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Published in:
1995 | OriginalPaper | Chapter
Schrödinger Operator
Authors : Kai Lai Chung, Zhongxin Zhao
Published in: From Brownian Motion to Schrödinger’s Equation
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
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Let D be a domain in ℝd (d≥ 1). We consider the following equation: $$ \frac{\Delta } {2}u(x) + q(x) = 0,x \in D, $$ where $$ \Delta = \sum\nolimits_{i = 1}^d {\partial ^2 /\partial x_i^2 } $$ is the Laplacian and q is a Borel measurable function on D. This equation is generally taken in the weak sense as discussed in Section 2.5. Thus (1) is satisfied when u ∈ Lloc1(D), qu ∈ Lloc1(D) and $$ \int {_{_D } u(x)\Delta \varphi (x)dx} = - 2\int {_D q(x)u(x)\varphi (x)dx} $$ for all φ ∈ Cc∞(D).