Point interaction Hamiltonians in bounded domains

Abstract

Making use of recent techniques in the theory of selfadjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a 3-D bounded domain with regular boundary. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator togheter with the corresponding related eigenfunctions. These operators are used to build up a model-system where the dynamics of a quantum particle depends on the state of a quantum bit.

A global, dynamical formulation of quantum confined systems

“…The problem of determining a dynamical formulation of quantum confinement can be addressed from the point of view of the study of s.a. extensions of symmetric restrictions [1,6,7,8] and is closely related with the subjects of point interaction Hamiltonians [7,9,10,11] and surface interactions [12]. Our results may be useful in this last context as well as for the deformation quantization of systems with boundaries [4].…”

Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions

“…with Dirichlet boundary condition at ∂Ω and some coupling ν α (of which α is a suitable renormalisation). In fact, H Ω α,y is rigorously defined as a self-adjoint extension in L 2 (Ω) of the Dirichlet Laplacian restricted to smooth functions vanishing on neighbourhoods of y, a construction obtained in [13] (see also [24]). Basic spectral properties and an amount of further results on H Ω α,y were established in [13,30,57].…”

“…In fact, H Ω α,y is rigorously defined as a self-adjoint extension in L 2 (Ω) of the Dirichlet Laplacian restricted to smooth functions vanishing on neighbourhoods of y, a construction obtained in [13] (see also [24]). Basic spectral properties and an amount of further results on H Ω α,y were established in [13,30,57]. As the above singular perturbation at y does not alter the lower semi-boundedness of the unperturbed Dirichlet Laplacian, it still makes sense to investigate the principal eigenvalue λ α 1 (Ω, y) of H Ω α,y .…”

“…In this respect, the point interaction modelled by H B α,y has the same effect of a negative and localised potential well (whence indeed the lowering of the lowest eigenvalue of H Ω α,y with respect to H Ω D , Remark 4.2). In fact, by inspection of the explicit eigenfunctions (computed in [13,Sect. 4]) one sees that they concentrate around y, analogously to the eigenfunctions of…”

On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region

“…to which one has to add eigenvalues of − D , the degenerate ones in any case and the nondegenerate ones, λ n, provided that the corresponding eigenfunction ψ n satisfies the condition (see, e.g. in [BFM07])…”