Abstract:
A Molchanov-type condition is considered in applications to ordinary differential operators of arbitrary order with complex-valued coefficients. It is proved to be a necessary condition for the compactness of the resolvent for a wide class of operators of this type. A counterexample is given showing that this condition does not suffice for the compactness of the resolvent for a Sturm–Liouville operator with nonnegative real part of the potential. Molchanov's criterion is generalized to potentials taking values in a sector bounded away from the negative half-axis and more narrow than a half-plane.
Bibliography: 18 titles.
Keywords:nonselfadjoint Sturm–Liouville operator, discreteness of spectrum, compactness of resolvent, Molchanov's criterion.
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