In order to solve the inverse spectral problem:
$$ -y''(x)+q(x)y(x)= \lambda _{n}y(x) $$
If we want to obtain $ q(x) $ what we should need about the spectrum?
a) The eigenvalue staircase $ N(\lambda)= \sum_{n=0}^{\infty}H( \lambda - \lambda _{n}) $
b) The fucntion $ E(t)= \sum_{n=0}^{\infty}e^{-t \lambda _{n}} $
c) The function $ C(t)= \sum_{n=0}^{\infty}cos(t \sqrt{\lambda}_{n}) $
d) The function $ \Delta (\lambda)= \prod_{n=0}^{\infty}(1- \frac{\lambda}{\lambda _{n}}) $
If I know any of these functions, could i recover the potential $ q(x) $ ?
Boundary conditions on the half line $ [0,\infty ) $ of the form $ y(0)+by'(0)=0$ or similar.