Feynman averaging of semigroups generated by Schrödinger operators

Abstract

The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

The method of Chernoff approximation

“…Then L is a second order uniformly elliptic operator and the family (F (t)) t≥0 in (10) has the following view: F (0) ∶= Id and for all t > 0 and all ϕ ∈ X…”

“…Case 1: X = C ∞ (R d ), (ξ t ) t≥0 is a Feller process whose generator L is given by ( 6) with A, B, C of the class C 2,α , A satisfies (11), and either N ≡ 0 or N ≠ 0 and the non-local term of L is a relatively bounded perturbation of the local part of L with some extra assumption on jumps of the process (see details in [22,24]). The family (F (t)) t≥0 is given by (10) (see also (17), or (12) in the corresponding particular cases) and D = C 2,α c (R d ). Further, Ω is a bounded C 4,α −smooth domain, Y = C 0 (Ω), BC are the homogeneous Dirichlet boundary/external conditions corresponding to killing of the process upon leaving the domain Ω.…”

“…Case 3: shifts and averaging. One more approach to construct Chernoff approximations for semigroups and Schrödinger groups generated by differential and pseudo-differential operators is based on shift operators (see [63,64]), averaging (see [10,57]) and their combination (see [10,9]). Let us demonstrate this method by means of simplest examples.…”

“…• averaging of semigroups (see Sec. 2.7, [10,57,10,9]); Moreover, Chernoff approximations have been obtained for some stochastic Schrödinger type equations in [55,54,56,37]; for evolution equations with the Vladimirov operator (this operator is a p-adic analogue of the Laplace operator) in [68,69,67,66,65]; for evolution equations containing Lévy Laplacians in [2,1]; for some nonlinear equations in [58].…”

Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with Respect to These Measures

“…Translation-invariant (or shift-invariant) measures are used for the study of solutions to differential equations based on mathematical expectations of functionals of random walks. Application of translation-invariant measures to representations of solutions to differential equations can be found in [2,4], where strongly continuous one-parameter operator semigroups that solve Cauchy problems for the diffusion equation, the fractional diffusion equation, and the Schrödinger equation with various Hamiltonians were obtained by averaging of random one-parameter families of operators of shifts by vectors of the coordinate space by measures defined on a set of shift operators. Thus approach to the study of properties of solutions to differential equations for functions on infinitedimensional spaces requires the analysis of measures on infinite-dimensional spaces that are invariant under shifts by vectors of this space or under other transformation groups (see [6]).…”