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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Abstract
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 Ldclhd 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 V1 and V2, 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 Ldclhd 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 V1 and V2 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.