Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians

TL;DR

This paper characterizes pseudo-Hermitian Hamiltonians, including nondiagonalizable ones, showing their spectral properties and relation to Hermitian operators under positive-semidefinite inner products, with implications for PT-symmetry and quantum cosmology.

Contribution

It provides two theorems characterizing pseudo-Hermitian Hamiltonians with discrete spectra, including nondiagonalizable cases, and links their spectral features to Hermiticity under positive-semidefinite inner products.

Findings

Pseudo-Hermitian Hamiltonians have spectra with real or complex-conjugate eigenvalues.

Diagonal blocks for complex-conjugate eigenvalues have equal dimensions.

Pseudo-Hermitian operators are Hermitian with respect to a positive-semidefinite inner product.

Abstract

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.