The eigenvalue of Schrodinger Equation

Question

I'm a student majoring in Mathematics.But now I'm studying the KDV equation which uses Schrodinger Equation. My question is that in time-independent Schrodinger Equation$$\psi_{xx}-(u-\lambda)\psi=0$$,and when $x\to|\infty|,u\to0,u_x\to0$,there are two questions that I have:

• Why are all the eigenvalues real?

• Why are there discrete eigenvalues for $\lambda<0$ and continuous eigenvalues for $\lambda>0$?

Answer 1

The eigenvalues are real because the Schroedinger operator $$u-\partial_{xx}$$ is self-adjoint.

For the second question, $\lambda<0$ corresponds to bound states for which $\psi \to 0$ as $x \to 0$ which are usually discrete, whereas $\lambda>0$ corresponds to scattering states. This is not rigorous. Perhaps someone can fill in a better answer.