A short proof of Zheludev’s theorem

Gesztesy, F.; Simon, B.

doi:10.1090/S0002-9947-1993-1096260-8

A short proof of Zheludev’s theorem
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by F. Gesztesy and B. Simon PDF

Trans. Amer. Math. Soc. 335 (1993), 329-340 Request permission

Abstract:

We give a short proof of Zheludev’s theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a Hill operator subject to certain short-range perturbations. As a by-product we simultaneously recover Rofe-Beketov’s result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed Hill operator. Our methods are operator theoretic in nature and extend to other one-dimensional systems such as perturbed periodic Dirac operators and weakly perturbed second order finite difference operators. We employ the trick of using a selfadjoint Birman-Schwinger operator (even in cases where the perturbation changes sign), a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps.

References

R. Blankenbecler, M. L. Goldberger, and B. Simon, The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians, Ann. Physics 108 (1977), no. 1, 69–78. MR 456018, DOI 10.1016/0003-4916(77)90351-7

The spectral theory of periodic differential equations

N. E. Firsova, A trace formula for a perturbed one-dimensional Schrödinger operator with a periodic potential. I, Problems in mathematical physics, No. 7 (Russian), Izdat. Leningrad. Univ., Leningrad, 1974, pp. 162–177, 184–185 (Russian). MR 0447695

Trace formula for a perturbed one-dimensional Schrödinger operator with a periodic potential

8

Riemann surface of quasimomentum and scattering theory for the perturbed Hill operator

11

Levinson formula for perturbed Hill operator

62

F. Gesztesy and B. Simon, On a theorem of Deift and Hempel, Comm. Math. Phys. 116 (1988), no. 3, 503–505. MR 937772, DOI 10.1007/BF01229205
D. B. Hinton, M. Klaus, and J. K. Shaw, On the Titchmarsh-Weyl function for the half-line perturbed periodic Hill’s equation, Quart. J. Math. Oxford Ser. (2) 41 (1990), no. 162, 189–224. MR 1053662, DOI 10.1093/qmath/41.2.189
M. Klaus, On the bound state of Schrödinger operators in one dimension, Ann. Physics 108 (1977), no. 2, 288–300. MR 503200, DOI 10.1016/0003-4916(77)90015-X
B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088, DOI 10.1515/9783110941937
Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106, DOI 10.1007/978-3-0348-5485-6
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
F. S. Rofe-Beketov, A perturbation of a Hill’s operator, that has a first moment and a non-zero integral, introduces a single discrete level into each of the distant spectral lacunae, Mathematical physics and functional analysis, No. 4 (Russian), Fiz.-Tehn. Inst. Nizkih Temperatur, Akad. Nauk Ukrain. SSR, Kharkov, 1973, pp. 158–159, 163 (Russian). MR 0477257

A test for the finiteness of the number of discrete levels introduced into gaps of a continuous spectrum by perturbations of a periodic potential

5

Barry Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288. MR 404846, DOI 10.1016/0003-4916(76)90038-5
Barry Simon, Brownian motion, $L^{p}$ properties of Schrödinger operators and the localization of binding, J. Functional Analysis 35 (1980), no. 2, 215–229. MR 561987, DOI 10.1016/0022-1236(80)90006-3

Eigenvalues of the perturbed Schrödinger operators with a periodic potential

Perturbation of the spectrum of the one-dimensional self-adjoint Schrödinger operator with a periodic potential