phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2012 - 3 issue » pp. 25-47

Archive of Issues

Russia Izhevsk
Year
2012
Issue
3
Pages
25-47
<<
>>
Section Mathematics
Title On the spectrum of a periodic Schrödinger operator with potential in the Morrey space
Author(-s) Danilov L.I.a
Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa
Abstract We consider the periodic Schrödinger operator $\widehat H_A+V$ in ${\mathbb R}^n,$ $n\geqslant 3.$ The vector potential $A$ is supposed to satisfy some conditions which are fulfilled whenever the potential $A$ belongs to the Sobolev class $H^q_{\mathrm {loc}}({\mathbb R}^n;{\mathbb R}^n),$ $q>\frac {n-1}2 ,$ and also in the case where $\sum \| A_N\|_{{\mathbb C}^n}<+\infty .$ Here $A_N$ are the Fourier coefficients of the potential $A.$ We prove absolute continuity of the spectrum of the periodic Schrödinger operator $\widehat H_A+V$ provided that the scalar potential $V$ belongs to the Morrey space $\mathfrak L^{ 2, p}({\mathbb R}^n),$ $p\in (\frac {n-1}2,\frac n2],$ and $$ \overline {\lim\limits_{r \to +0}}\ \sup\limits_{x \in {\mathbb R}^n} r^2 \biggl( \frac 1{v(B_r)} \int_{B_r(x)}|V(y)|^p dy\biggr) ^{1/p} \leqslant \varepsilon_0 , $$ where the number $\varepsilon_0=\varepsilon_0(n,p;A)>0$ depends on the vector potential $A,$ $B_r(x)$ is a closed ball of radius $r>0$ centered at the point $x\in {\mathbb R}^n,$ $v(B_r)$ is the $n$-dimensional volume of the ball $B_r=B_r(0).$ Let $K$ be the fundamental domain of the period lattice (which is common for the potentials $A$ and $V$), $K^*$ the fundamental domain of the reciprocal lattice. The operator $\widehat H_A+V$ is unitarily equivalent to the direct integral of operators $\widehat H_A(k)+V,$ $k\in 2\pi K^*,$ acting on the space $L^2(K).$ The last operators are also considered for complex vectors $k+ik^{ \prime }\in {\mathbb C}^n.$ To prove absolute continuity of the spectrum of the operator $\widehat H_A+V,$ we use the Thomas method. The main ingredient in the proof is the following inequality: $$ \| |\widehat H_0(k+ik^{ \prime })|^{-1/2} \bigl( \widehat H_A(k+ik^{ \prime })+V-\lambda \bigr) \varphi \|_{L^2(K)} \geqslant \widetilde C_1 \| |\widehat H_0(k+ik^{ \prime })|^{1/2}\varphi \|_{L^2(K)} ,\quad \varphi \in D(\widehat H_A(k+ik^{ \prime })+V), $$ which holds for some appropriate chosen complex vectors $k+ik^{\prime }\in {\mathbb C}^n$ (depending on $A$, $V,$ and the number $\lambda \in {\mathbb R}$) with sufficiently large imaginary part $k^{\prime },$ where $\widetilde C_1=\widetilde C_1 (n;A)>0$ and $\widehat H_0(k+ik^{\prime })$ is the operator $\widehat H_A(k+ik^{ \prime })$ for $A\equiv 0.$
Keywords Schrödinger operator, absolute continuity of the spectrum, periodic potential, Morrey space
UDC 517.958, 517.984.5
MSC 35P05
DOI 10.20537/vm120304
Received 23 December 2011
Language Russian
Citation Danilov L.I. On the spectrum of a periodic Schrödinger operator with potential in the Morrey space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 3, pp. 25-47.
References
  1. Danilov L.I. On the spectrum of the periodic Dirac operator, Theoret. and Math. Phys., 2000, vol. 124, no. 1, pp. 859-871.
  2. Danilov L.I. On absolute continuity of the spectrum of periodic Schrödinger and Dirac operators. I, Physical-Technical Institute of the Ural Branch of the Russian Academy of Sciences, Izhevsk, 2000, 76 p. Deposited at VINITI 15.06.2000, no. 1683-B00.
  3. Danilov L.I. Absolute continuity of the spectrum of a periodic Dirac operator, Differential Equations, 2000, vol. 36, no. 2, pp. 262-271.
  4. Danilov L.I. On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator, J. Phys. A: Math. Theor., 2009, vol. 42, 275204.
  5. Chanillo S., Sawyer E. Unique continuation for $\Delta +v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 1990, vol. 318, no. 1, pp. 275-300.
  6. Chiarenza F., Ruiz A. Uniform $L$$2$ -weighted Sobolev inequalities, Proc. Amer. Math. Soc., 1991, vol. 112, no. 1, pp. 53-64.
  7. Shen Z. The periodic Schrödinger operators with potentials in the Morrey class, J. Funct. Anal., 2002, vol. 193, pp. 314-345.
  8. Fefferman C. The uncertainty principle, Bull. Amer. Math. Soc. (N.S.), 1983, vol. 9, no. 2, pp. 129-206.
  9. Reed M., Simon B. Methods of modern mathematical physics. IV. Analysis of operators, New York - London: Academic Press, 1978. Translated under the title Metody sovremennoi matematicheskoi fiziki. IV. Analiz operatorov, Moscow: Mir, 1982. 428 p.
  10. Thomas L.E. Time dependent approach to scattering from impurities in a crystal, Commun. Math. Phys., 1973, vol. 33, pp. 335-343.
