Some Bound State Problems in Quantum Mechanics

We give a review of semi-classical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semi-classical bounds to multi-particle systems, in particular, large atoms and the stability of matter. In this survey, we focus on results for bound states of Schrödinger operators related to the semi-classical limit and Coulomb potentials. We will not discuss a large part of the existing literature on the general theory of bound states for Schrödinger operators. With no attempt on completeness, we would nevertheless like to mention at least some part of this literature: For one particle Schrödinger operators, see, for example, [19, 149]. Two-body cluster results are discussed in [4, 82, 140, 151, 152, 173], finiteness results of the discrete spectrum for N -particle systems can be found in [1, 40, 164], and for results on the Efimov effect see, for example, [3, 36, 123, 155, 160, 170]. 1. Semi-classical bounds for single particle Schrödinger operators The origin of semi-classical estimates can be traced back to the dawn of quantum mechanics in the beginning of the last century. Around 1912, Hermann Weyl published a series of papers [166, 167, 168], see also [169], on the frequencies of an oscillating membrane and the radiation inside a cavity, verifying a conjecture of Jeans and Lorenz on the connection between the asymptotic behavior of these frequencies and the volume of the cavity. While in itself being a classical problem, this work was the starting point 2000 Mathematics Subject Classification. 35J10,81Q10.