Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints
Langer, Matthias and Woracek, Harald; Binder, Ilia and Kinzebulatov, Damir and Mashreghi, Javad, eds. (2023) Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints. In: Function Spaces, Theory and Applications. Fields Institute Communications . Springer, Cham, Switzerland, pp. 105205. ISBN 9783031392702 (https://doi.org/10.1007/9783031392702_5)
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Abstract
Part I of this paper deals with twodimensional canonical systems y'(x)=zJH(x)y(x), x∈(a,b), whose Hamiltonian H is nonnegative 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. 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 that are obtained via the above construction (positive Borel measures with power growth at ±∞); (5) a global uniqueness theorem (if Weyl functions or spectral measures coincide, Hamiltonians essentially coincide); (6) a local uniqueness theorem (if Weyl functions coincide up to an exponentially small error, Hamiltonians essentially coincide up to a certain point). 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 Schrödinger equations, where coefficients are assumed to be singular but subject to growth restrictions. In the latter case potentials include Besseltype potentials but also highly oscillatory potentials. We obtain corresponding direct and inverse spectral theorems.


Item type: Book Section ID code: 87688 Dates: DateEvent8 July 2023PublishedSubjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 20 Dec 2023 11:51 Last modified: 08 Apr 2024 14:23 URI: https://strathprints.strath.ac.uk/id/eprint/87688