Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
نویسنده
چکیده
We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent. 1. H is pseudo-Hermitian; 2. The spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; 3. H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT -symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.
منابع مشابه
Erratum: Pseudo-hermiticity for a Class of Nondiagonalizable Hamiltonians
Theorem 2: Let H be as in Theorem 1 of Ref. [2]. Then H is pseudo-Hermitian if and only if it is Hermitian with respect to an inner product 〈〈 , 〉〉 that supports a positivesemidefinite basis [3] including the eigenvectors of H . In particular, for every eigenvector ψ of H , 〈〈ψ|ψ〉〉 ≥ 0; if the corresponding eigenvalue is real and nondefective (algebraic and geometric multiplicities are equal), ...
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