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IFAC-PapersOnLine

Staticization-based representations for Schrödinger equations driven by Coulomb potentials

William M. McEneaney, Peter M. Dower

Published Jan 1, 2019• 5 citations
Publisher

Abstract

A stochastic control representation for the Maslov dequantization of the solution of the Schrödinger equation is obtained. The representation employs complex-valued diffusion process and stationarity of an action functional. The use of stationarity removes the need for convexity of the action, and hence there is no restriction on the duration of the problem. Previous efforts along this line were restricted to potentials which had holomorphic extensions to the entire space. The use of a stationarity based representation for the Coulomb potential removes that restriction. The result is a representation in terms of iterated staticization, which is expected to yield exceptional computational benefits.

DOI
10.1016/j.ifacol.2019.07.005

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A stochastic control representation for the Maslov dequantization of the solution of the Schrödinger equation is obtained. The representation employs complex-valued diffusion process and stationarity of an action functional. The use of stationarity removes the need for convexity of the action, and hence there is no restriction on the duration of the problem. Previous efforts along this line were restricted to potentials which had holomorphic extensions to the entire space. The use of a stationarity based representation for the Coulomb potential removes that restriction. The result is a representation in terms of iterated staticization, which is expected to yield exceptional computational benefits.
A diffusion-based representation for the Maslov dequantization of the time-dependent Schrödinger equation is obtained, where the wave function corresponds to the statical value function of a complex-valued stochastic control problem. Whereas previous efforts along this line were restricted to holomorphic potentials, we here use an additional controller specifically constructed to generate the Coulomb potential via staticization. The solution to the Schrödinger equation is obtained from requantization of the representations. A demonstrative numerical example is included at the end.
Hamilton-Jacobi partial differential equations (HJ PDEs) arise in many scientific fields and applications, especially in mechanics and optimal control. Solutions to second order Hamilton-Jacobi partial differential equations (HJ PDEs) have controlled diffusion process representations. Of particular interest is the diffusion representation of an action functional associated with the solution of Schrödinger initial value problems (IVPs). In existing work that connects stochastic control problems to Schrödinger IVPs, one searches for the minimum of an action functional which is the payoff of the control problem. Issues arise, however, as the action functional loses convexity over longer time durations. The time duration where such representation is valid is in fact infinitesimal when the system dimension is infinite. In this dissertation, we present an approach inspired by the Principle of Stationary Action in physics that removes this limitation, which leads to exceptional computational benefits. Instead of searching for local minima of the action functional as in traditional optimal control problems, we look for its stationary values instead. We introduce the ``staticization" operator and stationary-action control problems. A stationary-action stochastic control representation for the dequantized Schrödinger IVP in a non-inertial frame where the potential field is a polynomial is given. A solution approximation as a series expansion in a small parameter is obtained through the use of complex-valued diffusion-process representations. Following this, the staticization operator is studied in detail. A new approach to solving conservative dynamical systems (e.g. paths of objects in gravitational or Coulomb potential field) using stationary-action control-theoretic methods with promising computational benefits is introduced. Two examples of application, the N-body problem and the Schrödinger IVP, are given. Lastly, we demonstrate the existence of strong solutions of a class of degenerate stochastic differential equations (SDEs) that arises in the stationary-action stochastic control representation of Schrödinger IVPs with the Coulomb potential, which is non-smooth and has branch cuts in the complex range. The SDEs we consider have drift terms that have discontinuities and singularities along some manifolds, and diffusion coefficients that are degenerate, and there is no previously existing results regarding the existence of strong solutions for such SDEs.
In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one-to-one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.
A class of functions ℋnF was chosen, which are boundary values of holomorphic functions. It was shown that if the potential V∈ℋnF and the initial wave function ψ∈ℋnF then the solution of the Schrödinger equation is ψt∈ℋnF. The unitary evolution ψ → ψt can be expressed by an integral with respect to the Wiener measure. To each ψt∈ℋnF is associated a complex Markov process qt fulfilling stochastic Hamilton equations. The time evolution of ψ as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to qt. Some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of Feynman’s sum over trajectories are discussed.
From a stochastic control perspective, the Schrödinger bridge is a density-valued continuous curve parametrized by time that connects a given pair of initial and terminal probability densities via minimum effort controlled Brownian motion. The control-affine Schrödinger bridge extends this idea to a generic control-affine It ôo diffusion, possibly with an additive state cost. In this work, we recast the necessary conditions of optimality for the control-affine Schrödinger bridge problem as a two point boundary value problem for a quantum mechanical Schrödinger PDE with complex potential. This complex-valued potential is a generalization of the realvalued Bohm potential in quantum mechanics. Our derived potential is akin to the optical potential in nuclear physics where the real part of the potential encodes elastic scattering (transmission of wave function), and the imaginary part encodes inelastic scattering (absorption of wave function). The key takeaway is that the process noise that drives the evolution of probability densities induces an absorbing medium in the evolution of wave function. These results make new connections between control theory and non-equilibrium statistical mechanics through the lens of quantum mechanics.