Tanaya Bhattacharyya

Bound and Scattering States of Extended Calogero Model with an Additional PT Invariant Interaction

Here we discuss two many-particle quantum systems, which are obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the Calogero model with and without confining potential. It is shown that the energy eigenvalues are real for both of these quantum systems. For the case of extended Calogero model with confining potential, we obtain discrete bound states satisfying generalised exclusion statistics. On the other hand, the extended Calogero model without confining term gives rise to scattering states with continuous spectrum. The scattering phase shift for this case is determined through the exchange statistics parameter. We find that, unlike the case of usual Calogero model, the exclusion and exchange statistics parameters differ from each other in the presence of PT invariant interaction.

PHASE SHIFT ANALYSIS OF PT-SYMMETRIC NONHERMITIAN EXTENSION OF AN-1 CALOGERO MODEL WITHOUT CONFINING INTERACTION

We discuss a manyparticle quantum system, which is obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the AN-1 rational Calogero model without confining potential. This model gives rise to scattering states with continuous real spectrum. The scattering phase shift is determined through the exchange statistics parameter. We find that, unlike the case of the usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.

Algebraic Bethe ansatz for a quantum integrable derivative nonlinear Schrödinger model

Abstract We find that the quantum monodromy matrix associated with a derivative nonlinear Schrodinger (DNLS) model exhibits U (2) or U (1,1) symmetry depending on the sign of the related coupling constant. By using a variant of quantum inverse scattering method which is directly applicable to field theoretical models, we derive all possible commutation relations among the operator valued elements of such monodromy matrix. Thus, we obtain the commutation relation between creation and annihilation operators of quasi-particles associated with DNLS model and find out the S -matrix for two-body scattering. We also observe that, for some special values of the coupling constant, there exists an upper bound on the number of quasi-particles which can form a soliton state for the quantum DNLS model.

Bound and anti-bound soliton states for a quantum integrable derivative nonlinear Schrödinger model

Abstract We find that localized quantum N-body soliton states exist for a derivative nonlinear Schrodinger (DNLS) model within an extended range of coupling constant (ξq) given by 0 q | 1 ℏ tan ( π N−1 ) . We also observe that soliton states with both positive and negative momentum can appear for a fixed value of ξq. Thus the chirality property of classical DNLS solitons is not preserved at the quantum level. Furthermore, it is found that the solitons with positive (negative) chirality have positive (negative) binding energy.

Jost solutions and quantum conserved quantities of an integrable derivative nonlinear Schrödinger model

We study differential and integral relations for the quantum Jost solutions associated with an integrable derivative nonlinear Schrodinger (DNLS) model. By using commutation relations between such Jost solutions and the basic field operators of DNLS model, we explicitly construct first few quantum conserved quantities of this system including its Hamiltonian. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This modified coupling constant plays a crucial role in our comparison between the results of algebraic and coordinate Bethe ansatz for the case of DNLS model. We also find out the range of modified coupling constant for which the quantum N-soliton state of DNLS model has a positive binding energy.

Quantum integrability of bosonic massive Thirring model in continuum

By using a variant of the quantum inverse scattering method, commutation relations between all elements of the quantum monodromy matrix of the bosonic massive Thirring (BMT) model are obtained. Using those relations, the quantum integrability of BMT model is established and the S-matrix of two-body scattering between the corresponding quasiparticles has been obtained. It is observed that for some special values of the coupling constant, there exists an upper bound on the number of quasiparticles that can form a quantum-soliton state of the BMT model. We also calculate the binding energy for a N-soliton state of the quantum BMT model.

Quantum Bound States for a Derivative Nonlinear Schrödinger Model and Number Theory

A derivative nonlinear Schrodinger model is shown to support localized N-body bound states for several ranges (called bands) of the coupling constant η. The ranges of η within each band can be completely determined using number theoretic concepts such as Farey sequences and continued fractions. For N≥3, the N-body bound states can have both positive and negative momenta. For η>0, bound states with positive momentum have positive binding energy, while states with negative momentum have negative binding energy.

Clusters of bound particles in the derivative δ-function Bose gas

Abstract In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative δ -function Bose gas in one dimension. It is shown that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. We also establish a connection between these special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles associated with the derivative δ -function Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant.

FERMIONIC DUAL OF ONE-DIMENSIONAL BOSONIC PARTICLES WITH DERIVATIVE DELTA FUNCTION POTENTIAL

We investigate the boson–fermion duality relation for the case of quantum integrable derivative δ-function Bose gas. In particular, we find a dual fermionic system with nonvanishing zero-range interaction for the simplest case of two bosonic particles with derivative δ-function interaction. The coupling constant of this dual fermionic system becomes inversely proportional to the product of the coupling constant of its bosonic counterpart and the center-of-mass momentum of the corresponding eigenfunction.

Multi-band structure of a coupling constant for quantum bound states of a generalized nonlinear Schrödinger model

By using the method of coordinate Bethe ansatz, we study N-body bound states of a generalized nonlinear Schrodinger model having two real coupling constants c and \eta. It is found that such bound states exist for all possible values of c and within several nonoverlapping ranges (called bands) of \eta. The ranges of \eta within each band can be determined completely using Farey sequences in number theory. We observe that N-body bound states appearing within each band can have both positive and negative values of the momentum and binding energy.