Perturbation theory of dense point spectra
Let H be a self-adjoint operator in a separable Hilbert space with pure point spectrum. Denote by \(\{e_ n\}\) a complete orthonormal set of eigenvectors of H. The author studies perturbations of H which are small in some appropriate sense. The main result of this paper are: 1. A bounded operator A is called strongly H-finite if \(\sum^{\infty}_{n=1}| Ae_ n| <\infty\) (\(| x|\) is the norm of the vector \(x\in H)\). Let \(A_ 1,...,A_ N\) be strongly H- finite. Then for a.e. \(\kappa =(\kappa_ 1...,\kappa_ n)\in {\mathbb{R}}^ N\), the operator \(H(\kappa)=H+\sum^{N}_{j=1}\kappa_ jA^*_ jA_ j\) has pure point spectrum. It follows that \(H+\kappa <\cdot,\phi >\phi\) is pure point for a.e. \(\kappa\) if \(\sum^{\infty}_{n=1}| <\phi,e_ n>| <\infty.\) 2. Let \(X_ 1,X_ 2,..\). be a sequence of independent identically distributed (i.i.d.) absolutely continuous random variables with density \(w(x)>0\). Let \(A_ 1,A_ 2,..\). be bounded operators with \(\sum^{\infty}_{n=1}\{\sum^{\infty}_{i=1}| A_ je_ n|^ 2\}^{1/2}<\infty\). Then the random operator \(H(\omega)=H+\sum^{\infty}_{j=1}(\tanh X_ j(\omega))A^*_ jA_ j\) has pure point spectrum almost surely (a.s.). 3. Let \(\{\phi_ n\}\) be a complete orthonormal set and \(\{a_ n\}^ a \)sequence of positive numbers with lim \(a_ n=0\). If H is self-adjoint and \(\sigma(H)\) has Lebesgue measure zero, then the random compact perturbation \(H+\sum^{\infty}_{n=1}a_ n \tanh (X_ n(\omega))<\cdot,\phi_ n>\phi_ n\) is pure point a.s. 4. Let \(Y_ j(\omega)\) be i.i.d. absolutely continuous random variables with density \(g(x) > 0\). Let \(H(\omega) = \text{diag}Y_ j(\omega)\), so that the eigenvalues of the diagonal unperturbed operator \(H(\omega)\) are random variables. If \(\sum_{n=1}^\infty| Ae_ n| < \infty\) and \(W\) is a bounded operator, then \(H(\omega) + A* WA\) has pure point spectrum a.s. These results are generalized to the case where \(H\) has pure point spectrum in an open interval \(J\).
sequence of independent identically distributed absolutely continuous random variables
self- adjoint operator in a separable Hilbert space with pure point spectrum
complete orthonormal set of eigenvectors
scientific article; zbMATH DE number 4054370
scientific article