Reconstruction of potential V(x)

[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 $$V(x)=ax^{2}$$

 

[azntoon]

??The answer is no. In general, it is not possible to determine the potential V(x) from the trace Tr(h(\hat H )). The trace includes information about the spectrum of the Hamiltonian, but does not contain enough information to uniquely determine the potential. To do this, one would need additional information, such as the eigenfunctions of the Hamiltonian or knowledge of the exact form of the Hamiltonian.