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To what extent can we derive the properties of a system given the existence of a hermitian operator with a particular spectrum?

For example, if we know that there exists a hermitian operator with eigenvalues equal to n+1/2 for all positive integers (in suitable units), can we conclude that there must exist hermitian operators x and p with eigenvalues spanning all real numbers with the appropriate commutation relations etc?

Of course, we usually work in the opposite direction; x and p (and their spectra/ commutation relations) are defined, then a Hamiltonian is defined as a function of them and we determine the spectrum of H from these assumptions.

To what extent can we ‘work backwards’ and derive that other operators with certain properties must exist, given the existence of a hermitian operator with a certain (positive definite/ discreet in this example) spectrum ?

Phil
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  • For quantum systems (from mechanics, not field theory), X, P and S are usually fundamental observables, since they are linked to physical observables and cannot be further "derived". But, OTOH, a quantum system is defined by its Hamiltonian plus boundary conditions for the states. So that H is a function of X, P, S. Angular momentum is also a derived observable, J = J(X, P, S). You cannot start, for example, with the ladder operators and derive the X, P, S, as they are not self-adjoint and are not measurable. You can start with H and N (the self-adjoint "number operator"), but, ctd. – DanielC Dec 18 '20 at 21:25
  • Ctd. As far I know, only with these, you cannot go backwards to define X, P, S and the CCRs. – DanielC Dec 18 '20 at 21:26
  • There is INVERSE SPECTRAL THEORY (I could not find a non-paywall source.) In your example they exist, but I don't think they are unique. Could it be a HO+1 plus a separate 1D system with eigenvalue 1/2? Or even and odd eigenvalues could both be HOs (with appropriate additive constants). – Keith McClary Dec 19 '20 at 04:25
  • Possible duplicates: https://physics.stackexchange.com/q/13480/2451 and links therein. – Qmechanic Dec 19 '20 at 05:18

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There is a famous question, which is can you hear the shape of a drum? This seems isomorphic to your question in the case of the nonrelativistic Schrodinger equation in two dimensions, so the answer would seem to be no. However, it seems possible that the answer is yes if you forbid some unphysical things like sharp corners in the boundary.

More trivially, you would definitely need some restrictions to make the problem interesting. For example, all nuclei have the same energy spectrum because their energy spectrum is continuous. To make their energy spectra different, you have to get rid of the center of mass motion, and then the spectra become discrete.

user283118
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