Hynek Kovařík

Stability of the Magnetic Schrödinger Operator in a Waveguide

Abstract The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right. *Also on leave of absence from Nuclear Physics Institute, Academy of Sciences, Rez, near Prague, Czech Republic.

Two-Dimensional Berezin-Li-Yau Inequalities with a Correction Term

We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas, [11].

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide i.e., a domain of the form Ωθ: = r θ ω × ℝ where ω ⊂ ℝ2 is a bounded domain, and r θ = r θ(x 3) is a rotation by the angle θ(x 3) depending on the longitudinal variable x 3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L 2(Ωθ). We suppose that the derivative of the rotation angle can be written as (x 3) = β − ϵ(x 3) with a positive constant β and ϵ(x 3) ∼ L|x 3|−α, |x 3| → ∞. We show that if L > 0 and α ∈ (0,2), or if L > L 0 > 0 and α = 2, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

Spectrum of the Magnetic Schrodinger Operator in a Waveguide with Combined Boundary Conditions

Abstract.We consider the magnetic Schrödinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann {For the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.

Edge currents in the absence of edges

We investigate a charged two-dimensional particle in a homogeneous magnetic field interacting with a periodic array of point obstacles. We show that while Landau levels remain to be infinitely degenerate eigenvalues, between them the system has bands of absolutely continuous spectrum and exhibits thus a transport along the array. We also compute the band functions and the corresponding probability current.

Spectrum of the Schrödinger Operator in a Perturbed Periodically Twisted Tube

We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of “slowing down” the twisting in the mean gives rise to a non-empty discrete spectrum

Hardy inequalities for Robin Laplacians

Abstract In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary conditions. We also study several extensions to non-convex and unbounded domains.

Heat kernels of two-dimensional magnetic Schrödinger and Pauli operators

We study the heat semigroup generated by two-dimensional Schrödinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some on-diagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the Aharonov-Bohm magnetic field. AMS 2000 Mathematics Subject Classification: 47D08, 35P05

On the exponential decay of magnetic stark resonances

We study the time decay of magnetic Stark resonant states. As a main result we prove that for sufficiently large time these states decay exponentially with the rate given by the imaginary parts of eigenvalues of certain non-selfadjoint operator. The proof is based on the method of complex translations.

A nonlinear Schrödinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schrodinger equation with a symmetric double-well potential represented by two Diracs δ. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem.