What Is Pt Symmetry?

The recently proposed complexification of Hamiltonians which keeps the spectra real (and is usually called PT symmetry) is re-interpreted as a certain natural linearalgebraic alternative to Hermiticity. PACS 03.65.Bz, 03.65.Ca 03.65.Fd March 15, 2008, whatis.tex file, J. Phys. A (lett.) Formalism of quantum mechanics is often illustrated by the harmonic oscillator H = p + q. Its eigenstates form a complete basis in Hilbert space. The Hamiltonian itself is Hermitian and commutes with the parity P. This means that we can split the basis in two subsets and write H = ∞ ∑ n=0 |n〉 ε n 〈n |+ ∞ ∑ m=0 |m〉 ε m 〈m | where P |n〉 = ± |n〉 and ε n = 4n + 2∓ 1. In the generalized, non-Hermitian quantum mechanics as proposed by Bender et al [1], the similar illustrative role is played by the imaginary cubic H(ω) = p + ω q + iq. Without a final rigorous mathematical proof, this Hamiltonian seems to generate the real, semi-bounded and discrete energies, at least in a certain range of energies and ω [2]. One may generalize the IC example and contemplate H = p + W (x) + iU(x) with any symmetric real well W (x) = W (−x) and with its purely imaginary antisymmetric complement iU(x) = −iU(−x). In place of the current Hermiticity of the Hamiltonians, the latter class of models satisfies a weaker condition which, presumably, implies the reality of the spectrum under certain circumstances. The condition is called PT symmetry and means just the commutativity [H(ω),PT ] = 0 (1) where T performs complex conjugation. The mathematical essence of the empirically discovered relation between the spectrum and symmetry (1) is not clarified yet [3]. In the present note we intend to contribute to the discussion by noticing that there exists a quite close relationship between the Hermiticity and PT symmetry conditions H = [ H ]+ andH = PT [ H ] PT ≡ [ H ]‡ , respectively. In our non-Hermitian model H we shall assume that the spectrum remains real. This means that the related Schrödinger equation H ψ(x) = E ψ(x) (2) is also satisfied by the functions PT ψ(x) = ψ(−x) and PT ψ(x) + ψ(x). In fact, the latter state has the spatially symmetric real part, the spatially antisymmetric imaginary part and is fully characterized by its positive PT parity, PT ψ(x) = +ψ(x). (3) 1 Such a behaviour (or, in effect, normalization) is particularly transparent and will be postulated everywhere in what follows. In a preparatory step, let us recollect the above-mentioned harmonic oscillators {|n〉} and/or any other orthonormalized basis with the property of the well defined parity. Its completeness enables us to expand the functions (3), |ψ〉 = ∞ ∑