Andrea Mantile

Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

Abstract The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on R n with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Kreĭn-like resolvent formulae where the reference operator coincides with the “free” operator with domain H 2 ( R n ) ; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ and δ ′ -type, assigned either on a ( n − 1 ) dimensional compact boundary Γ = ∂ Ω or on a relatively open part Σ ⊂ Γ . Schatten–von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.

ADIABATIC EVOLUTION OF 1D SHAPE RESONANCES: AN ARTIFICIAL INTERFACE CONDITIONS APPROACH

Artificial interface conditions parametrized by a complex number

Direct and Inverse Acoustic Scattering by a Collection of Extended and Point-Like Scatterers

\theta_{0}

Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

are introduced for 1D-Schr{o}dinger operators. When this complex parameter equals the parameter

Non-autonomous quantum systems with scale-dependent interface conditions

\theta\in i\R

Stability and control of a confined 1D quantum system with time-dependent delta potentials

of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale

An explicit model for the adiabatic evolution of quantum observables driven by 1D shape resonances

(h^N)_{N\in \N}

Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

as

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

h\to 0

Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

, according to