No abstract

The construction of Dirac delta type potentials has been achieved with the use of the theory of self adjoint extensions of non-self adjoint formally Hermitian (symmetric) operators. The application of this formalism to investigate the possible self adjoint extensions of the one dimensional kinematic operator K = −d 2 /dx 2 on the infinite square well potential is quite illustrative and has been given elsewhere. This requires the definition and use of four independent real parameters, which relate the boundary values of the wave functions at the walls. By means of a different approach, that fixes matching conditions at the origin for the wave functions, it is possible to define a perturbation of the type aδ(x) + bδ ′ (x), thus depending on two parameters, on the infinite square well. The objective of this paper is to investigate whether these two approaches are compatible in the sense that perturbations like aδ(x) + bδ ′ (x) can be fixed and determined using the first approach.

The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schrödinger's equation to have constant sign in its domain of definition. We extend this result by giving explicit formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions. Our results apply in particular to Wronskians of classical orthogonal polynomials, thus generalizing classical results by Karlin and Szegő. Our formulas hold under very mild conditions that are believed to hold for generic values of the parameters. In the Hermite case, our results allow to prove some conjectures recently formulated by Felder et al.