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Spectral deformations of one-dimensional Schrödinger operators. (English) Zbl 0951.34061

Summary: The authors provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operators \(H= -d^2/dx^2+ V\) in \(L^2(\mathbb{R})\). The technique is connected to Dirichlet data, that is, the spectrum of the operator \(H^D\) on \(L^2((-\infty,x_0))\oplus L^2((x_0,\infty))\) with a Dirichlet boundary condition at \(x_0\). The transformation moves a single eigenvalue of \(H^D\) and perhaps flips which side of \(x_0\) the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such as \(V(x)\to \infty\) as \(|x|\to\infty\), where \(V\) is uniquely determined by the spectrum of \(H\) and the Dirichlet data, the result implies that the specific Dirichlet data allowed are determined only by the asymptotics as \(E\to\infty\).

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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[1] Baumgartner, B., Level comparison theorems, Ann. Phys., 168, 484-526 (1986) · Zbl 0596.35024 · doi:10.1016/0003-4916(86)90041-2
[2] Belokolos, E. D.; Bobenko, A. I.; Enol’skii, V. Z.; Its, A. R.; Matveev, V. B., Algebro-Geometric Approach to Nonlinear Integrable Equations (1994), Berlin: Springer, Berlin · Zbl 0809.35001
[3] Bolle, D.; Gesztesy, F.; Grosse, H.; Schweiger, W.; Simon, B., Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics, J. Math. Phys., 28, 1512-1525 (1987) · Zbl 0643.47005 · doi:10.1063/1.527508
[4] Buys, M.; Finkel, A., The inverse periodic problem for Hill’s equation with a finite-gap potential, J. Differential Equations, 55, 257-275 (1984) · Zbl 0508.34013 · doi:10.1016/0022-0396(84)90083-4
[5] Crum, M. M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford, 6, 2, 121-127 (1955) · Zbl 0065.31901 · doi:10.1093/qmath/6.1.121
[6] Darboux, G., Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. (Paris), 94, 1456-1459 (1882) · JFM 14.0264.01
[7] Deift, P. A., Applications of a commutation formula, Duke Math. J., 45, 267-310 (1978) · Zbl 0392.47013 · doi:10.1215/S0012-7094-78-04516-7
[8] Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 121-251 (1979) · Zbl 0388.34005 · doi:10.1002/cpa.3160320202
[9] Eastham, M. S. P.; Kalf, H., Schrödinger-Type Operators with Continuous Spectra (1982), Boston: Pitman, Boston · Zbl 0491.35003
[10] Ehlers, F.; Knörrer, H., An algebro-geometric interpretation of the Bäcklund transformation of the Korteweg-de Vries equation, Comment. Math. Helv., 57, 1-10 (1982) · Zbl 0516.35071 · doi:10.1007/BF02565842
[11] Ercolani, N. M.; Flaschka, H., The geometry of the Hill equation and of the Neumann system, Philos. Trans. Roy. Soc. London Ser. A, 315, 405-422 (1985) · Zbl 0582.35105 · doi:10.1098/rsta.1985.0048
[12] Finkel, A.; Isaacson, E.; Trubowitz, E., An explicit solution of the inverse problem for Hill’s equation, SIAM J. Math. Anal., 18, 46-53 (1987) · Zbl 0622.34021 · doi:10.1137/0518003
[13] Firsova, N. E., On solution of the Cauchy problem for the Korteweg-de Vries equation with initial data the sum of a periodic and a rapidly decreasing function, Math. USSR Sbornik, 63, 257-265 (1989) · Zbl 0669.35106 · doi:10.1070/SM1989v063n01ABEH003272
