The One - Dimensional Schrödinger Equation With a Quasiperiodic Potential

Dinaburg, E. I.; Sinaĭ, Ya. G.

doi:10.1142/9789814317344_0046

Cited by 23 publications

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“…Let B(x, t), C(x, t) be solutions of the partial differential equations (11), (52), (12), and (53). Then the function X(x, t) = I + Proof.…”

Section: Definition 52 the Collection Of Operators Spaces And An Omentioning

confidence: 99%

“…The first equation (13) is immediate, since the last term depends on t only. In order to prove (54), we can calculate using (11), (52), (12), and (53) that Now present the main theorem, because the fact that X(x) is invertible for a fixed x implies that its norm is bounded from below and a small perturbation of it is still invertible.…”

Section: Definition 52 the Collection Of Operators Spaces And An Omentioning

confidence: 99%

“…Fadeyev inverse scattering theory 15 where the dρ(μ) is supported on the positive real line and has a finite number of point-mass measures on the negative real line, 4. periodic potentials 37 correspond to discrete spectral measures with accumulation point at infinity, and 5. quasi-periodic potentials. 5,12 While analyzing the KdV equation (1) with an analytic initial potential, one can apply Taylor series techniques to try to solve the equation. It turns out that the corresponding combinatorial problem is extremely difficult.…”

Section: Introductionmentioning

confidence: 99%

“…Suppose that B(x), C(x), X(x) satisfy (11),(12),(13), respectively, and X(x)(D(A ζ )) ⊆ D(A) for all x ∈ R. Then the Lyapunov equation(7) holds for at least one point x ∈ R if and only if it holds for all x ∈ R. In the case the operator X(x) is invertible and B(x), C(x), X(x) are part of an invertible node, Eq. (9) holds for at least one point x ∈ R, if and only if it holds for all x ∈ R.…”

mentioning

confidence: 99%

“…real-valued functions r 11 n , b12 n , d 22 n . More precisely, the conditions at 0 must be chosen so that i n H 22n (0) ∈ R, H 12 n + H 21 n = 0.…”

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Solution of the Korteweg-de Vries equation on the line with analytic initial potential

Melnikov

2014

Journal of Mathematical Physics

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Articles you may be interested inGeneralized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transform Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methodsWe present a theory of Sturm-Liouville non-symmetric vessels, realizing an inverse scattering theory for the Sturm-Liouville operator with analytic potentials on the line. This construction is equivalent to the construction of a matrix spectral measure for the Sturm-Liouville operator, defined with an analytic potential on the line. Evolving such vessels we generate Korteweg-de Vries (KdV) vessels, realizing solutions of the KdV equation. As a consequence, we prove the theorem as follows: Suppose that q(x) is an analytic function on R. Then there exists a closed subset ⊆ R 2 and a KdV vessel, defined on . For each x ∈ R one can find TThe potential q(x) is realized by the vessel for t = 0. Since we also show that if q(x, t) is a solution of the KdV equation on R × [0, t 0 ), then there exists a vessel, realizing it, the theory of vessels becomes a universal tool to study this problem. Finally, we notice that the idea of the proof applies to a similar existence of a solution for evolutionary nonlinear Schrödinger and Boussinesq equations, since both of these equations possess vessel constructions. C 2014 AIP Publishing LLC.differentiable potential by Gelfand-Levitan theory, 19 but not always it can be used to solve the KdV equation (1). For this situation in case var[dρ] < ∞ a solution of KdV in the first quadrant (x, t ≥ 0) is presented in Ref. 28.Although there is a good scattering theory of the SL equation (2) on the line (see Ref. 27, Chap. 2, p. 128, and Ref. 26, Chap. 2), even with arbitrary singularities, 13 the solutions of KdV, corresponding to them are not developed. In fact, the classes of initial potentials, for which solutions of (1) were presented using inverse scattering are as follows:Suppose that dρ = dρ + − dρ − for two positive measures dρ + , dρ − , creating two Hilbert spaces of column-functions H + , H − as above. We make the following definition. Definition 2.2. A Krein space Kρ, associated to the finite measure dρ is

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“…Let B(x, t), C(x, t) be solutions of the partial differential equations (11), (52), (12), and (53). Then the function X(x, t) = I + Proof.…”

Section: Definition 52 the Collection Of Operators Spaces And An Omentioning

confidence: 99%

Section: Definition 52 the Collection Of Operators Spaces And An Omentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…real-valued functions r 11 n , b12 n , d 22 n . More precisely, the conditions at 0 must be chosen so that i n H 22n (0) ∈ R, H 12 n + H 21 n = 0.…”

mentioning

confidence: 99%

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