Abstract:
This paper deals with the derivation of two-sided estimates for the trace of the difference of two semigroups generated by two Schrödinger operators in $L_{2}(\mathbb{R}^{3})$ with trace class difference of the resolvents. Use is made of a purely operator-theoretic technique. The results are stated in a rather general abstract form. The sharpness of our estimates is confirmed by the fact that they imply the asymptotic behavior of the trace of the difference of the semigroups as $t\to+0$. Our considerations are substantially based on the Krein–Lifshits formula and on the Birman–Solomyak representation for the spectral shift function.
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