arXiv subjects

Pavel Exner

Publications and source records attributed to Pavel Exner.

At least 55 records · Page 3Linked to original sources

arXiv2020

Note on a Product Formula Related to Quantum Zeno Dynamics

Given a nonnegative self-adjoint operator $H$ acting on a separable Hilbert space and an orthogonal projection $P$ such that $H_P := (H^{1/2}P)^*(H^{1/2}P)$ is densely defined, we prove that $\lim_{n\rightarrow \infty} (P\,\mathrm{e}^{-itH/n}P)^n = \mathrm{e}^{-itH_P}P$ holds in the strong operator topology. We also derive modifications of this product formula and its extension to the situation when $P$ is replaced by a strongly continuous projection-valued function satisfying $P(0)=P$.

math-ph↗

arXiv2020

Optimization of the lowest eigenvalue of a soft quantum ring

We consider the self-adjoint two-dimensional Schrödinger operator $H_μ$ associated with the differential expression $-Δ-μ$ describing a particle exposed to an attractive interaction given by a measure $μ$ supported in a closed curvilinear strip and having fixed transversal one-dimensional profile measure $μ_\bot$. This operator has nonempty negative discrete spectrum and we obtain two optimization results for its lowest eigenvalue. For the first one, we fix $μ_\bot$ and maximize the lowest eigenvalue with respect to shape of the curvilinear strip the optimizer in the first problem turns out to be the annulus. We also generalize this result to the situation which involves an additional perturbation of $H_μ$ in the form of a positive multiple of the characteristic function of the domain surrounded by the curvilinear strip. Secondly, we fix the shape of the curvilinear strip and minimize the lowest eigenvalue with respect to variation of $μ_\bot$, under the constraint that the total profile measure $α>0$ is fixed. The optimizer in this problem is $μ_\bot$ given by the product of $α$ and the Dirac $δ$-function supported at an optimal position.

math-ph↗

arXiv2020

Spectral properties of spiral-shaped quantum waveguides

We investigate properties of a particle confined to a hard-wall spiral-shaped region. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the thresholds. The subtle difference between the radial and perpendicular width implies, however, that in contrast to `less curved' waveguides, the discrete spectrum is empty in this case. We also discuss modifications such a multi-arm Archimedean spirals and spiral waveguides with a central cavity; in the latter case bound state already exist if the cavity exceeds a critical size. For more general spiral regions the spectral nature depends on whether they are `expanding' or `shrinking'. The most interesting situation occurs in the asymptotically Archimedean case where the existence of bound states depends on the direction from which the asymptotics is reached.

math-ph↗

arXiv2020

Approximation of point interactions by geometric perturbations in two-dimensional domains

We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.

math-ph↗

arXiv2020

Magnetic field influence on the discrete spectrum of locally deformed leaky wires

We consider magnetic Schrödinger operator $H=(i \nabla +A)^2-αδ_Γ$ with an attractive singular interaction supported by a piecewise smooth curve $Γ$ being a local deformation of a straight line. The magnetic field $B$ is supposed to be nonzero and local. We show that the essential spectrum is $[-\frac14α^2,\infty)$, as for the non-magnetic operator with a straight $Γ$, and demonstrate a sufficient condition for the discrete spectrum of $H$ to be empty.

math.SP↗

arXiv2020

Ring chains with vertex coupling of a preferred orientation

We consider a family of Schrödinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length $\ell>0$ with the vertex coupling which is non-invariant with respect to the time reversal. The spectral behavior of the model illustrates that the high-energy behavior of such vertices is determined by the vertex parity. The positive spectrum of the tightly connected chain covers the entire halfline while the one of the loose chain is dominated by gaps. In addition, there is a negative spectrum consisting of an infinitely degenerate eigenvalue in the former case, and of one or two absolutely continuous bands in the latter. Furthermore, we discuss the limit $\ell\to 0$ and show that while the spectrum converges as a set to that of the tight chain, as it should in view of a result by Berkolaiko, Latushkin, and Sukhtaiev, this limit is rather non-uniform.

math.SP↗

arXiv2020

Topologically induced spectral behavior: the example of quantum graphs

This review paper summarizes the contents of the talk given by the author at the 8th International Congress of Chinese Mathematicians. Using examples of Schrödinger operators on metric graphs, it is shown that a nontrivial topology of the configuration space can give rise to a rich variety of spectral types. In particular, it is shown that the spectrum may be of a pure point type or to have a Cantor structure. We also address the question about the number of open spectral gaps and show that it could be nonzero and finite. Finally, inspired by a recent attempt to model the anomalous Hall effect we analyze a vertex coupling which exhibits high-energy behavior determined by the vertex degree parity.

