Semiclassical bounds in measure spaces

Abstract: 

In quantum mechanics, a semiclassical approximation estimates a quantum expression such as the number of bound states in terms of the corresponding quantity in Newtonian mechanics.

One such well-known semiclassical bound is the Cwikel-Lieb-Rozenbljum (or Rozenbljum-Lieb-Cwikel) inequality

N(−Δ+V)≤cd∫RdV−(x)d/2dx,N(−Δ+V)≤cd∫RdV−(x)d/2dx,

estimating the number N(−Δ+V)N(−Δ+V) of bound states or negative eigenvalues of the Schr\" odinger operator −Δ+V−Δ+V with a potential VV in RdRd for d=3,4,…d=3,4,…. The result extends to a general setting for semigroups

S(t)=e−tH0S(t)=e−tH0, t≥0t≥0, on L2(μ)L2(μ) for a σσ-finite measure μμ, such that SS is dominated by a semigroup |S||S| of pointwise positive operators on L2(μ)L2(μ) and the spectrum of the selfadjoint operator H0H0 is bounded below.

Ingredients in the proof are an extension of the notion of the trace of an operator to the class of

pointwise positive operators with an integral kernel and the operator version of the Feynman-Kac formula due to I. Kluv\'anek (1978).

On Riemannian manifolds, the CLR inequality is known to be equivalent to Sobolev embedding estimates, raising the intriguing possibility of general Sobolev embedding estimates for abstract measure spaces which need not possess even a group structure or a Malliavin calculus. So called {\it logarithmic Sobolev inequalities} are known in

quantum field theory and statistical mechanics only for hypercontractive semigroups acting on linear function spaces and spaces of distributions.