaction on a contour

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive $δ'$-interaction of a fixed strength, the support of which is a $C^2$-smooth contour. Under the constraint of a fixed length of the contour, we prove that the lowest eigenvalue is maximized by the circle. The proof relies on the min-max principle and the method of parallel coordinates.

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arXiv2018

Optimization of the lowest eigenvalue for leaky star graphs

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive $δ$-interaction of a fixed strength, the support of which is a star graph with finitely many edges of an equal length $L \in (0,\infty]$. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle.

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arXiv2018

Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $\mathsf{H}$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $\mathsf{H}$ has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for $\mathsf{H}$ by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrödinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer $\mathcal{P}$ at infinity.

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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.

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arXiv2018

The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

We propose a unique way how to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimising a 'Hilbert-Schmidt distance' to the original inner product among the entire class of admissible inner products. We prove that either the minimiser exists and is unique, or it does not exist at all. In the former case we derive a system of Euler-Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supplied by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

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arXiv2018

Spectral isoperimetric inequalities for singular interactions on open arcs

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive $δ$-interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. As a consequence of the result for $δ$-interaction, we obtain that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length.

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arXiv2017

Asymptotics of the bound state induced by $δ$-interaction supported on a weakly deformed plane

In this paper we consider the three-dimensional Schrödinger operator with a $δ$-interaction of strength $α> 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,βf(x))$, where $β\in [0,\infty)$ and $f\colon \mathbb{R}^2\rightarrow\mathbb{R}$, $f\not\equiv 0$, is a $C^2$-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrödinger operator coincides with $[-\frac14α^2,+\infty)$. We prove that for all sufficiently small $β> 0$ its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit $β\rightarrow 0+$. In particular, this eigenvalue tends to $-\frac14α^2$ exponentially fast as $β\rightarrow 0+$.

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arXiv2017

Asymptotics of resonances induced by point interactions

We consider the resonances of the self-adjoint three-dimensional Schrödinger operator with point interactions of constant strength supported on the set $X = \{ x_n \}_{n=1}^N$. The size of $X$ is defined by $V_X = \max_{π\inΠ_N} \sum_{n=1}^N |x_n - x_{π(n)}|$, where $Π_N$ is the family of all the permutations of the set $\{1,2,\dots,N\}$. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius $R$ behaves asymptotically linear $\frac{W_X}π R + \mathcal{O}(1)$ as $R \to \infty$, where the constant $W_X \in [0,V_X]$ can be seen as the effective size of $X$. Moreover, we show that there exist configurations of any number of points such that $W_X = V_X$. Finally, we construct an example for $N = 4$ with $W_X < V_X$, which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.

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arXiv2017

Optimisation of the lowest Robin eigenvalue in the exterior of a compact set

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact planar set, subject to attractive Robin boundary conditions. Under either a constraint of fixed perimeter or area, we show that the maximiser within the class of exteriors of convex sets is always the exterior of a disk. We also argue why the results fail without the convexity constraint and in higher dimensions.

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arXiv2017

Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (arXiv:1608.04896, to appear in J. Convex Anal.), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball.

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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.

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arXiv2017

Spectral analysis of photonic crystals made of thin rods

In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in $L^2(\mathbb{R}^2)$. The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of Schroedinger operators with point interactions.

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arXiv2016

A spectral isoperimetric inequality for cones

In this note we investigate three-dimensional Schrödinger operators with $δ$-interactions supported on $C^2$-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.

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arXiv2016

On resonances and bound states of Smilansky Hamiltonian

We consider the self-adjoint Smilansky Hamiltonian $\mathsf{H}_\varepsilon$ in $L^2(\mathbb{R}^2)$ associated with the formal differential expression $-\partial_x^2 - \frac12\big(\partial_y^2 + y^2) - \sqrt{2}\varepsilon y δ(x)$ in the sub-critical regime, $\varepsilon \in (0,1)$. We demonstrate the existence of resonances for $\mathsf{H}_\varepsilon$ on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterise resonances for small $\varepsilon > 0$. In addition, we refine the previously known results on the bound states of $\mathsf{H}_\varepsilon$ in the weak coupling regime ($\varepsilon\rightarrow 0+$). In the proofs we use Birman-Schwinger principle for $\mathsf{H}_\varepsilon$, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

math-ph↗

arXiv2016

Spectral and resonance properties of Smilansky Hamiltonian

We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically. Furthermore, we show that the model has a rich resonance structure.

math-ph↗

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.

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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.

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