Debdeep Sinha, Pijush K. Ghosh
A class of non-local non-linear Schrodinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a \(d+1\) dimensional generalization of it, admits all the symmetries of the \(d+1\) dimensional Schrodinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the “pseudo-width” of the wave-packet for the special case when the system admits a \(O(2,1)\) conformal symmetry.
http://arxiv.org/abs/1408.0954
Exactly Solvable and Integrable Systems (nlin.SI); High Energy Physics – Theory (hep-th)
Pijush K. Ghosh
Examples of non-hermitian quantum systems admitting topological insulator phase are presented in one, two and three space dimensions. All of these non-hermitian Hamiltonians have entirely real bulk eigenvalues and unitarity is maintained with the introduction of appropriate inner-products in the corresponding Hilbert spaces. The topological invariant characterizing a particular phase is shown to be identical for a non-hermitian Hamiltonian and its hermitian counterpart, to which it is related through a non-unitary similarity transformation. A classification scheme for topological insulator phases in pseudo-hermitian quantum systems is suggested.
http://arxiv.org/abs/1109.1697
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)
Pijush K. Ghosh
A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.
http://arxiv.org/abs/1012.0907
Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)