  11. Birman M.Sh., Suslina T.A. Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, St. Petersburg Math. J., 2000, vol. 11, no. 2, pp. 203-232.
  12. Kuchment P., Levendorskii S. On the structure of spectra of periodic elliptic operators, Trans. Amer. Math. Soc., 2002, vol. 354, no. 2, pp. 537-569.
  13. Birman M.Sh., Suslina T.A. Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector valued potential, St. Petersburg Math. J., 1999, vol. 10, no. 4, pp. 579-601.
  14. Morame A. Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic potential, J. Phys. A: Math. Gen., 1998, vol. 31, pp. 7593-7601.
  15. Shen Z. Absolute continuity of periodic Schrödinger operators with potentials in the Kato class, Illinois J. Math., 2001, vol. 45, no. 3, pp. 873-893.
  16. Shterenberg R.G. Absolute continuity of the spectrum of a two-dimensional magnetic periodic Schrödinger operator with positive electric potential, Trudy S.-Peterburg. Mat. Obshch., 2001, vol. 9, pp. 199-233.
  17. Danilov L.I. On the spectrum of a two-dimensional periodic Schrödinger operator, Theoret. and Math. Phys., 2003, vol. 134, no. 3, pp. 392-403.
  18. Shterenberg R.G. Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with strongly subordinate magnetic potential, J. Math. Sci., 2005, vol. 129, pp. 4087-4109.
  19. Danilov L.I. On the absence of eigenvalues in the spectrum of two-dimensional periodic Dirac and Schrödinger operators, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., Izhevsk, 2004, no. 1 (29), pp. 49-84.
  20. Shen Z. On absolute continuity of the periodic Schrödinger operators, Int. Math. Res. Notices., 2001, no. 1, pp. 1-31.
  21. Shen Z., Zhao P. Uniform Sobolev inequalities and absolute continuity of periodic operators, Trans. Amer. Math. Soc., 2008, vol. 360, no. 4, pp. 1741-1758.
  22. Kenig C.E., Ruiz A., Sogge C.D. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 1987, vol. 55, pp. 329-347.
  23. Koch H., Tataru D. Sharp counterexamples in unique continuation for second order elliptic equations, J. Reine Angew. Math., 2002, issue 542, pp. 133-146.
  24. Sobolev A.V. Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math., 1999, vol. 137, pp. 85-112.
  25. Danilov L.I. On absolute continuity of the spectrum of a periodic Schrödinger operator, Math. Notes, 2003, vol. 73, no. 1, pp. 46-57.
  26. Tikhomirov M., Filonov N. Absolute continuity of the ''even" periodic Schrödinger operator with nonsmooth coefficients, St. Petersburg Math. J., 2005, vol. 16, no. 3, pp. 583-589.
  27. Suslina T.A., Shterenberg R.G. Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces, St. Petersburg Math. J., 2002, vol. 13, no. 5, pp. 859-891.
  28. Shen Z. Absolute continuity of generalized periodic Schrödinger operators, Contemp. Math., 2001, vol. 277, pp. 113-126.
  29. Friedlander L. On the spectrum of a class of second order periodic elliptic differential operators, Commun. Math. Phys., 2002, vol. 229, pp. 49-55.
  30. Danilov L.I. Absolute continuity of the spectrum of a multidimensional periodic magnetic Dirac operator, Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2008, no. 1, pp. 61-96.
  31. Danilov L.I. On absolute continuity of the spectrum of a d-dimensional periodic magnetic Dirac operator, Preprint arXiv: 0805.0399 [math-ph], 2008. http://arXiv.org/abs/0805.0399
  32. Danilov L.I. On absolute continuity of the spectrum of three- and four-dimensional periodic Schrödinger operators, J. Phys. A: Math. Theor., 2010, vol. 43, 215201.
  33. Kato T. Perturbation Theory for Linear Operators, Berlin: Springer, 1976. Translated under the title Teoriya vozmushchenii lineinykh operatorov, Moscow: Mir, 1972.
  34. Reed M., Simon B. Methods of modern mathematical physics. II. Fourier analysis. Self-adjointness, New York: Academic, 1975. Translated under the title Metody sovremennoi matematicheskoi fiziki. II. Garmonicheskii analiz. Samosopryazhennost', Moscow: Mir, 1978. 400 p.
  35. Kuchment P. Floquet theory for partial differential equations, Oper. Theory Adv. Appl., vol. 60. Basel: Birkhauser Verlag, 1993.
  36. Danilov L.I. The spectrum of the Dirac operator with periodic potential. VI, Physical-Technical Institute of the Ural Branch of the Russian Academy of Sciences, Izhevsk, 1996, 45 p. Deposited at VINITI 31.12.1996, no. 3855-B96.
  37. Filonov N., Sobolev A.V. Absence of the singular continuous component in the spectrum of analytic direct integrals, J. Math. Sci., 2006, vol. 136, pp. 3826-3831.
  38. Tomas P. A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 1975, vol. 81, pp. 477-478.
  39. Stein E.M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., vol. 43. Princeton: Princeton Univ. Press, 1993.
  40. Zygmund A. On Fourier coefficients and transforms of functions of two variables, Studia Math., 1974, vol. 50, pp. 189-201.
  41. Tao T. Recent progress on the restriction conjecture, Preprint arXiv: math/0311181 [math.CA], 2003. http://arXiv.org/abs/math/0311181
Full text
/doc/issues-2012/12-03-04.pdf
<< Previous article
Next article >>