Aubrey Truman

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.

Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold

We consider the limiting case λ→0 of the Cauchy problem, ∂gλ(x,t)/∂t = (1/2) λΔxgλ(x,t)+(V(x)/λ) gλ( x,t), with gλ (x,0) = exp{−S0(x)/λ}T0(x), V, S0 being real‐valued functions on N, T0 a complex‐valued function on N; V, S0, T0 being independent of λ, Δx being the Laplace–Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem  D2Z(s)/∂s2 = −∇ZV[Z(s)], s∈[0,t], with Z(t) = x and Z(0) = ∇S0(Z(0)).One of our results is equivalent to the fact that for short times Schrodinger quantum mechanics on the Riemannian manifold N tends to classical Newtonian mechanics on N as h/ tends to zero.

The Feynman maps and the Wiener integral

By introducing the family of Feynman maps Fs, we show that our earlier definition of the Feynman path integral F=F1 can be obtained as the analytic continuation of the Wiener integral E =F−i. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps Fs, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for F1=F.

Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation

We consider the limiting case λ→0 of the Cauchy problem ∂uλ/∂t= (λ/2μ) ∇2xuλ +[V (x)/λ]uλ, uλ(x,0) =exp[−S0(x)/λ]T0(x); S0, T0 independent of λ, for both real and pure imaginary λ. We prove two new theorems relating the limiting solution of the above Cauchy problem to the corresponding equations of classical mechanics μ (d2x/dτ2)(τ) =−∇xV[x (τ)], τ∈ (0,t). These relationships include the physical result quantum mechanics → classical mechanics as h/→0.

A note on almost sure exponential stability for stochastic partial functional differential equations

In a recent paper, Taniguchi (Stochastic Anal. Appl. 16 (5) (1998) 965-975) investigated the almost sure exponential stability of the mild solutions of a class of stochastic partial functional differential equations. Precisely, as small delay interval assumption is imposed, sufficient conditions are obtained there to ensure the almost sure exponential stability of the mild solutions of the given stochastic systems. Unfortunately, the main results derived by him are somewhat restrictive to be applied for practical purposes. In the note we shall prove that for a class of stochastic functional differential equations the small delay interval assumption imposed there is actually unnecessary and can be removed.

On stochastic diffusion equations and stochastic Burgers’ equations

In this paper we construct a strong solution for the stochastic Hamilton Jacobi equation by using stochastic classical mechanics before the caustics. We thereby obtain the viscosity solution for a certain class of inviscid stochastic Burgers’ equations. This viscosity solution is not continuous beyond the caustics of the corresponding Hamilton Jacobi equation. The Hopf–Cole transformation is used to identify the stochastic heat equation and the viscous stochastic Burgers’ equation. The exact solutions for the above two equations are given in terms of the stochastic Hamilton Jacobi function under a no‐caustic condition. We construct the heat kernel for the stochastic heat equation for zero potentials in hyperbolic space and for harmonic oscillator potentials in Euclidean space thereby obtaining the stochastic Mehler formula.

Feynman path integrals and quantum mechanics as h/→0

When the space of paths is a certain Hilbert space H, we show how to extend the Feynman path integral F of DeWitt and Albeverio and Hoegh‐Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of F and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.

Generalized Ito formulae and space-time Lebesgue-Stieltjes integrals of local times

Generalized Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded two-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.

Classical periodic solutions of the equal-mass 2n-body problem, 2n-ion problem and the n-electron atom problem

Abstract We present a new family of classical periodic orbits for physically interesting hamiltonian systems, such as the Zeeman effect hamiltonian for the n -electron atom.

On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender–Wu formula for x2N-anharmonic oscillators

Rigorous results on the Laplace expansions of conditional Wiener integrals with functional integrands having a finite number of global maxima are established. Applications are given to the Bender–Wu formula for the x2N ‐anharmonic oscillator.