arXiv subjects
Publications and source records attributed to Reido Kobayashi.
Given two potentials V0 and V1 together with a certain nodeless solution ϕ0 of V0, we form a composition of these two potentials. If V1 is exactly solvable, the composition is exactly solvable, too. By combining various solvable potentials in one-dimensional quantum mechanics, a huge variety of solvable compositions can be made.
In a previous paper$^1$, submitted to Journal of Physics A -- we presented an infinite class of potentials for which the radial Schrödinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be zero, but for the other part (also infinite), one can have any angular momentum. In the present paper, we study a simple subclass (also infinite) of the whole class for which the solution of the Schrödinger equation is simpler than in the general case. This subclass is obtained by combining another approach together with the general approach of the previous paper. Once this is achieved, one can then see that one can in fact combine the two approaches in full generality, and obtain a much larger class of potentials than the class found in ref. $^1$ We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity.
We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We apply then the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schrödinger equation with nonlocal potentials which are superposition of a local potential and separable potentials.