Riesz basis property of Hill operators with potentials in weighted spaces
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- by P. Djakov and B. Mityagin
- Trans. Moscow Math. Soc. 2014, 151-172
- DOI: https://doi.org/10.1090/S0077-1554-2014-00230-2
- Published electronically: November 4, 2014
Abstract:
Consider the Hill operator $L(v) = - d^2/dx^2 + v(x)$ on $[0,\pi ]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2$ there are one Dirichlet eigenvalue $\mu _n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda _n^-, \lambda _n^+$ (counted with multiplicity).
We describe classes of complex potentials $v(x)= \sum \nolimits _{\kern 1pt 2\mathbb {Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v)$ contains a Riesz basis if and only if \[ V(-2n) \asymp V(2n) \quad \text {as}~n \in 2\mathbb {N}~(\text {or}~n \in 1+ 2\mathbb {N}),\quad n \to \infty .\] For such potentials we prove that $\lambda _n^+ - \lambda _n^- \sim \pm 2\sqrt {V(-2n)V(2n)}$ and\[ \mu _n - \frac {1}{2}(\lambda _n^+ + \lambda _n^-) \sim -\frac {1}{2} (V(-2n) + V(2n)).\]
References
- Berkay Anahtarci and Plamen Djakov, Refined asymptotics of the spectral gap for the Mathieu operator, J. Math. Anal. Appl. 396 (2012), no. 1, 243–255. MR 2956958, DOI 10.1016/j.jmaa.2012.06.019
- Joseph Avron and Barry Simon, The asymptotics of the gap in the Mathieu equation, Ann. Physics 134 (1981), no. 1, 76–84. MR 626698, DOI 10.1016/0003-4916(81)90005-1
- Neşe Dernek and O. A. Veliev, On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator, Israel J. Math. 145 (2005), 113–123. MR 2154723, DOI 10.1007/BF02786687
- Plamen Djakov and Boris Mityagin, Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps, J. Funct. Anal. 195 (2002), no. 1, 89–128. MR 1934354, DOI 10.1006/jfan.2002.3975
- Plamen Djakov and Boris Mityagin, Spectral gaps of the periodic Schrödinger operator when its potential is an entire function, Adv. in Appl. Math. 31 (2003), no. 3, 562–596. MR 2006361, DOI 10.1016/S0196-8858(03)00027-7
- Plamen Djakov and Boris Mityagin, Spectral triangles of Schrödinger operators with complex potentials, Selecta Math. (N.S.) 9 (2003), no. 4, 495–528. MR 2031750, DOI 10.1007/s00029-003-0358-y
- P. Dzhakov and B. S. Mityagin, Instability zones of one-dimensional periodic Schrödinger and Dirac operators, Uspekhi Mat. Nauk 61 (2006), no. 4(370), 77–182 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 4, 663–766. MR 2279044, DOI 10.1070/RM2006v061n04ABEH004343
- Plamen Djakov and Boris Mityagin, Asymptotics of instability zones of the Hill operator with a two term potential, J. Funct. Anal. 242 (2007), no. 1, 157–194. MR 2274019, DOI 10.1016/j.jfa.2006.06.013
- Plamen Djakov and Boris Mityagin, Fourier method for one-dimensional Schrödinger operators with singular periodic potentials, Topics in operator theory. Volume 2. Systems and mathematical physics, Oper. Theory Adv. Appl., vol. 203, Birkhäuser Verlag, Basel, 2010, pp. 195–236. MR 2683242, DOI 10.1007/978-3-0346-0161-0_{9}
- Plamen Djakov and Boris Mityagin, Spectral gaps of Schrödinger operators with periodic singular potentials, Dyn. Partial Differ. Equ. 6 (2009), no. 2, 95–165. MR 2542499, DOI 10.4310/DPDE.2009.v6.n2.a1
- P. B. Dzhakov and B. S. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomials as potentials, Dokl. Akad. Nauk 436 (2011), no. 1, 11–13 (Russian); English transl., Dokl. Math. 83 (2011), no. 1, 5–7. MR 2810153, DOI 10.1134/S1064562411010017
- Plamen Djakov and Boris Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. 351 (2011), no. 3, 509–540. MR 2854104, DOI 10.1007/s00208-010-0612-5
- Plamen Djakov and Boris Mityagin, Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators, J. Funct. Anal. 263 (2012), no. 8, 2300–2332. MR 2964684, DOI 10.1016/j.jfa.2012.07.003
- Plamen Djakov and Boris Mityagin, Riesz bases consisting of root functions of 1D Dirac operators, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1361–1375. MR 3008883, DOI 10.1090/S0002-9939-2012-11611-9
