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Register a new userA global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy
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Added by
Evgeny Lakshtanov L
Publication date
2015
fields
Physics
and research's language is
English
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We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or gener
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We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrodinger equations and discrete nonlinear Schrodinger equations but it seems to be the first time to be applied to QW.
A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $zlog z$ type singularity on the jump contour is considered. The jump function and the jump contour are assumed to depend on a vector of external parameters $vecbeta$. We prove that if the RHP has a solution at some value $vecbeta_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $vecbeta_0$ and is smooth in $vecbeta$. This result is applied to the case of semiclassical focusing NLS.
The problem of building supersymmetry in the quantum mechanics of two Coulombian centers of force is analyzed. It is shown that there are essentially two ways of proceeding. The spectral problems of the SUSY (scalar) Hamiltonians are quite similar and become tantamount to solving entangled families of Razavy and Whittaker-Hill equations in the first approach. When the two centers have the same strength, the Whittaker-Hill equations reduce to Mathieu equations. In the second approach, the spectral problems are much more difficult to solve but one can still find the zero-energy ground states.
The Theory of (2+1) Systems based on 2D Schrodinger Operator was started by S.Manakov, B.Dubrovin, I.Krichever and S.Novikov in 1976. The Analog of Lax Pairs introduced by Manakov, has a form $L_t=[L,H]-fL$ (The $L,H,f$-triples) where $L=partial_xpartial_y+Gpartial_y+S$ and $H,f$-some linear PDEs. Their Algebro-Geometric Solutions and therefore the full higher order hierarchies were constructed by B.Dubrovin, I.Krichever and S.Novikov. The Theory of 2D Inverse Spectral Problems for the Elliptic Operator $L$ with $x,y$ replaced by $z,bar{z}$, was started by B.Dubrovin, I.Krichever and S.Novikov: The Inverse Spectral Problem Data are taken from the complex Fermi-Curve consisting of all Bloch-Floquet Eigenfunctions $Lpsi=const$. Many interesting systems were found later. However, specific properties of the very first system, offered by Manakov for the verification of new method only, were not studied more than 10 years until B.Konopelchenko found in 1988 analogs of Backund Transformations for it. He pointed out on the Burgers-Type Reduction. Indeed, the present authors quite recently found very interesting extensions, reductions and applications of that system both in the theory of nonlinear evolution systems (The Self-Adjoint and 2D Burgers Hierarhies were invented, and corresponding reductions of Inverse Problem Data found) and in the Spectral Theory of Important Physical Operators (The Purely Magnetic 2D Pauli Operators). We call this system GKMMN by the names of authors who studied it.
The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R simeq mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $mathbb{D}_h$ symmetries are presented.
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