Consider the Schrodinger Equation$$\psi_{xx}-(u-\lambda)\psi=0$$ with the condition

1.when $x\to|\infty|,u\to0,u_x\to0$

2.$\psi|_{x\to \infty}=0$ How to prove that spectrums are real?

3.$u(x,0)=f(x)$,$\Sigma_{i=0}^4\int_{-\infty}^{+\infty}|\frac{\partial^if}{\partial x^i}(x)|^2 dx<\infty$,$\int_{-\infty}^{+\infty}(1+|x|)|f(x)|<\infty$