Absolute summability of the trace relation for certain Schrödinger operators. (English) Zbl 0821.34076

Summary: A recently established general trace formula for one-dimensional Schrödinger operators is systematically studied in the context of short-range potentials, potentials which approach different spatial asymptotes sufficiently fast, and appropriate impurity (defect) interactions in one-dimensional solids. We prove the absolute summability of the trace formula and establish its connections with scattering quantities, such as reflection coefficients, in each case.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81U05 \(2\)-body potential quantum scattering theory
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[1] Aronszajn, N., Donoghue, W.F.: On exponential representations of analytic functions in the upper half-plane with positive imaginary part. J. Anal. Math.5, 321–388 (1957) · Zbl 0138.29502
[2] Bollé, D., Gesztesy, F., Wilk, S.F.J.: A complete treatment of low-energy scattering in one dimension. J. Operator Theory13, 3–31 (1985) · Zbl 0567.47008
[3] Bollé, D., Gesztesy, F., Klaus, M.: Scattering theory for one-dimensional systems withxV(x)=0. J. Math. Anal. Appl.122, 496–518 (1987) · Zbl 0617.35105
[4] Cohen, A., Kappeler, T.: Scattering and inverse scattering for steplike potentials in the Schrödinger equation. Indiana Univ. Math. J.34, 127–180 (1985) · Zbl 0553.34015
[5] Craig, W.: The trace formula for Schrödinger operators on the line. Commun. Math. Phys.126, 379–407 (1989) · Zbl 0681.34026
[6] Davies, E.B., Simon, B.: Scattering theory for systems with different spatial asymptotics to the left and right. Commun. Math. Phys.63, 277–301 (1978) · Zbl 0393.34015
[7] Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math.32, 121–251 (1979) · Zbl 0395.34019
[8] Dubrovin, B.A.: Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials. Funct. Anal. Appl.9, 215–223 (1975) · Zbl 0358.35022
[9] Firsova, N.E.: Levinson formula for perturbed Hill operator. Theoret. Math. Phys.62, 130–140 (1985) · Zbl 0573.34022
[10] Firsova, N.E.: The direct and inverse scattering problems for the one-dimensional perturbed Hill operator. Math. USSR Sbornik58, 351–388 (1987) · Zbl 0627.34028
[11] Flaschka, H.: On the inverse problem for Hill’s operator. Arch. Rat. Mech. Anal.59, 293–309 (1975) · Zbl 0376.34016
[12] Gel’fand, I.M., Levitan, B.M.: On a simple identity for eigenvalues of a second order differential operator. Dokl. Akad. Nauk SSSR88, 593–596 (1953) (Russian); English transl. in Gelfand, Izrail M., Collected papers Vol. I (Gindikin, S.G., Guillemin, V.W., Kirillov, A.A., Kostant, B., Sternberg, S., eds.) Berlin, Heidelberg, New York: Springer, 1987, pp. 457–461
[13] Gesztesy, F.: Scattering theory for one-dimensional systems with nontrivial spatial asymptotics. Schrödinger Operators, Aarhus 1985 (Balslev, E., ed.), Lecture Notes in Math. Vol.1218, Berlin, Heidelberg, New York: Springer, 1986, pp. 93–122
[14] Gesztesy, F.: New trace formulas for Schrödinger operators. Evolution Equations. (Ferreyra, G., Ruiz Goldstein, G., Neubrander, F., eds.), New York, Marcel Dekker, 1995, pp. 201–221 · Zbl 0811.34068
[15] Gesztesy, F., Simon, B.: A short proof of Zheludev’s theorem. Trans. Am. Math. Soc.335, 329–340 (1993) · Zbl 0770.34056
[16] Gesztesy, F., Simon, B.: The xi function. Acta. Math. (To appear) · Zbl 0885.34070
