Search

Historically, the first general method in the spectral analysis of non-selfadjoint differential operators was the Riesz integral, complemented by the refined technique of estimating the resolvent on the contours that divide the spectrum. Using this method, Lidskij (1962) proved the summation over groups (“with brackets”) of the spectral resolution of a general regular second order differential operator. Since then, the so-called “bases with brackets” have been studied extensively by his successors (see the references in Sadovnichij (1973)). Unfortunately, the arrangement of the “brackets”, that is, the combination into one group of the sets of eigenvectors and root vectors corresponding to some neighbouring points of the spectrum, is defined non-uniquely and, to a large extent, non-constructively. Hence, as a rule, the assertions concerning bases with brackets have the character of existence theorems.

We study the spectral problem $$\left. {\Delta \mathcal{U} + K^2 \mathcal{U} = 0, \frac{{\partial u}}{{\partial n}} - iK\mathcal{O}\mathcal{U}} \right|_{\partial \Omega } = 0, \mathfrak{S} \geqslant 0$$

CONTENTSIntroduction § 1. Boundary values for abstract operators and the Krein formula for generalized resolvents § 2. Zero-radius potentials in diffraction problems § 3. Quantum-mechanical problems with energy-dependent potentials Conclusion References

We study Bari and Riesz bases [1] of eigenspaces of contraction operators which are close to unitary. Subject to certain assumptions about the operator, we partition its spectrum into so-called Carleson series, in terms of which we establish new criteria for the basicity of the operator. Most completely studied are contractions with finite-dimensional deficiency operators I – T*T and I – TT* . As examples we consider classical bases of exponential functions in various function spaces.

The question of boundedness (compactness) of the imbedding of classes of smooth functions in a space of functions integrable with respect to some measure arises in many problems of spectral theory.

In a number of problems of operator theory, it is necessary to investigate integral operators of the form $$ \int {\int {\varphi (\lambda ,\mu )d{E_\lambda }Td{E_\mu }.} } $$ . Such integral operators have been studied in a number of articles by M. Sh. Birman and M. Z. Solomyak ([1], [2], [3]).