Methods of Functional Analysis
and Topology

We provide sufficient conditions for existence of a global diffeomorphism between tame Fréchet spaces. We prove a version of the mountain pass theorem which plays a key ingredient in the proof of the main theorem.

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space $\mathfrak H_1$ to a Hilbert space $\mathfrak H_2$ are considered in the Hilbert space $\mathfrak H_1\oplus\mathfrak H_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in $\mathfrak H_1 \oplus \mathfrak H_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative selfadjoint extensions $A$, in particular the Friedrichs and Krein-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that $A$ belongs to the class of extremal extensions of $S$ if and only if ${\rm dom\,} A \perp {\rm ran\,} A$. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of $S$ and semibounded parameters describing them if and only if the operator part of $R$ is bounded.

Entropy functionals and their extremal values for solving the Stieltjes matrix moment problem are defined and investigated for the first time. Explicit formulas for the extremal values of the entropy over the set of solutions of the Stieltjes matrix moment problem are obtained. A geometric interpretation in terms of Weyl matrix intervals is presented.

We consider one dimensional Schrödinger operators \begin{equation*} H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda \end{equation*} with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behavior of eigenvalues. Potentials $U$ and $V_\lambda$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_\lambda$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $\lambda\to 0$. We also construct two-term asymptotic formulas for the threshold eigenvalues.