Finite-dimensional perturbations of discrete operators and formulas for traces
Finite-dimensional perturbations \[ Lu=L_ ou+\sum^{r}_{k=1}(L_ 0u,f_ k)g_ k \] of a discrete self-adjoint operator \(L_ o\) in a Hilbert space H are considered. Let \(N(\lambda)=\sum_{\lambda_ n\leq \lambda}1=c\lambda +0(\lambda^ p)\), and \(f_ k\in D(L_ 0^{2q_ k})\), \(g_ k\in D(L_ 0^{2-q_ k})\) \((0\leq q_ k\leq 2)\). Then the operator L is discrete as well and there exists a sequence \(k_ n\to 0\) such that \(\lim_{n\to \infty}\sum^{k_ n}_{k=1}(\mu_ k-\lambda_ k)=\sum^{r}_{j=1}(L_ 0^{1-q_ j}g_ j,L_ 0^{q_ j}f_ j)\) \((\lambda_ k,\mu_ k\) are the eigenvalues of \(L_ 0\) and L, respectively). The complete proof of this theorem is written out, and three examples are given.
finite-dimensional perturbations of a discrete self-adjoint operator
scientific article; zbMATH DE number 4071703