[14] Flaschka, H.; McLaughlin, D. W.; Miura, R. M., Some comments on Bäcklund transformations, canonical transformations, and the inverse scattering method, 252-295 (1976), Berlin: Springer, Berlin · Zbl 0317.00006 · doi:10.1007/BFb0081172
[15] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Korteweg-de Vries equation and generalizations, VI. Methods for exact solution, Comm. Pure Appl. Math., 27, 97-133 (1974) · Zbl 0291.35012 · doi:10.1002/cpa.3160270108
[16] Gel’fand, I. M.; Levitan, B. M., On the determination of a differential equation from its spectral function, Amer. Math. Transl. Ser. 2, 1, 253-304 (1955) · Zbl 0066.33603
[17] Gesztesy, F.; Goldstein, J. A.; Kappel, F.; Schappacher, W., On the modified Korteweg-de Vries equation, Differential Equations with Applications in Biology, Physics, and Engineering, 139-183 (1991), New York: Marcel Dekker, New York · Zbl 0753.35078
[18] Gesztesy, F.; Fenstad, J. E.; Albeverio, S.; Holden, H.; Lindstrøm, T., Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equation, Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, 428-471 (1992), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0818.35103
[19] Gesztesy, F., A complete spectral characterization of the double commutation method, J. Funct. Anal., 117, 401-446 (1993) · Zbl 0813.34074 · doi:10.1006/jfan.1993.1132
[20] F. Gesztesy, M. Krishna and G. Teschl,On isospectral sets of Jacobi operators, to appear in Comm. Math. Phys. · Zbl 0881.58069
[21] F. Gesztesy, R. Nowell and W. Pötz,One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics, to appear in Adv. Differential Equations. · Zbl 0894.34077
[22] Gesztesy, F.; Schweiger, W.; Simon, B., Commutation methods applied to the mKdV-equation, Trans. Amer. Math. Soc., 324, 465-525 (1991) · Zbl 0728.35106 · doi:10.2307/2001730
[23] Gesztesy, F.; Simon, B., Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc., 348, 349-373 (1996) · Zbl 0846.34090 · doi:10.1090/S0002-9947-96-01525-5
[24] Gesztesy, F.; Simon, B., The xi function, Acta Math., 176, 49-71 (1996) · Zbl 0885.34070 · doi:10.1007/BF02547335
[25] Gesztesy, F.; Simon, B.; Teschl, G., Zeros of the Wronskian and renormalized oscillation theory, Amer. J. Math., 118, 571-594 (1996) · Zbl 0858.47027 · doi:10.1353/ajm.1996.0024
[26] Gesztesy, F.; Svirsky, R., (m)KdV-solitons on the background of quasi-periodic finite-gap solutions, Mem. Amer. Math. Soc., 118, 563-563 (1995) · Zbl 0855.35109
[27] Gesztesy, F.; Teschl, G., On the double commutation method, Proc. Amer. Math. Soc., 124, 1831-1840 (1996) · Zbl 0855.34028 · doi:10.1090/S0002-9939-96-03299-6
[28] Gesztesy, F.; Teschl, G., Commutation methods for Jacobi operators, J. Differential Equations, 128, 252-299 (1996) · Zbl 0854.34079 · doi:10.1006/jdeq.1996.0095
[29] Gesztesy, F.; Weikard, R.; Ames, W. F.; Harrell, E. M.; Herod, J. V., Spectral deformations and soliton equations, Differential Equations with Applications in Mathematical Physics, 101-139 (1993), Boston: Academic Press, Boston · Zbl 0795.35099
[30] Gesztesy, F.; Zhao, Z., On critical and subcritical Sturm-Liouville operators, J. Funct. Anal., 98, 311-345 (1991) · Zbl 0726.35119 · doi:10.1016/0022-1236(91)90081-F
[31] H. Grosse and A. Martin,Particle Physics and the Schrödinger Equation, Cambridge University Press, Cambridge, to appear. · Zbl 0946.81003
[32] Iwasaki, K., Inverse problem for Sturm-Liouville and Hill’s equations, Ann. Mat. Pure Appl., 149, 4, 185-206 (1987) · Zbl 0641.34012 · doi:10.1007/BF01773933
