Feynman integrals for non-smooth and rapidly growing potentials

The Feynman integral for the Schrödinger propagator is constructed as a generalized function of white noise, for a linear space of potentials spanned by finite signed measures of bounded support and Laplace transforms of such measures, i.e., locally singular as well as rapidly growing at infinity. Remarkably, all these propagators admit a perturbation expansion

Unified (q;α,β,γ;ν)(q;\alpha,\beta,\gamma;\nu)-deformation of one-parametric q-deformed oscillator algebras

We define a generalized $(q;\alpha,\beta,\gamma;\nu)$-deformed oscillator algebra and study the number of its characteristics. We describe the structure function of deformation, analyze the classification of irreducible representations and discuss the asymptotic spectrum behaviour of the Hamiltonian. For a special choice of the deformation parameters we construct the deformed oscillator with discrete spectrum of its "quantized coordinate" operator. We establish its connection with the (generalized) discrete Hermite I polynomials

Spectral theory and Wiener-Ito decomposition for the image of a Jacobi field

Assume that K: H → T is a bounded operator, where H and T are Hilbert spaces and ρ is a measure on the space H. Denote by ρ the image of the measure ρ under K . We study the measure ρ under the assumption that ρ is the spectral measure of a Jacobi field and obtain a family of operators whose spectral measure is equal to ρ. We also obtain an analog of the Wiener-Itô decomposition for ρ. Finally, we illustrate the results obtained by explicit calculations carried out for the case, where ρ is a Lévy noise measure