pseudo-Hermitian Hamiltonians Research Articles

Script PScript T-symmetric regularizations in supersymmetric quantum mechanics

Within supersymmetric quantum mechanics the necessary regularization of the poles of the superpotentials on the real line of coordinates x may be most easily mediated by a small constant shift of this axis into the complex plane. Detailed attention is paid here to the resulting recipe which works, in effect, with non-Hermitian (a.k.a. PT-symmetric or pseudo-Hermitian) Hamiltonians. Besides an exhaustive discussion of the role of the complex spike in harmonic oscillator, we mention also some applications concerning the regularized versions of the Smorodinsky-Winternitz and Calogero models and of the relativistic Klein-Gordon equation.

Physics of Nonhermitian Degeneracies

A summary, with references and additional comments, of a talk delivered at the Second International Workshop on Pseudohermitian Hamiltonians in Quantum Physics (Prague, 14-16 June 2004). After explaining some general features of nonhermitian degeneracies (‘exceptional points’), several applications are outlined: to multiple reflections in a pile of plates, linewidths of unstable lasers, atom diffraction by light, and crystal optics.

Proceedings of the 2nd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2004

Proceedings of the 2nd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2004

Gaussian-Random Ensembles of Pseudo-Hermitian Matrices

Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or well-known number sequences, may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros (1/2±it n would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian.

Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the $P$-unitary evolution of a physical system are also given.

Proceedings of the 1st International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2003

Proceedings of the 1st International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2003

Pseudo-Hermitian Hamiltonians and Generalized Involutory Symmetries

We propose definitions of generalized parity (P), time-reversal (T) and charge-conjugation (C) operators such, that any diagonalizable pseudo-Hermitian Hamiltonian is invariant under the involutory symmetries C, TP, and CPT. We inquire about the peculiarities of such symmetries showing that these constitute the P-unitary and P-antiunitary symmetry generators. Moreover, we give a necessary and sufficient condition for diagonalizable pseudo-Hermitian Hamiltonians to admit P-pseudounitary and P-pseudoantiunitary symmetries.

Experiments in PT-Symmetric Quantum Mechanics

Extended quantum mechanics using non-Hermitian (pseudo-Hermitian) Hamiltonians H = H ‡ is briefly reviewed. A few related mathematical experiments concerning supersymmetric regularizations, solvable simulations and large-N expansion techniques are summarized. We suggest that they could initiate a deeper study of nonlocalized structures (quasi-particles) and/or of their unstable and many-particle generalizations. Using the Klein-Gordon example for illustration, we show how the PT symmetry of its Feshbach-Villars Hamiltonian H FV might clarify experimental aspects of relativistic quantum mechanics.

Pseudo-Hermiticity and generalized PT- and CPT-symmetries

We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.

Complex optical potentials and pseudo-Hermitian Hamiltonians

Recently some authors have broadened the scope of canonical quantum mechanics by replacing the conventional Hermiticity condition on the Hamiltonian by a weaker requirement through the introduction of the notion of pseudo-Hermiticity. In the present Letter we investigate eigenvalues, transmission and reflection from complex optical potentials enjoying the property of pseudo-Hermiticity.

Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians

We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.

Pseudo-Hermitian Hamiltonians, time-reversal invariance and Kramers degeneracy

A necessary and sufficient condition in order that a (diagonalizable) pseudohermitian operator admits an antilinear symmetry T such that T^{2}=-1 is proven. This result can be used as a quick test on the T-invariance properties of pseudohermitian Hamiltonians, and such test is indeed applied, as an example, to the Mashhoon-Papini Hamiltonian.

Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.