arXiv subjects
Publications and source records attributed to Hynek Kovarik.
We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term.
We study Riesz means of the eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on bounded domains. We obtain an inequality with a sharp leading term and an additional lower order term, improving the result of Hanson and Laptev.
We consider a class of quasilinear operators on a bounded domain $Ω\subset \mathbb R^n$ and address the question of optimizing the first eigenvalue with respect to the boundary conditions, which are of the Robin-type. We describe the optimizing boundary conditions and establish upper and lower bounds on the respective maximal and minimal eigenvalue.
We consider a Schrödinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $\|e^{-itH(A,a)}\|_{L^1\to L^\infty}\lesssim t^{-n/2}$, then $ \||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\|_{L^1\to L^\infty}\lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$.
We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.
We consider two-dimensional Schroedinger operators with magnetic field. Under certain regularity and decay assumptions on the magnetic and electric field we show that the behavior of the corresponding resolvent at threshold and the behavior of the unitary group for large times are determined by the total flux of the associated magnetic field.
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.
We consider two-dimensional Schrödinger operators $H$ with Aharonov-Bohm magnetic field and an additional electric potential. We obtain an explicit leading term of the asymptotic expansion of the unitary group $e^{-i t H}$ for $t\to\infty$ in weighted $L^2$ spaces. In particular, we show that the magnetic field improves the decay of $e^{-i t H}$ with respect to the unitary group generated by non-magnetic Schrödinger operators, and that the decay rate in time is determined by the magnetic flux.
We consider Schrôdinger operators $H_α$ given by equation (1.1) below. We study the asymptotic behavior of the spectral density $E(H_α, λ)$ when $λ$ goes to $0$ and the $L^1\to L^\infty$ dispersive estimates associated to the evolution operator $e^{-i t H_α}$. In particular we prove that for positive values of $α$, the spectral density tends to zero as $λ\to 0$ with higher speed compared to the spectral density of Schrödinger operators with a short-range potential $V$. We then show how the long time behavior of $e^{-i t H_α}$ depends on $α$. More precisely we show that the decay rate of $e^{-i t H_α}$ for $t\to\infty$ can be made arbitrarily large provided we choose $α$ large enough and consider a suitable operator norm.
We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.
We consider a twisted quantum waveguide i.e. a domain of the form Ω_θ : = r_θω\times R, where ω\subset R^2 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 investigate the nature of the essential spectrum of the Dirichlet Laplacian H_θ, self-adjoint in L^2 (Ω_θ), and consider related scattering problems. First, we show that if the derivative of the difference θ_1 - θ_2 decays fast enough as |x_3| goes to infinity, then the wave operators for the operator pair (H_{θ_1}, H_{θ_2}) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ' = β- ε, with constant β\in R, and εwhich decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to εis an analytically fibered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of H_θ, is empty.
In this paper we study the eigenvalue sums of Dirichlet Laplacians on bounded domains. Among our results we establish an improvement of the Li-Yau bound in the presence of a constant magnetic field.
In this paper we consider a Robin-type Laplace operator on bounded domains. We study the dependence of its lowest eigenvalue on the boundary conditions and its asymptotic behavior in shrinking and expanding domains. For convex domains we establish two-sided estimates on the lowest eigenvalues in terms of the inradius and of the boundary conditions.
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.
We consider the heat semi-group generated by the Laplace operator on metric trees. Among our results we show how the behavior of the associated heat kernel depends on the geometry of the tree. As applications we establish new eigenvalue estimates for Schroedinger operators on metric trees.
We prove that the number of negative eigenvalues of two-dimensional magnetic Schroedinger operators is bounded from above by the strength of the corresponding electric potential. Such estimates fail in the absence of a magnetic field. We also show how the corresponding upper bounds depend on the properties of the magnetic field and discuss their connection with Hardy-type inequalities.
We consider the vibrational energy levels of the first two electronic states of the molecule ion $H_2^+$. The Born-Oppenheimer method applied to the case of the Stark effect on a $H_2^+$-like molecule gives existence of sharp resonances localized in the same interval of energy of the vibrational levels.
We study the heat semigroup generated by two-dimensional Schroedinger 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.