Quantum scattering

\(\varphi_{xx}+(\lambda-u)\varphi=0\)
looking for bounded functions on the whole line
- If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case the spectrum does no longer consist of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in quantum mechanics, since the one-dimensional Schrödinger equation is a special case of a S–L equation
discrete spectrum \(\lambda<0\)
continuous spectrum \(\lambda>0\)

\(E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\)

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