Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM?
Edit: I failed to make myself clear. In finite dimensions, there is a function of λ whose roots are the eigenvalues (or inverses of the eigenvalues) of a given operator U, namely, the characteristic polynomial det. Is there some way of regularising this determinant to do the same thing in infinite dimensions? In general? Or at least for unitary operators which describe the time evolution of a quantum mechanical system?
link to a related question What does a unitary transformation mean in the context of an evolution equation?
EDIT: Perhaps the question still is not clear. The question was, and still is, ¿if you regularise \det(I-\lambda U) as a complex valued function of \lambda, for U a unitary operator, will its zeroes be the values of \lambda such that I-\lambda U fails to be invertible? ¿Has a non-zero kernel? ¿Or does regularising the determinant lose touch with that property of the finite dimensional determinant?
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