[33] Jacobi, C. G. J., Zur Theorie der Variationsrechnung und der Differentialgleichungen, J. Reine Angew. Math., 17, 68-82 (1837) · ERAM 017.0558cj
[34] Kay, J.; Moses, H. E., Reflectionless transmission through dielectrics and scattering potentials, J. Appl. Phys., 27, 1503-1508 (1956) · Zbl 0073.22202 · doi:10.1063/1.1722296
[35] Kuznetsov, E. A.; Mikhailov, A. V., Stability of stationary waves in nonlinear weakly dispersive media, Soviet Phys. JETP, 40, 855-859 (1975)
[36] Leighton, W., On self-adjoint differential equations of second order, J. London Math. Soc., 27, 37-47 (1952) · Zbl 0048.06503 · doi:10.1112/jlms/s1-27.1.37
[37] Levitan, B. M., Inverse Sturm-Liouville Problems (1987), Utrecht: VNU Science Press, Utrecht · Zbl 0749.34001
[38] Levitan, B. M., Sturm-Liouville operators on the whole line, with the same discrete spectrum, Math. USSR Sbornik, 60, 77-106 (1988) · Zbl 0661.34017 · doi:10.1070/SM1988v060n01ABEH003157
[39] Marchenko, V. A., Sturm-Liouville Operators and Applications (1986), Basel: Birkhäuser, Basel · Zbl 0592.34011
[40] McKean, H. P., Variation on a theme of Jacobi, Comm. Pure Appl. Math., 38, 669-678 (1985) · Zbl 0591.34025 · doi:10.1002/cpa.3160380514
[41] McKean, H. P., Geometry of KdV (1): Addition and the unimodular spectral classes, Rev. Mat. Iberoamericana, 2, 235-261 (1986) · Zbl 0651.35074
[42] McKean, H. P., Geometry of KdV (2): Three examples, J. Statist. Phys., 46, 1115-1143 (1987) · Zbl 0689.35076 · doi:10.1007/BF01011159
[43] McKean, H. P.; Ehrenpreis, L.; Gunning, R. C., Is there an infinite-dimensional algebraic geometry? Hints from KdV, Theta Functions, 27-37 (1989), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 0699.14053
[44] McKean, H. P., Geometry of KdV (3): Determinants and unimodular isospectral flows, Comm. Pure Appl. Math., 45, 389-415 (1992) · Zbl 0774.35077 · doi:10.1002/cpa.3160450403
[45] McKean, H. P.; Trubowitz, E., The spectral class of the quantum-mechanical harmonic oscillator, Comm. Math. Phys., 82, 471-495 (1982) · Zbl 0493.34012 · doi:10.1007/BF01961236
[46] McKean, H. P.; van Moerbeke, P., The spectrum of Hill’s equation, Invent. Math., 30, 217-274 (1975) · Zbl 0319.34024 · doi:10.1007/BF01425567
[47] Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons (1984), New York: Consultants Bureau, New York · Zbl 0598.35002
[48] Pöschel, J.; Trubowitz, E., Inverse Spectral Theory (1987), Boston: Academic Press, Boston · Zbl 0623.34001
[49] Ralston, J.; Trubowitz, E., Isospectral sets for boundary value problems on the unit interval, Ergodic Theory & Dynamical Systems, 8, 301-358 (1988) · Zbl 0678.34025 · doi:10.1017/S0143385700009470
[50] Schmincke, U.-W., On Schrödinger’s factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A, 80, 67-84 (1978) · Zbl 0395.47022
[51] Simon, B.; Feldman, J.; Froese, R.; Rosen, L., Spectral, analysis of rank one perturbations and applications, CRM Proc. Lecture Notes, 109-149 (1995), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 0824.47019
[52] G. Teschl,Oscillation and renormalized oscillation theory for Jacobi operators, to appear in J. Differential Equations. · Zbl 0866.39002
[53] G. Teschl,Spectral deformations of Jacobi operators, preprint, 1996.
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