Uniform resolvent estimates for the discrete Schrödinger operator in dimension three

Resolvent Estimates of the Dirac Operator

Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities

Consider the Schrödinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h−1) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case wh...

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

We establish uniform error estimates of finite difference methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε (ε ∈ (0, 1]). When ε → 0+, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., 0 < ε 1, the solution of NLSW is perturbed from that of NLS with ...

Resolvent Estimates for Elliptic Finite Element Operators in One Dimension

We prove the analyticity (uniform in h ) of the semigroups generated on Lp(0, 1), 1 < p < oo , by finite element analogues Ah of a onedimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds ...

Uniform resolvent estimates for a non-dissipative Helmholtz equation

We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures o...