Abstract
Under certain assumptions (including \(d\ge 2)\) we prove that the spectrum of a scalar operator in \(\mathscr {L}^2({\mathbb {R}}^d)\)
covers interval \((\tau -\epsilon ,\tau +\epsilon )\), where \(A^0\) is an elliptic operator and B(x, hD) is a periodic perturbation, \(\varepsilon =O(h^\varkappa )\), \(\varkappa >0\).
Further, we consider generalizations.
This research was supported in part by National Science and Engineering Research Council (Canada) Discovery Grant RGPIN 13827.
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Bibliography
V. Guillemin, Some classical theorems in spectral theory revised, Seminar on Singularities of solutions of partial differential equations, Princeton University Press, NJ, 219–259 (1979).
G. Barbatis, L. Parnovski. Bethe - Sommerfeld conjecture for pseudo-differential perturbation, Comm.P.D.E. 34(4):383 - 418, (2009).
A. Sommerfeld, H. Bethe, Elektronentheorie der Metalle, in H. Geiger and K. Scheel, eds., Handbuch der Physik, Volume 24, Part 2, 333-622 (Springer, 1933). Later edition: Elektronentheorie der Metalle, Springer, 1967.
J.W.S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin, 1959.
B.E.J. Dahlberg, E. Trubowitz, A remark on two dimensional periodic potentials, Comment. Math. Helvetici 57:130–134 (1982).
M.S.P. Eastham, The spectral theory of periodic differential equations, Scottish Academice Press, 1973.
J. Feldman, H. Knörrer, E. Trubowitz, The perturbatively stable spectrum of a periodic Schrödinger operator, Invent. Math., 100:259–300 (1990).
J. Feldman, H. Knörrer, E. Trubowitz, Perturbatively unstable eigenvalues of a periodic Schrödinger operator, Comment. Math. Helvetici, 66:557–579 (1991).
V. Ivrii, Microlocal Analysis, Sharp Spectral, Asymptotics and Applications.
V. Ivrii. 100 years of Weyl’s law,Bull. Math. Sci., 6(3):379–452 (2016).
V. Ivrii. Complete semiclassical spectral asymptotics for periodic and almost periodic perturbations of constant operators, arXiv:1808.01619 (2018).
Y. E. Karpeshina, Perturbation theory for the Schrödinger operator with a periodic potential, Lecture Notes in Math. 1663, Springer Berlin 1997.
Y. E. Karpeshina, Spectral properties of periodic magnetic Schrödinger operator in the high-energy region. Two-dimensional case, Comm. Math. Phys., 251(3):473–514 (2004).
T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1980.
P. Kuchment, Floquet theory for partial differential equations, Birkhäuser, Basel, 1993.
S. Morozov, L. Parnovski, R. Shterenberg. Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators, Ann. Henri Poincar 15(2):263–312 (2014).
L. Parnovski, Bethe-Sommerfeld conjecture, Annales H. Poincaré, 9(3):457–508 (2008).
L. Parnovski, R. Shterenberg. Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schroedinger operator, Invent. Math., 176(2):275–323 (2009).
L. Parnovski, R. Shterenberg. Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators, Ann. of Math., Second Series, 176(2):1039–1096 (2012).
L. Parnovski, R. Shterenberg. Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrödinger operators, Duke Math. J., 165(3):509–561 (2016).
L. Parnovski, A. V. Sobolev, Bethe-Sommerfeld conjecture for polyharmonic operators, Duke Math. J., 107(2):209–238 (2001).
L. Parnovski, A. V. Sobolev, Perturbation theory and the Bethe-Sommerfeld conjecture, Annales H. Poincaré, 2:573–581 (2001).
L. Parnovski, A. V. Sobolev. Bethe-Sommerfeld conjecture for periodic operators with strong perturbations, Invent. Math., 181:467–540 (2010).
V.N. Popov, M. Skriganov, A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential, Zap. Nauchn. Sem. LOMI AN SSSR, 109:131–133 (1981) (Russian).
M. Reed M., B. Simon, Methods of Modern Mathematical Physics, IV, Academic Press, New York, 1975.
G. V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle, Trudy Moskov. Mat. Obshch. 36:59–84 (1978) (Russian).
M. Skriganov, Proof of the Bethe-Sommerfeld conjecture in dimension two, Soviet Math. Dokl. 20(1):89–90 1979).
M. Skriganov, Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators, Proc. Steklov Math. Inst. Vol. 171, 1984.
M. Skriganov, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Inv. Math. 80:107–121 (1985).
M. Skriganov, A. Sobolev, Asymptotic estimates for spectral bands of periodic Schrödinger operators, St Petersburg Math. J. 17(1):207–216 (2006).
M. Skriganov, A. Sobolev, Variation of the number of lattice points in large balls, Acta Arith. 120(3): 245–267 (2005).
A. V. Sobolev. Integrated density of states for the periodic Schrödinger operator in dimension two, Ann. Henri Poincaré. 6:31–84 (2005) .
A. V. Sobolev. Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one, Rev. Mat. Iberoam. 22(1):55–92 (2006).
A.V.Sobolev, Recent results on the Bethe-Sommerfeld conjecture, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, ix–xii, Proc. Sympos. Pure Math., 76, Part 1, Amer. Math. Soc., Providence, RI, 2007.
O.A. Veliev, Asymptotic formulas for the eigenvalues of the periodic Schrödinger operator and the Bethe-Sommerfeld conjecture, Functional Anal. Appl. 21(2):87–100 (1987).
O.A. Veliev, Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe-Sommerfeld Conjecture, Int. J. Contemp. Math. Sci., 2(2):19–87 (2007).
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Ivrii, V. (2019). Bethe-Sommerfeld Conjecture in Semiclassical Settings. In: Microlocal Analysis, Sharp Spectral Asymptotics and Applications V. Springer, Cham. https://doi.org/10.1007/978-3-030-30561-1_36
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