Khosrow Chadan

Inverse Problems in Quantum Scattering Theory

The physical importance of inverse problems in quantum scattering theory is clear since all the information we can obtain on nuclear, particle, and subparticle physics is gathered from scattering experiments. Exact and approximate methods of investigating scattering theory, inverse radial problems at fixed energy, inverse one-dimensional problems, inverse three-dimensional problems, and construction of the scattering amplitude from the cross section are presented. The methods often apply to other fields, e.g. applied mathematics and geophysics. The book will therefore be of interest to theoretical and mathematical physicists, nuclear particle physicists, and chemical physicists. For the second edition the chapters on one-dimensional and three-dimensional scattering problems have been rewritten and considerably expanded. Furthermore, two new chapters on spectral problems and on numerical aspects have been added; in the sections on classical methods the comments and references have been updated.

The Calogero bound for nonzero angular momentum

It is shown that the ‘‘Calogero’’ bound on the number of bound states in an attractive monotonous potential is not optimal for a strictly positive angular momentum l and a new bound including an extra additive term is proposed. It is Nl(V)<(2/π)∫0∞√‖V(r)‖dr+1−√1 +(2/π)2l(l+1). From this new bound it is possible to obtain a bound on the total number of bound states for arbitrary angular momentum. The situation for −1/2≤l<0 is investigated and a bound under the condition that r2V(r) has a single extremum is given. Consequences for zero angular momentum bound states in two dimensions are discussed.

Generalization of the Calogero–Cohn bound on the number of bound states

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)[r1−2p(−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.

New classes of potentials for which the radial Schrödinger equation can be solved at zero energy

In a previous paper (Chadan and Kobayashi 2006 J. Phys. A: Math. Gen. 39 3381), we presented an infinite class of potentials for which the radial Schrodinger equation at zero energy can be solved explicitly. 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 first a simple subclass (also infinite) of the whole class for which the solution of the Schrodinger 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 Chadan and Kobayashi (2006). We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity. The main result of the present paper is given in theorem 2.

New bounds on the number of bound states for Schrödinger operators

AbstractWe consider the Schrödinger operatorH = −Δ +V(|x|) onR3. Letnℓ denote the number of bound states with angular momentumℓ (not counting the 2ℓ + 1 degeneracy). We prove the following bounds onnℓ. LetV ⩽ 0 and d/dr r1-2p(-V)1 −p ⩽ 0 for somep ∈ [1/2, 1) then

The absolute definition of the phase-shift in potential scattering

n_\ell \leqslant p(1 - p)^{p - 1} (2\ell + 1)^{1 - 2p} \smallint _0^\infty ( - r^2 V)^p \frac{{dr}}{r}.

Sufficient Conditions for the Existence of Bound States in a Potential without Spherical Symmetry

This bound closes the gap between the celebrated bounds by Calogero (p = 1/2) and Bargmann (p = 1).

A sufficient condition for the existence of bound states in a potential

We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We then apply the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schrodinger equation with nonlocal potentials which are superpositions of a local potential and separable potentials.