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Construction of reflectionless potentials with infinite discrete spectrum. (English) Zbl 0857.34073
Theor. Math. Phys. 100, No. 2, 970-984 (1994) and Teor. Mat. Fiz. 100, No. 2, 230-247 (1994).
Summary: We investigate the one-dimensional Schrödinger operator. The condition that the potential be self-similar under Darboux transformations leads to transparent potentials with infinitely many eigenvalues.

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81U30 Dispersion theory, dispersion relations arising in quantum theory
Full Text: DOI
[1] A. Shabat, ?The infinite-dimensional dressing dynamical system,?Inverse Problems,8, 303-308 (1992). · Zbl 0762.35098
[2] F. Calogero and A. Degasperis,Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, Vol. 1, North Holland Publ. Co., Amsterdam (1982). · Zbl 0501.35072
[3] A. B. Shabat Potentials with vanishing reflection coefficient.Dynamics of Continuous Matter,5, 130-145 (1970).
[4] V. A. Marchenko, ?The Cauchy problem for the KdV equation with non-decreasing initial data,? In:What is integrability? (V. A. Zakharov, ed.), Springer-Verlag, Berlin (1991), pp. 273-318. · Zbl 0810.34090
[5] A. P. Veselov and A. B. Shabat The dressing dynamical system and spectral theory of Schrödinger operator.Funct. Anal. Appl.,27 (1993).
[6] A. Ramani, B. Grammaticos, and K. M. Tamizhmani, ?Painlevé analysis and singularity confinement: the ultimate conjecture,?J. Phys. A.: Math. Gen.,26, L53-L58 (1993). · Zbl 0773.35076
[7] G. Strang, ?Wavelets and dilation equations: a brief introduction.,?SIAM Review,31, 614-637 (1989). · Zbl 0683.42030
[8] A. Degasperis and A. Shabat, ?Construction of reflectionless potentials with infinitely many discrete eigenvalues,? In:Application of Analytic and Geometric Methods to Nonlinear Differential Equations, P. A. Clarkson (ed.),Proceedings of NATO Workshop, Exeter July 1992, Kluwer Academic Publ. (1993).
[9] T. Kato and J. B. McLeod, ?The functional-differential equationy?(x)=ay(?x)+by(x),?Bull Amer. Math. Soc.,77, 891-937 (1971). · Zbl 0236.34064
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