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## Main Question or Discussion Point

I have taken QM , and i find it very interesting but my question is , we have the Hamiltonian eigenvalue problem

[tex] i \partial _t \Psi (x,t) = \lambda _n \Psi (x,t)=(p^2+V(x))\Psi(x,t) [/tex]

of course in general, we know the potential V(x) but my question is the inverse, if we knew how the spectrum is or for example the (approximate) value of [tex] \sum_n e^{it \lambda _n} [/tex]

for example for Harmonic oscillator since all energies are lineal we know that

[tex] \sum_n e^{it \lambda _n}=(1-exp(it/2))^{-1} [/tex] from this could we deduce that V(x) is a quadratic potential x*x

[tex] i \partial _t \Psi (x,t) = \lambda _n \Psi (x,t)=(p^2+V(x))\Psi(x,t) [/tex]

of course in general, we know the potential V(x) but my question is the inverse, if we knew how the spectrum is or for example the (approximate) value of [tex] \sum_n e^{it \lambda _n} [/tex]

for example for Harmonic oscillator since all energies are lineal we know that

[tex] \sum_n e^{it \lambda _n}=(1-exp(it/2))^{-1} [/tex] from this could we deduce that V(x) is a quadratic potential x*x