H. D. Voulov

On a Hilbert space $\frak H$, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scale-invariant self-adjoint extension in $ \frak H$. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous three articles on the subject. First we study general symmetric and scale-invariant operators on a Hilbert space. We show that if the index of defect of the operator under consideration is $(1,1)$, then the operator either does not admit any scale-invariant self-adjoint extension, or it admits exactly one scale-invariant self-adjoint extension, or it admits exactly two scale-invariant self-adjoint extensions, or all self-adjoint extensions are scale invariant. We then apply these results to the differential operator and the corresponding difference operator under consideration. For the continuous case, we show that the interval of the parameter, for which the differential operator is not semi-bounded, contains an infinite sequence of values for which all self-adjoint extensions are scale-invariant, while for the remaining values of the parameter from that interval, there are no scale-invariant self-adjoint extensions. For the corresponding difference operator, we show that if it is not semi-bounded, then it does not admit any scale-invariant self-adjoint extension. We also show that both differential and difference operators, at value(s) of the parameter that cor espond to the endpoint(s) of the interval(s) of semi-boundedness, have exactly one scale-invariant self-adjoint extension.

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous article. For values of the parameter for which the difference operator is self adjoint, we show that the spectrum of the operator is discrete and simple. When $q$ approaches $1$, the spectrum fills the whole positive or negative semiaxis.