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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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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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