[17] Gesztesy, F., Holden, H., Simon, B., Zhao, Z.: A trace formula for multidimensional Schrödinger operators. Preprint · Zbl 0864.35030
[18] Gesztesy, F., Holden, H., Simon, B., Zhao, Z.: Trace formulae and inverse spectral theory for Schrödinger operators. Bull. Am. Math. Soc.29, 250–255 (1993) · Zbl 0786.34081
[19] Gesztesy, F., Holden, H., Simon, B., Zhao, Z.: Higher order trace relations for Schrödinger operators. Rev. Math. Phys. (To appear) · Zbl 0833.34084
[20] Gesztesy, F., Karwowski, W., Zhao, Z.: New types of soliton solutions. Bull. Am. Math. Soc.27, 266–272 (1992) · Zbl 0760.35039
[21] Gesztesy, F., Karwowski, W., Zhao, Z.: Limits of soliton solutions. Duke Math. J.68, 101–150 (1992) · Zbl 0811.35122
[22] Hochstadt, H.: On the determination of a Hill’s equation from its spectrum. Arch. Rat. Mech. Anal.19, 353–362 (1965) · Zbl 0128.31201
[23] Iwasaki, K.: Inverse problem for Sturm-Liouville and Hill equation. Ann. Mat. Pura Appl. Ser. 4,149, 185–206 (1987) · Zbl 0641.34012
[24] Kotani, S., Krishna, M.: Almost periodicity of some random potentials. J. Funct. Anal.78, 390–405 (1988) · Zbl 0644.60061
[25] Krein, M.G.: Perturbation determinants and a formula for the traces of unitary and self-adjoint operators. Sov. Math. Dokl.3, 707–710 (1962) · Zbl 0191.15201
[26] Kristensson, G.: The one-dimensional inverse scattering problem for an increasing potential, J. Math. Phys.27, 804–815 (1986) · Zbl 0594.47006
[27] Levitan, B.M.: On the closure of the set of finite-zone potentials, Math. USSR Sbornik51, 67–89 (1985) · Zbl 0589.34026
[28] Levitan, B.M.: Inverse Sturm-Liouville Problems. Utrecht: VNU Science Press, 1987 · Zbl 0749.34001
[29] Marchenko, V.A.: Sturm-Liouville Operators and Applications. Basel: Birkhäuser, 1986 · Zbl 0592.34011
[30] McKean, H.P., van Moerbeke, P.: The spectrum of Hill’s equation. Invent. Math.30, 217–274 (1975) · Zbl 0319.34024
[31] Rofe-Beketov, F.S.: A test for the finiteness of the number of discrete levels introduced into gaps of a continuous spectrum by perturbations of a periodic potential. Sov. Math. Dokl.5, 689–692 (1964) · Zbl 0117.06004
[32] Rofe-Beketov, F.S.: Perturbation of a Hill operator having a first moment and nonzero integral creates one discrete level in distant spectral gaps. Mat. Fizika i Funkts. Analiz (Khar’kov)19, 158–159 (1973) (Russian)
[33] Simon. B.: Spectral analysis of rank one perturbations and applications. Lecture given at the 1993 Vancouver Summer School, Proceedings on Mathematical Quantum Theory II: Schrödinger Operators (CRM Proceedings and Lecture Notes). (Feldman, J., Froese, R., Rosen, L.M., eds.). To appear
[34] Trubowitz, E.: The inverse problem for periodic potentials. Commun. Pure Appl. Math.30, 321–337 (1977) · Zbl 0403.34022
[35] Venakides, S.: The infinite period limit of the inverse formalism for periodic potentials. Commun. Pure Appl. Math.41, 3–17 (1988) · Zbl 0663.34028
[36] Zheludev, V.A.: Eigenvalues of the perturbed Schroedinger operator with a periodic potential. Topics in Mathematical Physics (M.Sh. Birman, ed.), Vol. 2, New York: Consultants Bureau, 1968, pp. 87–101
[37] Zheludev, V.A.: Perturbation of the spectrum of the one-dimensional self-adjoint Schrödinger operator with a periodic potential. Topics in Mathematical Physics (M.Sh. Birman, ed.), Vol. 4, New York: Consultants Bureau, 1971, pp. 55–75
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