arXiv subjects
Publications and source records attributed to Pavel Exner.
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.
In this paper the spectral properties of Dirac operators $A_η$ with electrostatic $δ$-shell interactions of constant strength $η$ supported on compact smooth surfaces in $\mathbb{R}^3$ are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of $A_η$ are investigated. In particular, it turns out that the discrete spectrum of $A_η$ inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of $A_η$ and the free Dirac operator $A_0$ is trace class, and in the nonrelativistic limit $A_η$ converges in the norm resolvent sense to a Schrödinger operator with an electric $δ$-potential of strength $η$.
The aim of the paper is to investigate resonances in quantum graphs with a general self-adjoint coupling in the vertices and their trajectories with respect to varying edge lengths. We derive formulae determining the Taylor expansion of the resonance pole position up to the second order which represent, in particular, a counterpart to the Fermi rule derived recently by Lee and Zworski for graphs with the standard coupling. Furthermore, we discuss the asymptotic behavior of the resonances in the high-energy regime in the situation where the leads are attached through $δ$ or $δ_\mathrm{s}'$ conditions, and we prove that in the case of $δ_\mathrm{s}'$ coupling the resonances approach to the real axis with the increasing real parts as $\mathcal{O}\big((\mathrm{Re\,}k)^{-2}\big)$.
We show that a Schrödinger operator $A_{δ, α}$ with a $δ$-interaction of strength $α$ supported on a bounded or unbounded $C^2$-hypersurface $Σ\subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator $A_{δ, α}$ with a singular interaction is regarded as a self-adjoint realization of the formal differential expression $-Δ- α\langle δ_Σ, \cdot \rangle δ_Σ$, where $α\colonΣ\rightarrow \mathbb{R}$ is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.
We analyze a family of singular Schrödinger operators with local singular interactions supported by a hypersurface $Σ\subset \mathbb{R}^n, n \ge 2$, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of $Σ$ the interaction is characterized by four real parameters, the earlier studied case of $δ$- and $δ'$-interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.
We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - λ(x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $λ\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $λ$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $λ$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.
We discuss spectral properties of a charged quantum particle confined to a chain graph consisting of an infinite array of rings under influence of a magnetic field assuming a $δ$-coupling at the points where the rings touch. We start with the situation when the system has a translational symmetry and analyze spectral consequences of perturbations of various kind, such as a local change of the magnetic field, of the coupling constant, or of a ring circumference. A particular attention is paid to weak perturbations, both local and periodic; for the latter we prove a version of Saxon-Hutner conjecture.
The main question studied in this paper concerns the weak-coupling behavior of the geometrically induced bound states of singular Schrödinger operators with an attractive $δ$ interaction supported by a planar, asymptotically straight curve $Γ$. We demonstrate that if $Γ$ is only slightly bent or weakly deformed, then there is a single eigenvalue and the gap between it and the continuum threshold is in the leading order proportional to the fourth power of the bending angle, or the deformation parameter. For comparison, we analyze the behavior of a general geometrical induced eigenvalue in the situation when one of the curve asymptotes is wiggled.
We analyze spectral properties of a leaky wire model with a potential bias. It describes a two-dimensional quantum particle exposed to a potential consisting of two parts. One is an attractive $δ$-interaction supported by a non-straight, piecewise smooth curve $\mathcal{L}$ dividing the plane into two regions of which one, the `interior', is convex. The other interaction component is a constant positive potential $V_0$ in one of the regions. We show that in the critical case, $V_0=α^2$, the discrete spectrum is non-void if and only if the bias is supported in the interior. We also analyze the non-critical situations, in particular, we show that in the subcritical case, $V_0<α^2$, the system may have any finite number of bound states provided the angle between the asymptotes of $\mathcal{L}$ is small enough.
We analyze spectral properties of a quantum graph in the form of a ring chain with a $δ$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the chain exhibits a translational symmetry and study the discrete spectrum in the gaps resulting from changing a finite number of vertex coupling constants. In particular, we discuss in details some examples such as perturbations of one or two vertices, weak perturbation asymptotics, and a pair of distant perturbations.
We consider a singular Schrödinger operator in $L^2(\mathbb{R}^2)$ written formally as $-Δ- βδ(x-γ)$ where $γ$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the $j$th negative eigenvalue of this operator behaves in the strong-coupling limit, $β\to +\infty$, asymptotically as \[ E_j(β)=-\frac{β^2}{4} +μ_j +\mathcal{O}\Big(\dfrac{\logβ}β\Big), \] where $μ_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{κ(s)^2}{4}\, \] on $L^2(0,L)$ with $κ(s)$ being the signed curvature of $γ$ at the point $s\in(0,L)$.
We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than $π$, and it is void if the domain is concave.
We investigate non-equilibrium particle transport in the system consisting of a geometric scatterer and two leads coupled to heat baths with different chemical potentials. We derive expression for the corresponding current the carriers of which are fermions and analyze numerically its dependence of the model parameters in examples, where the scatterer has a rectangular or triangular shape.
We analyze a family of singular Schrödinger operators describing a Neumann waveguide with a periodic array of singular traps of a $δ'$ type. We show that in the limit when perpendicular size of the guide tends to zero and the $δ'$ interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.
We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a $δ$ type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.
We investigate the spectral properties of self-adjoint Schrödinger operators with attractive $δ$-interactions of constant strength $α> 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum is given by $[-α^2/4,+\infty)$ and that the discrete spectrum is infinite and accumulates to $-α^2/4$. Furthermore, an asymptotic estimate of these eigenvalues is obtained.
Robin problem for the Laplacian in a bounded planar domain with a smooth boundary and a large parameter in the boundary condition is considered. We prove a two-sided three-term asymptotic estimate for the negative eigenvalues. Furthermore, improving the upper bound we get a two term asymptotics in terms of the coupling constant and the maximum of the boundary curvature.
We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive $δ'$ interaction supported by a smooth surface in $\R^3$, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrödinger type operator with an effective potential expressed in terms of the interaction support curvatures.