Exactly Solvable Problems in One Dimension

Francesco Calogero

Marko Horbatsch

Avinash Khare, Rajat K. Bhaduri

We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.

Avinash Khare, J. M. Richard

The Hall-Post inequalities provide lower bounds on N -body energies in terms of N '-body energies with N ' < N . They are rewritten and generalized to be tested with exactly solvable models of Calogero-Sutherland type in one and higher dimensions. The bound for N spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillator potential.

Alberto Galindo, P. Pascual

R. Shankar