Level crossing and the space of operators commuting with the Hamiltonian

T.P. Valkering, W.J. Caspers

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    Abstract

    The space of n-dimensional hermitean matrices that commute with a given hermitean matrix A + hB, h being a real parameter, is discussed. In particular a basis in this space is constructed consisting of polynomials in h of the lowest possible total degree. The sum of the degrees of the elements of this minimal basis equals 1/2n((n-1) - q , q being the number of linearly independent linear relations between the symmetrized products of A and B of order 0,…,n − 1. These linear relations determine the values of h for which crossing occurs, the total number of crossings for each value, and in some cases the order of the different crossings. A discussion of the noncrossing rule concludes this paper.
    Original languageEnglish
    Pages (from-to)113-138
    JournalPhysica
    Volume63
    Issue number1
    DOIs
    Publication statusPublished - 1973

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