Schrödinger Operator

Abstract

Let D be a domain in ℝ^d (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 ∈ L ¹_loc (D), qu ∈ L ¹_loc (D) and

$$ \int {_{_D } u(x)\Delta \varphi (x)dx} = - 2\int {_D q(x)u(x)\varphi (x)dx} $$

for all φ ∈ C ^∞_c (D).