A short proof of Zheludev's theorem

Authors:
F. Gesztesy and B. Simon

Journal:
Trans. Amer. Math. Soc. **335** (1993), 329-340

MSC:
Primary 34L40; Secondary 34L10, 47E05, 49R05, 81Q10

DOI:
https://doi.org/10.1090/S0002-9947-1993-1096260-8

MathSciNet review:
1096260

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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**, https://doi.org/10.1016/0003-4916(77)90351-7**[2]**M. S. P. Eastham,*The spectral theory of periodic differential equations*, Scottish Academic Press, Edinburgh, 1973.**[3]**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****[4]**-,*Trace formula for a perturbed one-dimensional Schrödinger operator with a periodic potential*. II, Problemy Mat. Fiz.**8**(1976), 158-171. (Russian)**[5]**-,*Riemann surface of quasimomentum and scattering theory for the perturbed Hill operator*, J. Soviet Math.**11**(1979), 487-497.**[6]**-,*Levinson formula for perturbed Hill operator*, Theoret. and Math. Phys.**62**(1985), 130-140.**[7]**F. Gesztesy and B. Simon,*On a theorem of Deift and Hempel*, Comm. Math. Phys.**116**(1988), no. 3, 503–505. MR**937772****[8]**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**, https://doi.org/10.1093/qmath/41.2.189**[9]**M. Klaus,*On the bound state of Schrödinger operators in one dimension*, Ann. Physics**108**(1977), no. 2, 288–300. MR**503200**, https://doi.org/10.1016/0003-4916(77)90015-X**[10]**B. M. Levitan,*Inverse Sturm-Liouville problems*, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR**933088****[11]**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****[12]**Michael Reed and Barry Simon,*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419****[13]**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****[14]**-,*A test for the finiteness of the number of discrete levels introduced into gaps of a continuous spectrum by perturbations of a periodic potential*, Soviet Math. Dokl.**5**(1964), 689-692.**[15]**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**, https://doi.org/10.1016/0003-4916(76)90038-5**[16]**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**, https://doi.org/10.1016/0022-1236(80)90006-3**[17]**V. A. Zheludev,*Eigenvalues of the perturbed Schrödinger operators with a periodic potential*, Topics in Mathematical Physics (M. Sh. Birman, ed.), Vol. 2, Consultants Bureau, New York, 1968, pp. 87-101.**[18]**-,*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, Consultants Bureau, New York, 1971, pp. 55-75.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34L40,
34L10,
47E05,
49R05,
81Q10

Retrieve articles in all journals with MSC: 34L40, 34L10, 47E05, 49R05, 81Q10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1096260-8

Article copyright:
© Copyright 1993
American Mathematical Society