Non-local interactions in quantum mechanics modelled by shifted Dirac delta functions

We introduce a new interaction formulated in terms of two shifted delta functions with non-local boundary conditions. For this purpose we use a method of self-adjoint extensions for a particular simmetric operator. We obtain bound states of the associated quantum mechanical problem and study its symmetry properties. We compare the results with the ground state of the quantum mechanical problem with a local interaction in terms of two delta functions.


Introduction
The Dirac delta distribution as a potential in quantum physics allows to construct approximate models with great historical and physical interest [1,2,3,4,5,6,7,8,9]. One of them is an approximate model of an hydrogen molecule ion, H + 2 . The ion is commonly formed in molecular clouds in space. The first successful quantum mechanical treatment of H + 2 was published by O. Burrau in 1927 [1]. The electron of the molecule moves in an attractive potential generated by the two protons. The quantum analysis of the electron arises from the solution of the Schrödinger equation, where the potential can be approximated by two delta "functions" with negative coefficients describing the attractive interaction.
Another interesting application arises from an approximation to the Kronig-Penney [3] model, which describes some basic quantum effects in the conduction of electrical charges in metals. The potential is expressed as an infinite sequence of Dirac deltas with a finite separation between them, a combination of Dirac deltas. The delta interactions have been also used in toy models of quantum wires or nanowires.
In this work we introduce a new non-local interaction of the type of two delta functions. We will determine the bound states, in particular the ground state of the quantum mechanical hamiltonian and study its symmetry properties. A main motivation for this work is the formulation of non-local field theory, based on string field theory [10].

Preliminaries
Let us consider the differential operator D = −d 2 ·/dt 2 acting on the real line R . with the domain  = c = const. In this case the first generalized derivative of y(t) has a jump and, so, the second one has a generalized summand with the delta-function. Thus the corresponding extension D can be naturally presented in the form If c < 0, then A has the negative eigenvalue λ = −c 2 that corresponds to the eigenfunction y(t) = e −c|t| . These facts are well known and can be find in the book of Albeverio et al [11].
In the present paper we study some generalization of the described above scheme for two boundary problems for points −h, h and a behaviour of the corresponding extension if h → 0. We shall show that this behaviour involves the derivatives of the delta-function.
Note that even for the one-point problem there are some self-adjoint extensions with one or two negative eigenvalues, naturally involve not only the delta-function but its first derivative. One of such extension is given by the boundary conditions (α > 0, β > 0) The eigenvalues for this extension are −α 2 and −β 2 , the corresponding eigenfunctions are e −α|t| and Sgn (t)e −β|t| respectively. The extension D has the representation

Continuous function
Let the differential operator D = −d 2 · /dt 2 act on the real line R . and have the domain Then for the adjoint operator D * we have Seeking self-adjoint extensions of D let us suppose that It easy to check that the corresponding extension will be self-adjoint if and only if the matrix is symmetric. Note that the first derivative of y(t) can have jumps in the points −h and h, so the second one (if y(t) is considered as a generalized function) two shifted δ-summands: Thus, the corresponding extension D h of the operator D can be re-written as Note that due to presence of α 12 and α 21 in (7) the interaction in question is not local.

Proposition 1 Let an extension D satisfies the conditions (4) and (5). Then D has no nonnegative eigenvalues and the number of its negative eigenvalues doesn't exceed the number of negative eigenvalues (taking in account the multiplicity) of matrix A from (6).
Proof of Proposition 1 Let us suppose that D has a positive eigenvalue ω 2 , ω > 0. Then a corresponding eigenfunction y(t) has a form Let us find the extension D h of D such that it has two eigenvalues λ = −α 2 and μ = −β 2 . In order to do it we construct a matrix A such that for every positive h the function In the same way let us demand that for every positive h the function would be an eigenfunction of the operator D h . According to (5) The system (9), (11) has a solution and we have , Then the expression (7) is converted to .
Let us consider some particular cases. The interaction in (5) is not, generally speaking, local, because the jump of the derivative y in the point −h depends of the values y not only in the point −h but also in the point h, etc. This interaction is local if and only if a 12 = a 21 = 0, that means .
The latter brings β < β · (1 + e −2αh ) = α · (1 − e −2βh ) < α, so in the case of locality the bound state is given by (8) . If α = β then a 12 > 0 and the bound state is a linear combination of (10) (8). It can be asymmetric. If we consider (8)+(10) we obtain a lefthanded valued eigenfunction and if we take (8)-(10) we get a right handed valued eigenfunction. Note that a 12 → 0 and a 11 → ∞ if α → ∞, so for relatively big α a violence of locality in (5) thus, Let us introduce some notations. We put and denote by N h the linear span of f h (t) and g h (t). Then (note that this decomposition is not orthogonal) Let Then for every f (t) ∈ L 2 (R . ) the decomposition (15) the form Thus (see (14)), The direct calculation brings so for small h we have ν h = 2h β+γ − 2h 2 3 + . . . . The latter equalities bring