A global, dynamical formulation of quantum confined systems

A brief review of some recent results on the global self-adjoint formulation of systems with boundaries is presented. We specialize to the 1-dimensional case and obtain a dynamical formulation of quantum confinement.


Introduction
To obtain a confined version (for instance, to Ω 1 ) of the system described by H 0 , the standard approach is to determine the s.a. realizations of the operator H 0 in L 2 (Ω 1 ). It is well known, however, that this formulation displays several inconsistencies [1,2,3], the main issues being the ambiguities besetting the physical predictions (when there are several possible self-adjoint realizations of H 0 in L 2 (Ω 1 )), the lack of self-adjoint (s.a.) formulations of some important observables in L 2 (Ω 1 ) and the difficulties in translating this approach to other (non-local) formulations of quantum mechanics, like the deformation formulation [4]. These problems are well illustrated by textbook examples [1,4,5].
Our aim here is to present an alternative approach to quantum confinement. This formulation consists in determining all s.a. Hamiltonian operators H : D(H) ⊂ L 2 (IR d ) −→ L 2 (IR d ) -defined on a dense subspace D(H) of the global Hilbert space L 2 (IR d ) -which dynamically confine the system to Ω 1 (or Ω 2 ) while reproducing the action of H 0 in an appropriate subdomain. More precisely, let P Ω k be the projector operator onto Ω k , k = 1, 2, i.e.
where χ Ω k is the characteristic function of Ω k : χ Ω k (x) = 1 if x ∈ Ω k and χ Ω k (x) = 0, otherwise. Our aim is to determine all linear operators H : D(H) ⊂ L 2 (IR d ) → L 2 (IR d ) that satisfy the following three properties: (i) H is self-adjoint on L 2 (IR d ).
Moreover, for the 1-dimensional case, we want to recast the operators H in the form H = H 0 + B BC , where B BC is a distributional boundary potential (that may depend on the particular boundary conditions satisfied by the domain of H) and H is s.a. on its maximal domain. This formulation is global, because the system is defined in L 2 (IR d ), and the confinement is dynamical, i.e. it is a consequence of the initial state and of the Hamiltonian H. Indeed, from (i) and (ii) it follows that P Ω k commutes with all the spectral projectors of H and so also with the operator exp{iHt} for t ∈ IR. Hence, if ψ is an eigenstate of P Ω k it will evolve to exp{iHt}ψ, which is again an eigenstate of P Ω k with the same eigenvalue. In other words, P Ω k is a constant of motion and a wave function confined to Ω 1 (or to Ω 2 ) will stay so forever.
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].
In this paper we shall provide a concise review of the solutions to the above problems. The reader should refer to [13,14] for a detail presentation, including proofs of the main theorems, the extension of the boundary potential formulation to higher dimensions and some applications to particular systems.
2 Confining Hamiltonians defined on L 2 (IR d ) We start by introducing some relevant notation. Let X, Y ⊂ V be two subspaces of a vector space V such that X ∩ Y = {0}, then their direct sum is denoted by X ⊕ Y .
For simplicity let us assume that D(Ω k ) ⊂ L 2 (Ω k ) ∩ D(H 0 ), k = 1, 2 (where D(Ω k ) is the space of infinitely smooth functions t : IR d → C with support on a compact subset of Ω k ) and let us define the operators: which are symmetric. Let also H S † k be the adjoint of H S k . Our main result characterizes the operators H : D(H) ⊂ L 2 (IR d ) → L 2 (IR d ), associated to a s.a. H 0 , and satisfying properties (i) to (iii). The condition (stated in the theorem) that [H 0 , P Ω k ]ψ = 0, ∀ψ ∈ D(Ω 1 ) ∪ D(Ω 2 ), and the assumption that H S 1 and H S 2 have s.a. extensions are the minimal requirements for the existence of operators H satisfying (i) to (iii). Proofs of these results are given in [13,14].
We now focus on the case where d = 1, Ω 1 = IR − and is the set of absolutely continuous functions on IR and V (x) is a regular potential. We shall assume it to be i) real, ii) locally integrable and satisfying iii) V (x) > −kx 2 , k > 0 for sufficiently large |x|. The conditions on V (x) are such that H 0 : D(H 0 ) ⊂ L 2 (IR) → L 2 (IR) is the unique s.a. realization of the differential expression − d 2 dx 2 +V (x) on L 2 (IR) [15] and ensure that all s.a. realizations of − where λ k ∈ IR ∪ {∞}, k = 1, 2 and the case λ k = ∞ corresponds to Dirichlet boundary conditions. Moreover Hence, all s.a. confining Hamiltonians of the form H 1 ⊕H 2 are s.a. restrictions of H S † . To proceed let us define the operators (k = 1, 2 and n = 0, 1): is the space of Schwartz distributions on IR and δ (0) (x) = δ(x) and δ (1) (x) = δ ′ (x) are the Dirac measure and its first distributional derivative. We can now recast the operators (5) in the additive form H = H 0 +B BC :
The proof is given in [13].