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Ullmann, J. (2020): Generalization of Weyl's asymptotic formula for the relative trace of singular potentials. In: Journal of Mathematical Physics, Vol. 61, No. 5, 052102
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Weyl's asymptotic formula states that in the semiclassical limit h down arrow 0, the sum Tr[-h2 Delta+V]- of negative eigenvalues of a Schrodinger operator is given by Ldclh-d integral[V]-1+d/2 and an error of order o(h(-d)), whenever the integral of [V]-1+d/2 is finite. In this paper, we show that if we are given two Schrodinger operators with potentials V-1 and V-2, their difference Tr[-h2 Delta+V1]--Tr[-h2 Delta+V2]- still follows a semiclassical asymptotic, i.e., it is, up to o(h(-d)), given by Ldclh-d times the integral over the difference [V1]-1+d/2-[V2]-1+d/2 if certain conditions implying the finiteness of that integral are fulfilled. This holds even if V-1 and V-2 do not fulfill the integrability conditions of Weyl's formula, and the traces Tr[-h2 Delta+V1]- and Tr[-h2 Delta+V2]- are of higher order than O(h(-d)) each, and it is a generalization of Weyl's formula in those cases.