Critical points for a one-dimensional Schrödinger operator with finite number of point delta- interactions and number of eigenvalues

Abstract

For a one-dimensional Schrödinger operator with finite number n of point delta-interactions, the parameters are the n intensities, n − 1 intercenter distances and the mass. A point Pc in parameter space is a critical point if a continuous path P(t), with P(0) = Pc, can be found such as one eigenvalue of the Hamiltonian tends to zero when t tends to zero. The number of eigenvalues changes only at a critical point. The critical points are determined as the roots of the determinant of an (n − 1)-order real symmetric tridiagonal matrix whose entries are simple functions of the parameters. A simple algorithm is given for the determination of the number of eigenvalues from the parameters in the general case. For the two particular cases, namely equidistant centers and equal intensity parameters for the first case, and equidistant centers and alternating intensity parameters for the second case, the critical points and number of eigenvalues are determined analytically.