## The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that.(English)Zbl 1222.47072

Funct. Anal. Appl. 41, No. 3, 217-236 (2007); translation from Funkts. Anal. Prilozh. 41, No. 3, 60-83 (2007).
The author studies a new approach to the construction of an asymptotic expansion of the function $$\xi(\lambda)$$ for $$\lambda\to\infty$$. The existence of such an expansion is shown using the formula of Birman-Krein (higher energetical asymptotic expansion of $$S(\lambda)$$). The coefficients are effectively calculated. Moreover, a trace identity is given for entire and half orders as well as characterised in terms of invariants of the heat equation. The whole paper is based on Krein’s theory applied to the Schrödinger operator.

### MSC:

 47F05 General theory of partial differential operators 35P20 Asymptotic distributions of eigenvalues in context of PDEs
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### References:

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