Equality Between the Spectrum of PT-Symmetric Shrödinger Operators and Their Adjoint

In classical spectral theory, self-adjoint differential operators satisfy the relation σ(L)=σ(L∗). However, this is not necessarily true for non-self-adjoint operators. In this study, we show that a similar spectral equality holds for non-self-adjoint with periodic PT-symmetric complex valued potential. These operators are often described in the literature as having self-adjoint-like spectral characteristics; however, here, we show that the reality of the spectrum depends on the isospectral relationship between the operator and its adjoint. Moreover, we prove that the adjoint operator L∗ inherits PT-symmetry and analyze its spectral properties under quasi-periodic boundary conditions, extending prior studies from the original operator to its adjoint.