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Nonlinear models and bifurcation trees in quantum mechanics: a review of recent results

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In this talk we discuss some recent results I obtained for a class of nonlinear models in quantum mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrodinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the Stark–Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs; for a detailed treatment we refer to Sacchetti (Phys Rev E 95:062212, 2017, SIAM J Math Anal 50(6):5783–5810, 2018) where this model has been studied.

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Authors and Affiliations

  1. Department of Physics, Informatics and Mathematics, University of Modena e Reggio Emilia, Modena, Italy

    Andrea Sacchetti

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  1. Andrea Sacchetti
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Correspondence to Andrea Sacchetti.

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Communicated by Salvatore Rionero.

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This paper is partially supported by GNFM-INdAM. This paper is dedicated to Professor Tommaso Antonio Ruggeri on occasion of his 70th birthday and it is an extract of the talk given by the author at the Conference Wascom 2017 (Bologna) in honor to Professor Tommaso Antonio Ruggeri.

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Sacchetti, A. Nonlinear models and bifurcation trees in quantum mechanics: a review of recent results. Ricerche mat 68, 883–898 (2019). https://doi.org/10.1007/s11587-019-00443-1

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