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A characterization of the Hamiltonian. (English) Zbl 1138.81410

Summary: The classical Hamiltonian \(\frac 12(-\frac{\text d^2}{\text dx^2}+x^2)\) of the very classical quantum harmonic oscillator, which is regarded as a germ of the most of what comes about in quantum mechanics, can be sublimed to an abstract operator in a separable Hilbert space. Having this done one may ask for a condition which would allow it to be identified among operators of a suitable class. This class is that corresponding to three diagonal matrices and the property which makes the action successful is a kind of diagonal invariance (up to change of basis) within the class in question.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N50 Applications of operator theory in the physical sciences
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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References:

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