Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

Barbatis, G., Parnovski, L.: Bethe-Sommerfeld conjecture for pseudo-differential perturbation. Commun. Partial Differ. Equ. 34(4), 383–418 (2009)

Birman, M.Ş., Solomyak, M.Z.: Spectral Theory of Self-adjoint Operators in Hilbert Space. Reidel, Dordrecht (1987)

Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Springer, Berlin (1959)

Dahlberg, B.E.J., Trubowitz, E.: A remark on two dimensional periodic potentials. Comment. Math. Helv. 57, 130–134 (1982)

Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Scottish Academic, Edinburgh (1973)

Feldman, J., Knörrer, H., Trubowitz, E.: The perturbatively stable spectrum of a periodic Schrödinger operator. Invent. Math. 100, 259–300 (1990)

Feldman, J., Knörrer, H., Trubowitz, E.: Perturbatively unstable eigenvalues of a periodic Schrödinger operator. Comment. Math. Helv. 66, 557–579 (1991)

Helffer, B., Mohamed, A.: Asymptotics of the density of states for the Schrödinger operator with periodic electric potential. Duke Math. J. 92, 1–60 (1998)

Karpeshina, Y.E.: Perturbation series for the Schrödinger operator with a periodic potential near planes of diffraction. Commun. Anal. Geom. 4(3), 339–413 (1996)

Karpeshina, Y.E.: Perturbation Theory for the Schrödinger Operator with a Periodic Potential. Lecture Notes in Math., vol. 1663. Springer, Berlin (1997)

Karpeshina, Y.E.: Spectral properties of periodic magnetic Schrödinger operator in the high-energy region. Two-dimensional case. Commun. Math. Phys. 251(3), 473–514 (2004)

Kuchment, P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993)

Mohamed, A.: Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential. J. Math. Phys. 38(8), 4023–4051 (1997)

Parnovski, L.: Bethe-Sommerfeld conjecture. Ann. Henri Poincaré 9(3), 457–508 (2008)

Parnovski, L., Sobolev, A.V.: Bethe-Sommerfeld conjecture for polyharmonic operators. Duke Math. J. 107(2), 209–238 (2001)

Parnovski, L., Sobolev, A.V.: Perturbation theory and the Bethe-Sommerfeld conjecture. Ann. Henri Poincaré 2, 573–581 (2001)

Popov, V.N., Skriganov, M.: A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential. Zap. Nauchn. Semin. LOMI AN SSSR 109, 131–133 (1981) (Russian)

Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV. Academic Press, New York (1975)

Rozenbljum, G.V.: Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle. Tr. Mosk. Mat. Obshch. 36, 59–84 (1978) (Russian)

Skriganov, M.: Proof of the Bethe-Sommerfeld conjecture in dimension two. Sov. Math. Dokl. 20(1), 89–90 (1979)

Skriganov, M.: Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators. Proc. Steklov Math. Inst. 171 (1984)

Skriganov, M.: The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential. Invent. Math. 80, 107–121 (1985)

Skriganov, M., Sobolev, A.: Asymptotic estimates for spectral bands of periodic Schrödinger operators. St. Petersburg Math. J. 17(1), 207–216 (2006)

Sobolev, A.V.: Integrated density of states for the periodic Schrödinger operator in dimension two. Ann. Henri Poincaré 6, 31–84 (2005)

Sobolev, A.V.: Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one. Rev. Mat. Iberoam. 22(1), 55–92 (2006)

Sobolev, A.V.: Recent results on the Bethe-Sommerfeld conjecture. In: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday. Proc. Sympos. Pure Math., Part 1, vol. 76, pp. ix–xii. Am. Math. Soc., Providence (2007)

Sommerfeld, A., Bethe, H.: Elektronentheorie der Metalle. In: Geiger, H., Scheel, K. (eds.) Handbuch der Physik, Part 2, vol. 24, pp. 333–622. Springer, Berlin (1933). Later edition: Elektronentheorie der Metalle. Springer (1967)

Veliev, O.A.: Asymptotic formulas for the eigenvalues of the periodic Schrödinger operator and the Bethe-Sommerfeld conjecture. Funct. Anal. Appl. 21(2), 87–100 (1987)

Veliev, O.A.: On the spectrum of multidimensional periodic operators. Theory Funct. Funct. Anal. Appl. Kharkov Univ. 49, 17–34 (1988) (in Russian)

Veliev, O.A.: Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe-Sommerfeld conjecture. Int. J. Contemp. Math. Sci. 2(2), 19–87 (2007)