Fundamental solutions for differential equations associated with the number operator

Author:
Yuh Jia Lee

Journal:
Trans. Amer. Math. Soc. **268** (1981), 467-476

MSC:
Primary 35R15; Secondary 28C20, 46G99, 58D20, 60J99

DOI:
https://doi.org/10.1090/S0002-9947-1981-0632538-9

Correction:
Trans. Amer. Math. Soc. **276** (1983), 621-624.

MathSciNet review:
632538

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Abstract | References | Similar Articles | Additional Information

Abstract: Let $(H, B)$ be an abstract Wiener space. If $u$ is a twice $H$-differentiable function on $B$ such that $Du(x) \in {B^{\ast }}$ and ${D^2}u(x)$ is of trace class, then we define $\mathfrak {N}u(x) = - \Delta u(x) + (x, Du(x))$, where $\Delta u(x) = {\operatorname {trace} _H} {D^2}u(x)$ is the Laplacian and $( \cdot , \cdot )$ denotes the $B$-${B^{\ast }}$ pairing. The closure $\overline {\mathfrak {N}}$ of $\mathfrak {N}$ is known as the number operator. In this paper, we investigate the existence, uniqueness and regularity of solutions for the following two types of equations: (1) ${u_t} = - \mathfrak {N}u$ (initial value problem) and (2) ${\mathfrak {N}^k}u = f(k \geqslant 1)$. We show that the fundamental solutions of (1) and (2) exist in the sense of measures and we represent their solutions by integrals with respect to these measures.

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Keywords:
Abstract Wiener space,
Wiener measure,
<IMG WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$H$">-differentiation,
number operator,
fundamental solution

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© Copyright 1981
American Mathematical Society