arXiv subjects

Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 73 records · Page 4Linked to original sources

arXiv2018

Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions

This paper summarizes the contents of a plenary talk at the Pan African Congress of Mathematics held in Rabat in July 2017. We provide a survey of recent results on spectral properties of Schrödinger operators with singular interactions supported by manifolds of codimension one and of Robin billiards with the focus on the geometrically induced discrete spectrum and its asymptotic expansions in term of the model parameters.

math-ph↗

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.

math.SP↗

arXiv2018

A family of quantum graph vertex couplings interpolating between different symmetries

The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the $δ$ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family in terms of circulant matrices, we analyze the spectral and scattering property of such vertices, and investigate the band spectrum of the corresponding square lattice graph.

math-ph↗

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.

math.SP↗

arXiv2018

A geometric Iwatsuka type effect in quantum layers

We study motion of a charged particle confined to Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.

math-ph↗

arXiv2018

Smilansky-Solomyak model with a $δ'$-interaction

We investigate a strongly singular version of the model of irreversible dynamics proposed by Smilansky and Solomyak in which the interaction responsible for an abrupt change of the spectrum is of $δ'$ type. We determine the spectrum in both the subcritical and supercritical regimes and discuss its character as well as its asymptotic properties of the discrete spectrum in terms of the coupling constant.

math-ph↗

arXiv2017

Geometrically induced spectral effects in tubes with a mixed Dirichlet-Neumann boundary

We investigate spectral properties of the Laplacian in $L^2(Q)$, where $Q$ is a tubular region in $\mathbb{R}^3$ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.

math.SP↗

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 $β\rightarrow 0+$.

math.SP↗

arXiv2017

Aharonov and Bohm vs. Welsh eigenvalues

We consider a class of two-dimensional Schrödinger operator with a singular interaction of the $δ$ type and a fixed strength $β$ supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov-Bohm flux $α\in [0,\frac12]$ in the center. It is shown that if $β\ne 0$, there is a critical value $α_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $α<α_\mathrm{crit} $, while for $α\geα_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $α\in (0,\frac12)$ and $|β|$ small enough.

math.SP↗

arXiv2017

Periodic quantum graphs from the Bethe-Sommerfeld perspective

The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is finite. To the date its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the `Bethe-Sommerfeld property', that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with the vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a $δ$ coupling at the vertices and a suitable irrational ratio of the edges. Our result allows to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to the date.

math-ph↗

arXiv2017

Quantum graphs with vertices of a preferred orientation

Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.

math-ph↗

arXiv2017

A magnetic version of the Smilansky-Solomyak model

We analyze spectral properties of two mutually related families of magnetic Schrödinger operators, $H_{\mathrm{Sm}}(A)=(i \nabla +A)^2+ω^2 y^2+λy δ(x)$ and $H(A)=(i \nabla +A)^2+ω^2 y^2+ λy^2 V(x y)$ in $L^2(R^2)$, with the parameters $ω>0$ and $λ<0$, where $A$ is a vector potential corresponding to a homogeneous magnetic field perpendicular to the plane and $V$ is a regular nonnegative and compactly supported potential. We show that the spectral properties of the operators depend crucially on the one-dimensional Schrödinger operators $L= -\frac{\mathrm{d}^2}{\mathrm{d}x^2} +ω^2 +λδ(x)$ and $L (V)= - \frac{\mathrm{d}^2}{\mathrm{d}x^2} +ω^2 +λV(x)$, respectively. Depending on whether the operators $L$ and $L(V)$ are positive or not, the spectrum of $H_{\mathrm{Sm}}(A)$ and $H(V)$ exhibits a sharp transition.

math.SP↗

arXiv2017

Quantum graphs with the Bethe-Sommerfeld property

In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned $δ$-coupling at the vertices.

quant-ph↗

arXiv2017

A regular analogue of the Smilansky model: spectral properties

We analyze spectral properties of the operator $H=\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} +ω^2y^2-λy^2V(x y)$ in $L^2(\mathbb{R}^2)$, where $ω\ne 0$ and $V\ge 0$ is a compactly supported and sufficiently regular potential. It is known that the spectrum of $H$ depends on the one-dimensional Schrödinger operator $L=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ω^2-λV(x)$ and it changes substantially as $\infσ(L)$ switches sign. We prove that in the critical case, $\infσ(L)=0$, the spectrum of $H$ is purely essential and covers the interval $[0,\infty)$. In the subcritical case, $\infσ(L)>0$, the essential spectrum starts from $ω$ and there is a non-void discrete spectrum in the interval $[0,ω)$. We also derive a bound on the corresponding eigenvalue moments.

math-ph↗

arXiv2017

Cantor spectra of magnetic chain graphs

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $δ$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

math-ph↗

arXiv2017

On the spectrum of leaky surfaces with a potential bias

We discuss operators of the type $H = -Δ+ V(x) - αδ(x-Σ)$ with an attractive interaction, $α>0$, in $L^2(\mathbb{R}^3)$, where $Σ$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $V$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0=α^2$. We show that $σ_\mathrm{disc}(H)$ is then empty if the bias is supported in the `exterior' region, while in the opposite case isolated eigenvalues may exist.

math-ph↗

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.

math.SP↗

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↗