This paper describes fundamental contributions made by M. G. Krein and his colleagues, including the authors, to the theory of the spectral shift function. Numerous papers are cited involving the shift function. The paper is dedicated to the memory of M. G. Krein.
Given two selfadjoint operators \(H\) and \(H^ 0\) on a separable Hilbert space \({\mathfrak H}\) the spectral shift function \(\xi(\ell)\) is a function related to the change in the spectral density in going from \(H^ 0\) to \(H\). By definition \(\xi(\ell)\) is called a spectral shift function if for all \(\varphi\) in some appropriate class of functions \[ \text{trace }(\varphi(H)-\varphi(H^ 0))=\int\varphi'(\ell)\xi(\ell)d\ell \] [I. M. Lifshits, Usp. Mat. Nauk 7, No. 1(47), 171-180 (1952; Zbl 0046.212)].
Let \(\Delta(z)=\text{det}(I+(H-H^ 0)R_ z(H^ 0))\), \(R_ z(H^ 0)=(zI-H^ 0)^{-1}\) and let \(W_ 1(T)\), \(T=\{z\mid\| z\|=1\}\), be the set of functions on \(T\) whose derivatives may be expanded into an absolutely convergent Fourier series. One of the main theorems proved in the paper is the following:
Theorem. For all \(z\) in the resolvent set of both \(H\) and \(H^ 0\) let the operator \(R_ z(H)-R_ z(H^ 0)\) be in the trace class of bounded operators on \({\mathfrak H}\). Let \(\varphi\) be a function of the form
\(\varphi(\ell)=\psi((\ell-a)(\ell-\overline{a})^{-1})\), \(\psi\in W_ 1(T)\), \(\text{Im } a>0\).
Then there exists a spectral shift function \(\xi(\ell)\) which satisfies (i) for all \(\varphi\). Also \(\xi(\ell)\) satisfies
(ii) \(\int| \xi(\ell)| (1+\ell^ 2)^{-1} d\ell<\infty\),
(iii) \(\xi(\ell)=(1/\pi)\arg_{\varepsilon\searrow 0} \Delta (\ell+i\varepsilon)\) (Theorem 6.6).
The spectral shift function \(\xi(\ell)\) is connected to the scattering operator \(S\) associated with \(H\) and \(H^ 0\). If \(S\) is written as a direct integral \(S=\int\oplus S(\ell)d\ell\) then
(iv) \(\text{det }S(\ell) =\exp(-2\pi i\xi(\ell))\), \(\ell\in \sigma(H^ 0)\).
If \(U U_ 0^{-1}\) is an operator of negative (positive) type, \(U=\exp (-iHt)\), \(U_ 0= \exp(-iH^ 0 t)\) and if \(H\), \(H^ 0\) satisfy the condition stated in the above theorem then \(S(\ell)\) is an operator of negative (positive) type (Theorem 8.3).
Many applications of the spectral shift function involve scattering theory. For the non-relativistic two body problem the spectral shift function determines asymptotically the phase shift of the scattered wave (Part 1 of Section 9). More recently the spectral shift function has been employed together with the Witten index in supersymmetric scattering theory (Part 5 of Section 9).
The reader will find this paper a current and useful source for basic theory and applications of the spectral shift function.