M. Langer and H. Woracek,
Direct and inverse spectral theorems for a class of canonical systems with
two singular endpoints,
in "Function Spaces, Theory and Applications",
Fields Institute Communications, vol. 87, Springer, Cham, 2023, pp. 105–205.
Abstract:
Part I of this paper deals with two-dimensional canonical systems
y'(x)=zJH(x)y(x),
x∈(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.
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 Bessel-type potentials but also
highly oscillatory potentials.
We obtain corresponding direct and inverse spectral theorems.