Analytic Feynman Integral and a Change of Scale Formula for Wiener Integrals of an Unbounded Cylinder Function

We investigate the behavior of the unbounded cylinder function F x = ∫ 0 T α 1 t d x t 2 k ⋅ ∫ 0 T α 2 t d x t 2 k ⋅ ⋯ ⋅ ∫ 0 T α n t d x t 2 k , k = 1,2 , … whose analytic Wiener integral and analytic Feynman integral exist, we prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral, and we prove a change of scale formula for the Wiener integral about the unbounded function on the Wiener space C 0 0 , T .

Integral Transforms on a Function Space with Change of Scales Using Multivariate Normal Distributions

Using simple formulas for generalized conditional Wiener integrals on a function space which is an analogue of Wiener space, we evaluate two generalized analytic conditional Wiener integrals of a generalized cylinder function which is useful in Feynman integration theories and quantum mechanics. We then establish various integral transforms over continuous paths with change of scales for the generalized analytic conditional Wiener integrals. In these evaluation formulas and integral transforms we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms. Consequently the transforms in the present paper can be expressed in terms of the generalized cylinder function itself.

BESSEL FUNCTIONS IN A QUANTUM-BILLIARD CONFIGURATION PROBLEM

In studying various quantum-billiard configurations, R. L. Liboff (J. Math. Phys.35 (1994) 2218), was led to investigate the vanishing of f(ν)=j2ν,1 - jν2, where jμk is the kth positive zero of the Bessel function Jμ(x). Here we show that the even more general function fα(ν)=cαν,k - cν,k+l is increasing and vanishes once (and only once) in 0<ν<∞, provided α≥π/2 and [Formula: see text], k, l=1,2,3,…. As usual, cμn is the nth positive zero of the cylinder function Cμ(x)=Jμ (x) cos θ - Yμ(x) sin θ. Specialized to Liboff's case, f(ν), this yields not only the existence of a zero of f(ν) but also its uniqueness.

EXPONENTIAL INTEGRABILITY FOR IMAGES OF ORNSTEIN–UHLENBECK OPERATORS ACTING ON CYLINDER FUNCTIONS ON LOOP SPACES

Let E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric (TSS) connection. Let L be the Ornstein–Uhlenbeck (O-U) operator on the loop space E, and f be a cylinder function on E. We first extend the expression of Lf, proved by Enchev and Stroock for the Levi–Cività connection, to a general TSS connection, and then prove that if [Formula: see text], ε |Lf|2 is exponential integrable for some constant ε := ε (f)>0.