Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-positive type

The two-dimensional Hamiltonian system (*)  y'(x)=zJH(x)y(x),  x∈(a,b), where the Hamiltonian H takes non-negative 2x2-matrices as values, and $J:= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, has attracted a lot of interest over the past decades.  Special emphasis has been put on operator models and direct and inverse spectral theorems.  Weyl theory plays a prominent role in the spectral theory of the equation, relating the class of all equations (*) to the class N0 of all Nevanlinna functions via the construction of Titchmarsh–Weyl coefficients. In connection with the study of singular potentials, an indefinite (Pontryagin space) analogue of equation (*) was proposed, where the 'general Hamiltonian' is allowed to have a finite number of inner singularities. Direct and inverse spectral theorems, relating the class of all general Hamiltonians to the class <N∞ of all generalized Nevanlinna functions, were established. In the present paper, we investigate the spectral theory of general Hamiltonians having a particular form, namely, such which have only one singularity and the interval to the left of this singularity is a so-called indivisible interval.  Our results can comprehensively be formulated as follows. — We prove direct and inverse spectral theorems for this class, i.e. we establish an intrinsic characterization of the totality of all Titchmarsh–Weyl coefficients corresponding to general Hamiltonians of the considered form. —  We determine the asymptotic growth of the fundamental solution when approaching the singularity. —  We show that each solution of the equation has 'polynomially regularized' boundary values at the singularity. Besides the intrinsic interest and depth of the presented results, our motivation is drawn from forthcoming applications: the present theorems form the core for our study of Sturm–Liouville equations with two singular endpoints and our further study of the structure theory of general Hamiltonians (both to be presented elsewhere)

Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints

Part I of this paper deals with two-dimensional canonical systems $y'(x)=yJH(x)y(x)$, $x\in(a,b)$, whose Hamiltonian $H$ is non-negative and locally integrable, and where Weyl's limit point case takes place at both endpoints $a$ and $b$. We investigate a class of such systems defined by growth restrictions on H towards a. For example, Hamiltonians on $(0,\infty)$ of the form $H(x):=\begin{pmatrix}x^{-\alpha}&0\\ 0&1\end{pmatrix}$ where $\alpha<2$ are included in this class. We develop a direct and inverse spectral theory parallel to the theory of Weyl and de Branges for systems in the limit circle case at $a$. Our approach proceeds via - and is bound to - Pontryagin space theory. It relies on spectral theory and operator models in such spaces, and on the theory of de Branges Pontryagin spaces. The main results concerning the direct problem are: (1) showing existence of regularized boundary values at $a$; (2) construction of a singular Weyl coefficient and a scalar spectral measure; (3) construction of a Fourier transform and computation of its action and the action of its inverse as integral transforms. The main results for the inverse problem are: (4) characterization of the class of measures occurring above (positive Borel measures with power growth at $\pm\infty$); (5) a global uniqueness theorem (if Weyl functions or spectral measures coincide, Hamiltonians essentially coincide); (6) a local uniqueness theorem. In Part II of the paper the results of Part I are applied to Sturm--Liouville equations with singular coefficients. We investigate classes of equations without potential (in particular, equations in impedance form) and Schr\"odinger equations, where coefficients are assumed to be singular but subject to growth restrictions. We obtain corresponding direct and inverse spectral theorems

Spectral multiplicity of selfadjoint Schrodinger operators on star-graphs with standard interface conditions

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of Theorem of I.S. Kac

Karamata's theorem for regularised Cauchy transforms

We prove Abelian and Tauberian theorems for regularised Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularised Cauchy transform