Spectral Analysis of a Dissipative Singular Schrödinger Operator in Terms of a Functional Model

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.