On the Domains of Bessel Operators

We consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ ( m 2 - 1 4 ) 1 x 2 , often called the Bessel operator. We assume that m is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $$|\mathrm{Re}(m)|<1$$ | Re ( m ) | < 1 and of its unique closed realization for $$\mathrm{Re}(m)>1$$ Re ( m ) > 1 coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$ Re ( m ) = 1 the minimal second-order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.

Classical and Quantum Markov Processes Associated with q-Bessel Operators

We study the fundamental properties of classical and quantum Markov processes generated by q-Bessel operators and their extension to the algebra of all bounded operators on the Hilbert space [Formula: see text]. In particular, we find a suitable generalized Gorini–Kossakowski–Sudarshan–Lindblad representation for the infinitesimal generator of q-Bessel operator and show that both the classical and quantum Markov processes are transient for α > 0 and recurrent for α = 0. We also show that they do not admit invariant states and, moreover that the support projection of any initial state instantaneously fills the full space.