Eigenvalues of Schrödinger operators on bounded intervals

Before explaining how resonances defined in §1 are related to the long time behaviour of scattered waves we discuss the more familiar case of eigenvalues and eigenfuctions. The displacement, $w(t, x)$, of a point $x$ on vibrating string, $[-L, L]$, at time $t$, is approximately given by the solution of the wave equation:

$$\begin{array}{c} (\partial^2_t - \partial^2_x) w(t, x) = 0, \... ... w(0, x) = w_0(x), \partial_t w(0, x) = w_1(x). \end{array}$$ (3)

We can generalize this by introducing a potential and considering the wave equation for a Schrödinger operator, $H_V = -\partial_x^2 + V(x)$,

$$\begin{array}{c} (\partial^2_t + H_V) w(t, x) = 0, \vert ... ... w(0, x) = w_0(x), \partial_t w(0, x) = w_1(x). \end{array}$$ (4)

The Dirichlet (zero boundary condition) realization of the operator $H_V$ on $[-L, L]$ has a discrete spectrum

$$E_N < E_{N-1} < \cdots < E_1 < 0 \leq \lambda_1^2 < \cdots < \lambda_n^2 < \cdots ,$$

with the corresponding eigenfunctions,

$$\begin{array}{c} H_V v_k = E_k v_k, \ v_k(\pm L) = 0, \... ...= 0, \ \int_{-L}^L \vert u_n(x)\vert^2 dx = 1. \end{array}$$

The solution, $w(t, x)$, of (4) can be expanded in terms of the eigenvalues and eigenfunctions:

$$w(t,x) = \sum^N_{k=1} \cosh(t\sqrt{-E_k}) a_k v_k(x) + \sum^N_{k=1} (\sinh (t\sqrt{-E_k})/\sqrt{-E_k}) b_k v_k(x)$$

$$\quad + \sum^\infty_{j=0}\cos(t\lambda_j) c_j u_j(x) + \sum^\infty_{j=0}\lambda_j^{-1} \sin(t \lambda_j) d_j u_j(x)$$ (5)

where

$$\begin{split}a_k &= \int^L_{-L} w_0(x)\overline{v_k(x)} dx ... ...ad d_j = \int^L_{-L} w_1(x)\overline{u_j(x)} dx . \end{split}$$ (6)

This decomposition is the basis of harmonic analysis, signal processing, and many other things.