Oleg G. Smolyanov

Hamiltonian Feynman path integrals via the Chernoff formula

The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrodinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrodinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.

Hamilton–Jacobi–Bellman equations for quantum optimal feedback control

We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton?Jacobi?Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton?Pontryagin set-up. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback. A controlled qubit with a feedback is considered as example.

Continuous Quantum Measurement: Local and Global Approaches

In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schrodinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menskis formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schrodinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator.

The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold

We construct the surface measure on the space C([0; 1]; M) of paths in a compact Riemannian manifold M without boundary embedded into R n which is induced by the usual at Wiener measure on C([0; 1]; R n ) conditioned to the event that the Brownian particle does not leave the tubular -neighborhood of M up to time 1. We prove that the limit as ! 0 exists, the limit measure is equivalent to the Wiener measure on C([0; 1]; M), and we compute the corresponding density explicitly in terms of scalar and mean curvature.

Feynman Formulas for Particles with Position-Dependent Mass

In this paper, we obtain Feynman formulas for solutions to equations describing the diffusion of particles with mass depending on the particle position and to Schrodinger-type equations describing the evolution of quantum particles with similar properties. Such particles (to be more precise, quasi-particles) arise in, e.g., models of semiconductors. Tens of papers studying such models have been published (see [1] and the references therein), but representations of solutions to the arising Schrodinger- and heat-type equations, which go back to Feynman, have not been considered so far. One of the possible reasons is that the traditional application of Feynman’s approach involves integrals with respect to diffusion processes whose transition probabilities have no explicit representation (in terms of elementary functions) in the situation under consideration. In this paper, instead of these transition probabilities, we use their approximations, which can be expressed in terms of elementary functions. Apparently, similar approximations were first applied in [4, 5] to study the diffusion and the quantum evolution of particles of constant mass on Riemannian manifolds. It turns out that the central idea of the approach developed in [4, 5] can also be applied (after appropriate modifications) to the situation considered in this paper. In what follows, we assume that solutions to the Cauchy problems for the equations under examination exist and are unique; thus, we can and shall consider not only solutions to equations but also the corresponding semigroups. Somewhat changing the terminology of [2, 6], we define a real (complex) Schrodinger semigroup as e – tH (respectively, e ith ), where H is a self-adjoint positive operator on a Hilbert space or the generator of a diffusion process.

Feynman formulas as a method of averaging random Hamiltonians

We propose a method for finding the mathematical expectation of random unbounded operators in a Hilbert space. The method is based on averaging random one-parameter semigroups by means of the Feynman-Chernoff formula. We also consider an application of this method to the description of various operations that assign quantum Hamiltonians to the classical Hamilton functions.

LAGRANGIAN AND HAMILTONIAN FEYNMAN FORMULAE FOR SOME FELLER SEMIGROUPS AND THEIR PERTURBATIONS

A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite-dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman–Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.

LAGRANGIAN FEYNMAN FORMULAS FOR SECOND-ORDER PARABOLIC EQUATIONS IN BOUNDED AND UNBOUNDED DOMAINS

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

Schrödinger-Belavkin equations and associated Kolmogorov and Lindblad equations

A new derivation of the Schrödinger-Belavkin equation is proposed. The Feynman-Kac formulas are proved for the solutions of the Schrödinger-Belavkin equation and for the solutions of the corresponding Kolmogorov and Lindblad equations.

The Probabilistic Feynman–Kac Formula for an Infinite-Dimensional Schrödinger Equation with Exponential and Singular Potentials

Using the probabilistic Feynman–Kac formula, the existence of solutions of the Schrödinger equation on an infinite dimensional space E is proven. This theorem is valid for a large class of potentials with exponential growth at infinity as well as for singular potentials. The solution of the Schrödinger equation is local with respect to time and space variables. The space E can be a Hilbert space or other more general infinite dimensional spaces, like Banach and locally convex spaces (continuous functions, test functions, distributions). The specific choice of the infinite dimensional space corresponds to the smoothness of the fields to which the Schrödinger equation refers. The results also express an infinite-dimensional Heisenberg uncertainty principle: increasing of the field smoothness implies increasing of divergence of the momentum part of the quantum field Hamiltonian.