Chains of Non-regular de Branges Spaces

We consider canonical systems with singular left endpoints, and discuss the concept of a scalar spectral measure and the corresponding generalized Fourier transform associated with a canonical system with a singular left endpoint. We use the framework of de Branges’ theory of Hilbert spaces of entire functions to study the correspondence between chains of non-regular de Branges spaces, canonical systems with singular left endpoints, and spectral measures.

We find sufficient integrability conditions on a Hamiltonian H which ensure the existence of a chain of de Branges functions in the first generalized Pólya class with Hamiltonian H. This result generalizes de Branges’ Theorem 41, which showed the sufficiency of stronger integrability conditions on H for the existence of a chain in the Pólya class. We show the conditions that de Branges came up with are also necessary. In the case of Krein’s strings, namely when the Hamiltonian is diagonal, we show our proposed conditions are also necessary.

We also investigate the asymptotic conditions on chains of de Branges functions as t approaches its left endpoint. We show there is a one-to-one correspondence between chains of de Branges functions satisfying certain asymptotic conditions and chains in the Pólya class. In the case of Krein’s strings, we also establish the one-to-one correspondence between chains satisfying certain asymptotic conditions and chains in the generalized Pólya class.