math.SP↗

arXiv2020

Schrödinger operators with a switching effect

This paper summarizes the contents of a plenary talk given at the 14th Biennial Conference of Indian SIAM in Amritsar in February 2018. We discuss here the effect of an abrupt spectral change for some classes of Schrödinger operators depending on the value of the coupling constant, from below bounded and partly or fully discrete, to the continuous one covering the whole real axis. A prototype of such a behavior can be found in Smilansky-Solomyak model devised to illustrate that an an irreversible behavior is possible even if the heat bath to which the systems is coupled has a finite number of degrees of freedom and analyze several modifications of this model, with regular potentials or a magnetic field, as well as another system in which $x^py^p$ potential is amended by a negative radially symmetric term. Finally, we also discuss resonance effects in such models.

math-ph↗

arXiv2020

Topological bulk-edge effects in quantum graph transport

We examine quantum transport in periodic quantum graphs with a vertex coupling non-invariant with respect to time reversal. It is shown that the graph topology may play a decisive role in the conductivity properties illustrating this claim with two examples. In the first, the transport is possible at high energies in the bulk only being suppressed at the sample edges, while in the second one the situation is opposite, the transport is possible at the edge only.

math-ph↗

arXiv2019

An optimization problem for finite point interaction families

We consider the spectral problem for a family of $N$ point interactions of the same strength confined to a manifold with a rotational symmetry, a circle or a sphere, and ask for configurations that optimize the ground state energy of the corresponding singular Schrödinger operator. In case of the circle the principal eigenvalue is sharply maximized if the point interactions are distributed at equal distances. The analogous question for the sphere is much harder and reduces to a modification of Thomson problem; we have been able to indicate the unique maximizer configurations for $N=2,\,3,\,4,\,6,\,12$. We also discuss the optimization for one-dimensional point interactions on an interval with periodic boundary conditions. We show that the equidistant distributions give rise to maximum ground state eigenvalue if the interactions are attractive, in the repulsive case we get the same result for weak and strong coupling and we conjecture that it is valid generally.

math.SP↗

arXiv2019

Spectral asymptotics of the Laplacian on Platonic solids graphs

We investigate the high-energy eigenvalue asymptotics quantum graphs consisting of the vertices and edges of the five Platonic solids considering two different types of the vertex coupling. One is the standard $δ$-condition, the other is the preferred-orientation one introduced in [ET18]. The aim is to provide another illustration of the fact that the asymptotic properties of the latter coupling are determined by the vertex parity by showing that the octahedron graph differs in this respect from the other four for which the edges at high energies effectively disconnect and the spectrum approaches the one of the Dirichlet Laplacian on an interval.

math.SP↗

arXiv2019

Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maxi\-mized when the arms are arranged in one of the known five sharp configurations known as solutions of the closely related Thomson problem.

math-ph↗

arXiv2019

On Dirac operators in $\mathbb{R}^3$ with electrostatic and Lorentz scalar $δ$-shell interactions

In this article Dirac operators $A_{η, τ}$ coupled with combinations of electrostatic and Lorentz scalar $δ$-shell interactions of constant strength $η$ and $τ$, respectively, supported on compact surfaces $Σ\subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $δ$-potentials are modelled by coupling conditions at $Σ$. In the proof of the self-adjointness of $A_{η, τ}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{η, τ}$ is possible. In particular, the essential spectrum of $A_{η, τ}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{η, τ}$ is computed and it is discussed that for some special interaction strengths $A_{η, τ}$ is decoupled to two operators acting in the domains with the common boundary $Σ$.

math.SP↗

arXiv2019

Spectral geometry in a rotating frame: properties of the ground state

We investigate spectral properties of the operator describing a quantum particle confined to a planar domain $Ω$ rotating around a fixed point with an angular velocity $ω$ and demonstrate several properties of its principal eigenvalue $λ_1^ω$. We show that as a function of rotating center position it attains a unique maximum and has no other extrema provided the said position is unrestricted. Furthermore, we show that as a function $ω$, the eigenvalue attains a maximum at $ω=0$, unique unless $Ω$ has a full rotational symmetry. Finally, we present an upper bound to the difference $λ_{1,Ω}^ω- λ_{1,B}^ω$ where the last named eigenvalue corresponds to a disk of the same area as $Ω$.

math.SP↗

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.

math.SP↗

arXiv2018

Gap control by singular Schrödinger operators in a periodically structured metamaterial

We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ of $\varepsilon\mathbb{Z}^n$-periodic Schrödinger operators with $δ'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}^\varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

math.SP↗

arXiv2018

Scattering on leaky wires in dimension three

We consider the scattering problem for a class of strongly singular Schrödinger operators in $L^2(\mathbb{R}R^3)$ which can be formally written as $H_{α,Γ}= -Δ+ δ_α(x-Γ)$ where $α\in\mathbb{R}$ is the coupling parameter and $Γ$ is an infinite curve which is a local smooth deformation of a straight line $Σ\subset\mathbb{R}^3$. Using Kato-Birman method we prove that the wave operators $Ω_\pm(H_{α,Γ}, H_{α,Σ})$ exist and are complete.

math-ph↗

arXiv2018

Spectral Theory of Infinite Quantum Graphs

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

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