Search results for "Inverse scattering"
showing 10 items of 34 documents
Complex, energy-independent, local potential reproducing an absorptive phase shift and a bound state
1994
The triton binding energy, and the partly real and partly complex neutron-deuteron doublet channel elastic scattering phase shifts, calculated by means of the exact three-body theory, are used as input in the fixed-[ital l] inverse scattering theory of Marchenko. The local potentials obtained hereby are independent of energy, and complex. Their strong imaginary part reflects the strong absorption in the doublet channel arising from the opening of the deuteron breakup channel. For total orbital angular momentum [ital l] different from zero the potentials are unique, reproducing the input phase shift in the whole energy region. For [ital l]=0 where there exists, in addition, a bound state we …
Fixed angle inverse scattering in the presence of a Riemannian metric
2020
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [23,24] from the Euclidean case to certain Riemannian metrics.
Three dimensional reductions of four-dimensional quasilinear systems
2017
In this paper we show that integrable four dimensional linearly degenerate equations of second order possess infinitely many three dimensional hydrodynamic reductions. Furthermore, they are equipped infinitely many conservation laws and higher commuting flows. We show that the dispersionless limits of nonlocal KdV and nonlocal NLS equations (the so-called Breaking Soliton equations introduced by O.I. Bogoyavlenski) are one and two component reductions (respectively) of one of these four dimensional linearly degenerate equations.
Unitarity of the SoV Transform for the Toda Chain
2014
The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler "separated" form. In order to realise such a program, one should construct the map explicitly and then show that it is unitary. In the present paper, we develop a technique which allows one to prove the unitarity of this map in the case of the quantum Toda chain. Our proof solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it should be readily transposable to other quantum integrable models solvable by the …
Quantum and Classical Statistical Mechanics of the Integrable Models in 1 + 1 Dimensions
1990
In a short but remarkable paper Yang and Yang [1] showed that the free energy of a model system consisting of N bosons on a line with repulsive δ-function interactions was given by a set of coupled integral equations. The Yangs’ chosen model is in fact the repulsive version of the quantum Nonlinear Schrodinger (NLS) model. We have shown that with appropriate extensions and different dispersion relations and phase shifts similar formulae apply to ‘all’ of the integrable models quantum or classical. These models include the sine-Gordon (s-G) and sinh-Gordon (sinh-G) models, the two NLS models (attractive and repulsive), the Landau-Lifshitz (L-L’) model which includes all four previous models,…
Yang-Baxter equation and reflection equations in integrable models
1996
The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and on the half-line using the Zamolodchikov-Faddeev algebra. Due to the vertex-IRF model correspondence the face model analogue of the ZF-algebra and the IRF reflection equation are written down as well as the $Z_2$-graded and colored algebra forms of the YBE and RE.
A Look at Some Remarkable Mathematical Techniques
1996
The nonlinear equations that we have encountered in the previous chapters can be solved by using mathematical techniques such as the powerful inverse scattering transform (IST) (Gardner et al. 1967) and the remarkable Hirota method (Hirota 1971). Specifically, in addition to the one-soliton solutions, explicit multisoliton solutions representing the interaction of any number of solitons can be constructed. Moreover, in several cases a precise prediction, closely related to experiments, can be made by the IST of the nonlinear response of the physical system, that is, of the number of solitons that can emerge from a finite initial disturbance (Zakharov, 1980. Ablowitz and Segur 1981; Calogero…
The fixed angle scattering problem with a first order perturbation
2021
We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by $2n$ measurements up to a natural gauge. We also show that one can recover the full first order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and M. Salo to Hamiltonians with first order perturbations, and it is based on wave equation methods and Carleman estimates.
The su(1,1) Tavis-Cummings model
1998
A generic su(1,1) Tavis-Cummings model is solved both by the quantum inverse method and within a conventional quantum-mechanical approach. Examples of corresponding quantum dynamics including squeezing properties of the su(1,1) Perelomov coherent states for the multiatom case are given.
Noncompact Topological Quantum Groups
1995
A star-product construction of quantum semisimple real Lie groups is performed for the noncompact case.