Exponential Decay of Eigenfunctions
We take a pause from our development of the theory of linear operators to present a first application to Schrödinger operators. Let us recall from the Introduction that a Schrödinger operator is a linear operator on the Hilbert space L 2 (ℝn) of the form H = -△ + V, where and the potential V is a real-valued function. The general problem we study here is as follows. Suppose that L is a linear operator on L 2(ℝn) with eigenvalue λ and corresponding eigenfunction ψ, that is, a function ψ ∈ L 2(ℝn) such that Lψ= λψ Since ψ ∈ L 2(ℝn), it has some average decay as. How is this decay determined by the operator L? In the case that L is a Schrödinger operator, we would like to know how the behavior of the potential V, as determines the decay of an eigenfunction. This can be answered very nicely provided we content ourselves with upper bounds on the rate of decay. We will also use this discussion to introduce various geometric ideas concerning Schrödinger operators. These ideas will play important roles in the later chapters on semiclassical analysis.
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