Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of nonpositive type
Langer, Matthias and Woracek, Harald (2013) Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of nonpositive type. Operators and Matrices, 7 (3). pp. 477555. ISSN 18463886 (https://doi.org/10.7153/oam0729)
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Abstract
The twodimensional Hamiltonian system (*) y'(x)=zJH(x)y(x), x∈(a,b), where the Hamiltonian H takes nonnegative 2x2matrices 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 socalled 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).
ORCID iDs
Langer, Matthias ORCID: https://orcid.org/0000000188137914 and Woracek, Harald;

Item type: Article ID code: 45120 Dates: DateEvent30 September 2013PublishedKeywords: Hamiltonian system with inner singularity, Titchmarsh–Weyl coefficient, inverse problem, asymptotics of solutions, Probabilities. Mathematical statistics, Mathematics(all) Subjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 11 Oct 2013 13:33 Last modified: 25 May 2023 09:13 URI: https://strathprints.strath.ac.uk/id/eprint/45120