Real Spectra of PT-symmetric Operators and Perturbation Theory

F. Cannata, M. V. Ioffe, Rajkumar Roychoudhury +1 more

Emanuela Caliceti, S. Graffi, Johannes Sjöstrand

A criterion is formulated for existence and another for the non-existence of complex eigenvalues for a class of non-self-adjoint operators in Hilbert space invariant under a particular discrete symmetry. Applications to the PT -symmetric Schrödinger operators are discussed.

Miloslav Znojil

PT symmetric quantum mechanics admits non-Hermitian Hamiltonians with real spectra but, as long as the evolution is pseudo-unitary, the probabilistic interpretation of wave functions is lost. We offer an effective Hermitization of the theory using the Feshbach-type elimination of ghosts. The energy-dependence of the resulting Hermitian effective Hamiltonians Heff is a nonlinearity which, in a strong-coupling regime, explains the spontaneous PT symmetry breaking and the simultaneous loss of the Hermiticity of Heff while the pseudo-unitary evolution law persists. PACS 03.65.Bz, 03.65.Ca 03.65.Fd The concept of extended, non-Hermitian quantum mechanics [1] with real spectra and PT symmetric Hamiltonians (P means parity and T denotes time reversal) grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [2] have discovered that a

Bryce Seligman DeWitt

Vyacheslav S. Polikanov

Emanuela Caliceti, S. Graffi

A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the $PT$-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.