Geodesic
Spectral Theory of the Nonstationary Schr\"odinger Equation with a~Bidimensionally Perturbed One-Dimensional Potential
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 10-53

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We derive and describe in detail the extension of the inverse scattering transform method to the case of linear spectral problems with potentials that do not decay in some space directions. Our presentation is based on the extended resolvent approach. As a basic example, we consider the nonstationary Schrödinger equation with a potential that is a perturbation of a generic one-dimensional potential by means of a decaying function of two variables. We give the corresponding modifications of the Jost solutions and the spectral data and derive their properties and characterization equations. 
@article{TM_2005_251_a2,
     author = {M. Boiti and F. Pempinelli and A. K. Pogrebkov and B. Prinari},
     title = {Spectral {Theory} of the {Nonstationary} {Schr\"odinger} {Equation} with {a~Bidimensionally} {Perturbed} {One-Dimensional} {Potential}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {10--53},
     publisher = {mathdoc},
     volume = {251},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2005_251_a2/}
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M. Boiti; F. Pempinelli; A. K. Pogrebkov; B. Prinari. Spectral Theory of the Nonstationary Schr\"odinger Equation with a~Bidimensionally Perturbed One-Dimensional Potential. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 10-53. http://geodesic.mathdoc.fr/item/TM_2005_251_a2/