Operators continuously dependent on parameters

Discussion in 'Undergraduate Math' started by Mark, Jul 3, 2004.

  1. Mark

    Mark Guest

    What is the exact definition for that notion?

    In classical mechanics, the notion of a continuously parametrized
    operator may be viewed (?) simply as the classical Hamiltonian
    containing some parameter(s) such that the resulting Hamiltonian is
    continuously dependent on it/them.

    In quantum mechanics, I could probably require |x0-x|<d(e) =>
    |A(x0)-A(x)|<e, where the right norm is an operator norm.

    Does that somehow fit together? If yes, how? Almost all of my
    textbooks use continuously depending operators but fail to explain
    appropriate details. I took a course on functional analysis but that
    did not cover that aspect. My (mathematically oriented) books do not
    even care to mention it.

    Additionally, I read in B.L. van der Waerden's book on group theory in
    quantum mechanics (1932), that in the case of the quantum mechanical
    energy-eigenvalue problem, the eigenvalues are continuously and
    differentiably dependent on parameters entering the Hamiltonian
    continuously. He also stated that he is not sure if that has been
    proved as true. Has it?

    Best Regards,
    Mark, Jul 3, 2004
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