pseudoHermitian

A pseudo-Hermitian operator is a linear operator on a Hilbert space that satisfies a specific condition related to its adjoint. In quantum mechanics, Hermitian operators are crucial because their eigenvalues are real, representing observable quantities. However, sometimes non-Hermitian operators are studied, and the concept of pseudo-Hermiticity provides a way to relate them to Hermitian operators. A pseudo-Hermitian operator A is defined by the existence of an invertible, self-adjoint operator $\eta$ such that $A^\dagger = \eta A \eta^{-1}$. Here, $A^\dagger$ denotes the adjoint of operator A. The operator $\eta$ is sometimes called the metric operator.

The significance of pseudo-Hermitian operators lies in the fact that if an operator A is pseudo-Hermitian, then it can be transformed into a truly Hermitian operator by a similarity transformation. Specifically, the operator $H = \eta^{1/2} A \eta^{-1/2}$ is Hermitian. This means that while the original operator A might not have real eigenvalues, its transformed counterpart H does. This connection allows for the interpretation of physical observables in systems described by pseudo-Hermitian Hamiltonians. The development of pseudo-Hermitian quantum mechanics has opened avenues for studying systems with complex potentials or non-self-adjoint dynamics that still possess a consistent probabilistic interpretation.