• Corpus ID: 21732847

Darboux transformation for the discrete Schr\"odinger equation

@article{Aktosun2018DarbouxTF,
  title={Darboux transformation for the discrete Schr\"odinger equation},
  author={Tuncay Aktosun and Abdon E. Choque-Rivero and Vassilis G. Papanicolaou},
  journal={arXiv: Mathematical Physics},
  year={2018}
}
The discrete Schr\"odinger equation on a half-line lattice with the Dirichlet boundary condition is considered when the potential is real valued, is summable, and has a finite first moment. The Darboux transformation formulas are derived from first principles showing how the potential and the wavefunction change when a bound state is added or removed from the discrete spectrum of the corresponding Schr\"odinger operator without changing the continuous spectrum. This is done by explicitly… 
View PDF on arXiv
3 Citations
Methods Citations
1

3 Citations

BOUND STATES OF THE DISCRETE SCHRÖDINGER EQUATION WITH COMPACTLY SUPPORTED POTENTIALS

The discrete Schrödinger operator is considered on the half-line lattice n ∈ {1, 2, 3, . . . } with the Dirichlet boundary condition at n = 0. It is assumed that the potential belongs to class Ab,…

On the bound states of the discrete Schr\"odinger equation with compactly supported potentials

The discrete Schrodinger operator with the Dirichlet boundary condition is considered on the half-line lattice $n\in \{1,2,3,\dots\}.$ It is assumed that the potential belongs to class $\mathcal…

Inverse problems for Jacobi operators with finitely supported perturbations

We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse…

References

SHOWING 1-10 OF 21 REFERENCES

Inverse problem with transmission eigenvalues for the discrete Schrödinger equation

The discrete Schrodinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery…

Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey-Wilson polynomials

Consequences of the Darboux transformations for the finite-difference Schrodinger equations and three-term recurrence relations for orthogonal polyno- mials are considered. An equivalence of the…

Inverse problems for selfadjoint Schrödinger operators on the half line with compactly supported potentials

For a selfadjoint Schrodinger operator on the half line with a real-valued, integrable, and compactly supported potential, it is investigated whether the boundary parameter at the origin and the…

Darboux Transformations and Solitons

In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partial…

BOUND STATES OF THE DISCRETE SCHRÖDINGER EQUATION WITH COMPACTLY SUPPORTED POTENTIALS

The discrete Schrödinger operator is considered on the half-line lattice n ∈ {1, 2, 3, . . . } with the Dirichlet boundary condition at n = 0. It is assumed that the potential belongs to class Ab,…

Darboux transformations for a system of coupled discrete Schrödinger equations

Darboux transformations and a factorization procedure are presented for a system of coupled finite-difference Schrödinger equations. The conformity between generalized Darboux transformations and the…

Half-line Schrödinger operators with no bound states

acting in L2([0, c~)) with the boundary condition r For convenience, we require that the potential, V, be uniformly locally square integrable. We write l~(L 2) for the Banach space of such functions.…

Bound states of discrete Schrödinger operators with super-critical inverse square potentials

We consider discrete one-dimensional Schrodinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete…

The Inverse Problem in the Quantum Theory of Scattering...

This report is a translation from the Russian of a survey article by L. D. Faddeyev, which appeared in Uspekhi Matem. Nauk., 14, 57 (1959). Our own interest in this article lies in its relevance to…

Sum rules for Jacobi matrices and their applications to spectral theory

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms.…