September 2014
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Month September 2014

Metric operators, generalized hermiticity and lattices of Hilbert spaces

Jean-Pierre Antoine, Camillo Trapani

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.
Mathematical Physics (math-ph)

Non-Hermitian oscillators with \(T_d\) symmetry

Paolo Amore, Francisco M. Fernández, Javier Garcia

We analyse some PT-symmetric oscillators with \(T_d\) symmetry that depend on a potential parameter \(g\). We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of \(g\). Pairs of eigenvalues coalesce at exceptional points \(g_c\); their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of g as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues for sufficiently small nonzero values of \(g\).
Quantum Physics (quant-ph)

Spinor structure and internal symmetries

V. V. Varlamov

Space-time and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space time discrete symmetries P, T and their combination PT are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation C is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation C allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Quotient representations of the Lorentz group and their possible relations with P- and CP-violations are considered. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincar\’{e} group and SU(3)-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of SU(3)). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of SU(3)-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin allows one to take a new look at the problem of mass spectrum of elementary particles.
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th)

Exceptional points, phase rigidity and nonlinear Schrodinger equation

Hichem Eleuch, Ingrid Rotter

The natural environment of a localized quantum system is the continuum of scattering wavefunctions into which the system is embedded. It can be changed by external fields, however never be deleted. The control of the system’s properties by varying a certain parameter provides us information on the system. It is, in many cases, counterintuitive and points to the same phenomena in different systems in spite of the specific differences between them. In our paper, we use a schematic model in order to simulate the main features of small open quantum systems. At low level density, the system is described well by standard Hermitian quantum physics while fundamental differences appear at high level density due to the non-Hermiticity of the Hamiltonian which cannot be neglected under this condition. The influence of exceptional points, the phase rigidity of the wavefunctions and the nonlinearities in the equations are discussed by means of different numerical and (when possible) analytical results. The transition from a closed system at low level density to an open one at high level density occurs smoothly.
Quantum Physics (quant-ph)

Asymmetric transmission through a flux-controlled non-Hermitian scattering center

X. Q. Li, X. Z. Zhang, G. Zhang, Z. Song

We study the possibility of asymmetric transmission induced by a non-Hermitian scattering center embedded in a one-dimensional waveguide, motivated by the aim of realizing quantum diode in a non-Hermitian system. It is shown that a PT symmetric non-Hermitian scattering center always has symmetric transmission although the dynamics within the isolated center can be unidirectional, especially at its exceptional point. We propose a concrete scheme based on a flux-controlled non-Hermitian scattering center, which comprises a non-Hermitian triangular ring threaded by an Aharonov-Bohm flux. The analytical solution shows that such a complex scattering center acts as a diode at the resonant energy level of the spectral singularity, exhibiting perfect unidirectionality of the transmission. The connections between the phenomena of the asymmetric transmission and reflectionless absorption are also discussed.
Quantum Physics (quant-ph)

Chiral interactions of light induced by low-dimensional dynamics in complex potentials

Sunkyu Yu, Hyun Sung Park, Xianji Piao, Bumki Min, Namkyoo Park

Chirality is a universal feature in nature, as observed in fermion interactions and DNA helicity. Much attention has been given to the chiral interactions of light, not only regarding its physical interpretation but also focusing on intriguing phenomena in excitation, absorption, generation, and refraction. Although recent progress in metamaterials and 3-dimensional writing technology has spurred artificial enhancements of optical chirality, most approaches are founded on the same principle of the mixing of electric and magnetic responses. However, due to the orthogonal form of electric and magnetic fields, intricate designs are commonly required for mixing. Here, we propose an alternative route to optical chirality, exploiting the nonmagnetic mixing of amplifying and decaying electric modes based on non-Hermitian theory. We show that a 1-dimensional helical eigenmode can exist singularly in a complex anisotropic material, in sharp contrast to the 2-dimensional eigenspaces employed in previous approaches. We demonstrate that exceptional interactions between propagating chiral waves result from this low-dimensionality, for example, one-way reflectionless chiral conversions and chirality reversal, each occurring for circular and linear polarization. Our proposal and experimental verification with complex polar meta-molecules not only provide a significant step for low-dimensional chirality, but also enable the dynamics of optical spin black hole.
Optics (physics.optics)