Suppose that f is a Lipschitz function on R with ‖f‖Lip ≤ 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let p ∈ (1,∞) and suppose that x ∈ B(H) is an operator such that the commutator [A, x] is contained in the Schatten class Sp. It is proved by the last two authors, that then also [f(A), x] ∈ Sp and there exists a constant Cp independent of x and f such that ‖[f(A), x]‖p ≤ ...

We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for cla...

The spectral problem (A + V (z))ψ = zψ is considered with A, a self-adjoint operator. The perturbation V (z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A ′ : V (z) = −B(A ′ − z) −1 B *. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A ′ are separated. Conditions are formulated when the p...

We review Fock space based quantum probability and in particular the theory of stop times based on it. In the Fock space H = F(L2(R+)), a stop time may be defined as as a positive self-adjoint operator T = ∫ R+ λ dE(λ) whose spectral resolution (E(λ):λ ∈ R+) is adapted to the natural filtration based on the splittings F(L(R)) = F(L([0, λ[)⊗F(L([λ,∞[) Then if K is one of the basic quantum martin...

In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. We consider the quantum dynamics of a real massless scalar field coupled to gravity in this framework. A Hamiltonian operator for the scalar field can be well defined in the coupled diffeomorphism invariant Hilbert space, which is both ...

The spectral problem (A+V (z))ψ = zψ is considered with A, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation V (z) is assumed to depend on the energy z as resolvent of another self-adjoint operator A ′ : V (z) = −B(A ′ −z) −1 B *. It is supposed that operator B has a finite Hilbert-Schmidt norm and spectra of operators A and A ′ are separated. The conditions are form...

For a von Neumann algebra M acting on a Hilbert space H with a cyclic and separating vector ξ0, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (M, ξ0). For a general Lindblad type generator L of a conservative quantum dynamical semigroup on M, we give sufficient conditions so that the operator H induced by L via the symmetric embedding of M...

We consider dynamics of a quantum scalar field, minimally coupled to classical gravity, in the near-horizon region of a Schwarzschild black-hole. It is described by a static Klein-Gordon operator which in the near-horizon region reduces to a scale invariant Hamiltonian of the system. This Hamiltonian is not essentially self-adjoint, but it admits a one-parameter family of self-adjoint extension...

It was observed in [Su5] that the spectrum of a periodic Schrodinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only f...

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For arbitrary self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism...