# Reconstruction of potential V(x)

1. Aug 21, 2009

### zetafunction

the idea is, let us suppose we know the trace

$$Tr(h(\hat H ))= \sum_{n=0}^{\infty}h(E_n )$$

here 'h' can be a real or complex exponential of the form exp(-ax) and 'H' is the usual Hamiltonian operator

$$H=p^2 + V(x)$$

what information about the spectrum of Hamiltonian would i need in order to obtain V(x) ??

for example: for the Harmonic Oscillator in Planck's unit so h=1 and w=1 i have that

$$\sum _{n=0}^{\infty} exp(-s(n+1/2))= \frac{exp(-s/2)}{1-exp(-s)}$$

then from the expression above could i conclude that potential goes like $$V(x)=ax^{2}$$