Parity-Time Symmetry

Parity-time symmetry (PT symmetry) is invariance of an operator under the combined action of a parity operation and an antilinear time-reversal operation (Bender 2005). In the standard finite-dimensional setting, let be a real orthogonal matrix satisfying , and let act by componentwise complex conjugation. A matrix has PT symmetry iff

(1)

where the bar conjugates each matrix entry and is the identity matrix. This is the condition that commute with .

If , then

(2)

Thus its eigenvalues are real or occur in complex conjugate pairs. If an eigenvector also satisfies with , its eigenvalue is real. PT symmetry alone does not imply that every eigenvalue is real.

For example, consider

			(3)
			(4)

where . These matrices satisfy the PT condition, but is a Hermitian matrix only when . Its eigenvalues are

(5)

The figure above shows the real parts and imaginary parts of these eigenvalues. They are real for and form a nonreal complex conjugate pair for . At , is a nonzero nilpotent matrix with just one independent eigenvector. These parameter values are exceptional points. This is a special case of the two-dimensional model of Bender et al. (2003). Even when its eigenvalues are real, a non-Hermitian PT-symmetric matrix need not generate a unitary matrix through the matrix exponential in the usual inner product.

For a Schrödinger operator with formal expression on the real line, the corresponding action is . The potential must satisfy

(6)

The operator domain and boundary conditions must also be invariant under this action. For example, with gives a PT-symmetric extension of the quartic oscillator (Trott 2006). Symmetry of the differential expression alone is not a proof of reality of the operator spectrum.