### A generalization of Gordon's theorem and applications to quasiperiodic Schrödinger operators.

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We study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained.

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator...

We introduce a Lie algebra, which we call adelic $W$-algebra. Then we construct a natural bosonic representation and show that the points of the Calogero-Moser spaces are in 1:1 correspondence with the tau-functions in this representation.

In this article, we consider the operator $L$ defined by the differential expression $$\ell \left(y\right)=-{y}^{\text{'}\text{'}}+q\left(x\right)y,\phantom{\rule{1.0em}{0ex}}-\infty <x<\infty $$ in ${L}_{2}(-\infty ,\infty )$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition $$\underset{-\infty <x<\infty}{sup}\left\{exp\left(\u03f5\sqrt{\left|x\right|}\right)\left|q\left(x\right)\right|\right\}<\infty ,\phantom{\rule{1.0em}{0ex}}\u03f5>0$$ holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria...