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

Abstract: Part I of this paper deals with two-dimensional canonical systems $y'(x)=zJH(x)y(x)$, $x\in(a,b)$, 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 $a$. 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 $\pm\infty$); (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.