Regularized function of spectral shift for the one-dimensional Schrödinger operator with slowly decreasing potential. (Russian) Zbl 0581.47034

Koplienko, L. S.

Sib. Mat. Zh. 26, No. 3(151), 72-77 (1985).

The paper refers to the one-dimensional Schrödinger operator with slowly decreasing potential, i.e. like \(X^{-\alpha}\) for \(\alpha >1/2\). One proves that: if \(H_ 0\) is an autoadjoint operator generated by \(- d^ 2/dx^ 2\) under the boundary condition \(\psi (0)=0\) and V is the multiplication operator on real function v with \[ | d^ jv(x)/dx^ j| \leq C(1+x)^{-\alpha -j},\quad \alpha >1/2,\quad j=0,1,2 \] then to \(H_ 0\) and V corresponds a function \(\eta\) of spectral shift which is differentiable and for which holds \(\eta '(\lambda)=\pi^{-1}\theta (\sqrt{\lambda})\) if \(\lambda >0\), and \(=-N(\lambda)\) if \(\lambda <0\), N(\(\lambda)\) and \(\theta\) (\(\sqrt{\lambda})\) being defined in the proof.

Reviewer: C.Simionescu

MSC:

47F05	General theory of partial differential operators
47A10	Spectrum, resolvent

Keywords:

one-dimensional Schrödinger operator with slowly decreasing potential; spectral shift