We discuss direct and inverse spectral theory for a Sturm–Liouville type problem with a quadratic dependence on the eigenvalue parameter, −f ′′ + 1 4 f = z ωf + zυf, which arises as the isospectral problem for the conservative Camassa–Holm flow. In order to be able to treat rather irregular coefficients (that is, when ω is a real-valued Borel measure on R and υ is a non-negative Borel measure o...

The self-adjointness of an evolution operator corresponding to the model of the flat FRW universe with massless scalar field and cosmological constant quantized in the framework of loop quantum cosmology is studied in the case > 0. It is shown that, for < c 10:3‘ 2 Pl , the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for c the operator i...

We show that the particle motion in Bohmian mechanics as the solution of an ordinary differential equation exists globally, i.e., the singularities of the velocity field and infinity will not be reached in finite time for typical initial configurations and a large class of potentials, including the physically most relevant potential of N -particle Coulomb interaction with arbitrary charges and ...

In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Newmann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamil...

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded operator-valued symbols. Our applies, in particular, to symbols Lipschitz continuous derivatives, which is the natural class Hamiltonian functions for classical mechanics. For this we extend Berger-Coburn estimate case vector-valued Segal-Bargmann spaces. Finally, apply our result (operator-valued) qu...

A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can shown that symmetry could omitted by using criterion for graph and adjoint graph. Namely, S is densely defined closed if only . In more general setup, we consider relations instead operators prove this situation similar result holds. We give necessary...