Realizing Discontinuous Wave Functions with Renormalized Short-Range Potentials

It is shown that a potential consisting of three Dirac's delta functions on the line with disappearing distances can give rise to the discontinuity in wave functions with the proper renormalization of the delta function strength. This can be used as a building block, along with the usual Dirac's delta, to construct the most general three-parameter family of point interactions, which allow both discontinuity and asymmetry of the wave function, as the zero-size limit of self-adjoint local operators in one-dimensional quantum mechanics. Experimental realization of the Neumann boundary is discussed. KEYWORDS: point interaction, self-adjoint extension, $\delta'$ potential, wave function discontinuity, Neumann boundary PACS Nos: 3.65.-w, 11.10.Gh, 68.65+gComment: 4 pages, ReVTeX double column format with an epsf figure, expanded reference

Supersymmetry and discrete transformations on S^1 with point singularities

We investigate N-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer n, N=2n supercharges are explicitly constructed and a class of point singularities compatible with supersymmetry is clarified. Key ingredients in our construction are n sets of discrete transformations, each of which forms an su(2) algebra of spin 1/2. The degeneracy of the spectrum and spontaneous supersymmetry breaking are briefly discussed.Comment: 11 pages, 3 figure

Separation of variables in perturbed cylinders

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients. Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically yielding the eigenvalues of the operator. The same approach allows us to find complex resonances arising in some non-compact domains. We discuss numerical examples related to quantum waveguide problems.Comment: LaTeX 2e, 18 pages, 6 figure

On the eigenvalue spacing distribution for a point scatterer on the flat torus

We study the level spacing distribution for the spectrum of a point scatterer on a flat torus. In the 2-dimensional case, we show that in the weak coupling regime the eigenvalue spacing distribution coincides with that of the spectrum of the Laplacian (ignoring multiplicties), by showing that the perturbed eigenvalues generically clump with the unperturbed ones on the scale of the mean level spacing. We also study the three dimensional case, where the situation is very different.Comment: 25 page