- Plamen Djakov and Boris Mityagin, Divergence of spectral decompositions of Hill operators with two exponential term potentials, J. Funct. Anal. 265 (2013), no. 4, 660–685. MR 3062541, DOI 10.1016/j.jfa.2013.05.003
- Plamen Djakov and Boris Mityagin, Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions, J. Approx. Theory 164 (2012), no. 7, 879–927. MR 2922621, DOI 10.1016/j.jat.2012.03.013
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. MR 1009164
- Fritz Gesztesy and Vadim Tkachenko, A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions, J. Differential Equations 253 (2012), no. 2, 400–437. MR 2921200, DOI 10.1016/j.jde.2012.04.002
- R. O. Hryniv and Ya. V. Mykytyuk, 1-D Schrödinger operators with periodic singular potentials, Methods Funct. Anal. Topology 7 (2001), no. 4, 31–42. MR 1879483
- T. Kappeler and B. Mityagin, Gap estimates of the spectrum of Hill’s equation and action variables for KdV, Trans. Amer. Math. Soc. 351 (1999), no. 2, 619–646. MR 1473448, DOI 10.1090/S0002-9947-99-02186-8
- T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math. Anal. 33 (2001), no. 1, 113–152. MR 1857991, DOI 10.1137/S0036141099365753
- N. B. Kerimov and Kh. R. Mamedov, On the Riesz basis property of root functions of some regular boundary value problems, Mat. Zametki 64 (1998), no. 4, 558–563 (Russian, with Russian summary); English transl., Math. Notes 64 (1998), no. 3-4, 483–487 (1999). MR 1687220, DOI 10.1007/BF02314629
- G. M. Kesel′man, On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vysš. Učebn. Zaved. Matematika 1964 (1964), no. 2 (39), 82–93 (Russian). MR 0166630
- B. M. Levitan and I. S. Sargsyan, Vvedenie v spektral′nuyu teoriyu. Samosopryazhennye obyknovennye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0299863
- A. S. Makin, On the convergence of expansions in root functions of a periodic boundary value problem, Dokl. Akad. Nauk 406 (2006), no. 4, 452–457 (Russian). MR 2347327
- 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
- V. P. Mihaĭlov, On Riesz bases in ${\cal L}_{2}(0,\,1)$, Dokl. Akad. Nauk SSSR 144 (1962), 981–984 (Russian). MR 0149334
- B. S. Mityagin, Convergence of expansions in eigenfunctions of the Dirac operator, Dokl. Akad. Nauk 393 (2003), no. 4, 456–459 (Russian). MR 2088512
- M. A. Naĭmark, Lineĭ nye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
- George Pólya and Gabor Szegő, Problems and theorems in analysis. I, Classics in Mathematics, Springer-Verlag, Berlin, 1998. Series, integral calculus, theory of functions; Translated from the German by Dorothee Aeppli; Reprint of the 1978 English translation. MR 1492447, DOI 10.1007/978-3-642-61905-2
- A. M. Savchuk and A. A. Shkalikov, Sturm-Liouville operators with distribution potentials, Tr. Mosk. Mat. Obs. 64 (2003), 159–212 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2003), 143–192. MR 2030189
- O. A. Veliev and A. A. Shkalikov, On the Riesz basis property of eigen- and associated functions of periodic and anti-periodic Sturm-Liouville problems, Mat. Zametki 85 (2009), no. 5, 671–686 (Russian, with Russian summary); English transl., Math. Notes 85 (2009), no. 5-6, 647–660. MR 2572858, DOI 10.1134/S0001434609050058
Bibliographic Information
- P. Djakov
- Affiliation: Sabanci University, Orhanli, Istanbul, Turkey
- Email: djakov@sabanciuniv.edu
- B. Mityagin
- Affiliation: Department of Mathematics, The Ohio State University
- Email: mityagin.1@osu.edu
- Published electronically: November 4, 2014
- Additional Notes: P. Djakov acknowledges the hospitality of the Department of Mathematics of the Ohio State University, July–August 2013.
B. Mityagin acknowledges the support of the Scientific and Technological Research Council of Turkey and the hospitality of Sabanci University, May–June, 2013. - © Copyright 2014 P. Djakov and B. Mityagin
- Journal: Trans. Moscow Math. Soc. 2014, 151-172
- MSC (2010): Primary 47E05, 34L40, 34L10
- DOI: https://doi.org/10.1090/S0077-1554-2014-00230-2
- MathSciNet review: 3308607
Dedicated: Dedicated to the memory of Boris Moiseevich Levitan on the occasion of the 100th anniversary of his birthday