The realization of an elliptic operator A under suitable boundary conditions is considered and the dependence of the square-root of A from the various conditions is studied.
We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics...
Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are developed for solving the arising systems of convection-diffusion-dispersion-reaction equations, and the received results of several discretization methods are presented. We concentrate on linear reaction systems, which can be solved analytically. In the numerical methods, we use large time-steps to achieve long simulation times of about 10 000 years. We propose...
2000 Mathematics Subject Classification: 35L15, 35B40, 47F05.We prove dispersive estimates for solutions to the wave equation with a real-valued potential V.
For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, , α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean...
After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator . If is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775,...
We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic...