No abstract

For pt.I see ibid., vol.5, p.257 (1988). The authors generalize the inverse problem in the coupling constant to the case of two potentials, one of them being known. Identifying this last potential with energy (constant potential) they obtain the solution of the inverse problem at non-zero energy. This generalizes the previous results obtained for the zero energy case. The interval in which they consider the Schrodinger equation may be finite or infinite, and the potential may be singular at the origin. Several soluble examples are given.

It is shown that for the Calogero-Cohn type upper bounds on the number of bound states of a negative spherically symmetric potential V (r), in each angular momentum state, that is, bounds containing only the integral ∞ 0 |V (r)| 1/2 dr, the condition V ′ (r) ≥ 0 is not necessary, and can be replaced by the less stringent condition (d/dr)[r 1−2p (−V ) 1−p ] ≤ 0, 1/2 ≤ p < 1, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on p and ℓ, and tend to the standard value for p = 1/2.

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.