nonHermitian

Non-Hermitian, occasionally written as nonHermitian, refers to operators that are not equal to their Hermitian adjoint. In quantum mechanics and linear algebra, a Hermitian (self-adjoint) operator H satisfies H† = H. Non-Hermitian operators do not satisfy this, and can have complex eigenvalues. They arise naturally in effective descriptions of open systems where gain, loss, or leakage is present.

Spectrally, non-Hermitian operators may have complex eigenvalues and non-orthogonal eigenvectors. Right eigenvectors satisfy H|ψ⟩ = λ|ψ⟩, left eigenvectors satisfy ⟨φ|H = λ⟨φ|, and the two sets form a biorthogonal basis. The operator may be diagonalizable or contain Jordan blocks, reflecting non-normality (H H† ≠ H† H).

Some non-Hermitian Hamiltonians with PT symmetry can exhibit real spectra in certain regimes (unbroken PT symmetry); beyond a threshold, eigenvalues become complex conjugate pairs. More generally, pseudo-Hermiticity and metric operators can render the spectrum real with respect to a modified inner product: there exists an η such that H† = η H η−1 with η positive-definite.

Applications include open quantum systems, optics and photonics with balanced gain and loss, lasers, and metamaterials. The presence of exceptional points—where eigenvalues and eigenvectors coalesce—gives rise to enhanced response and nontrivial dynamics, highlighting distinctive features of non-Hermitian physics.