Mixtures of potentials in quantum mechanics

Abstract

A global geometrical approximation theory for the spectra of Schrödinger operators H =− D2 + vf(x) is discussed. The potential f(x) is composed either of sums, or of continuous mixtures, of soluble potentials. In both cases it is proved that the kinetic-potential formalism [J. Math. Phys. 24, 324 (1983); 25, 2078 (1984)] automatically yields optimal energy lower bounds. The examples f(x)=‖x‖+x2 and f(x)=−∫basech2(tx)dt are treated in detail.