Riesz basis property of Hill operators with potentials in weighted spaces

P. Djakov ; B. Mityagin

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 181-204.

Résumé

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 $v(x)=\sum_{2\mathbb{Z}}V(k)e^{ikx}$ 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 $$ V(-2n)\asymp V(2n) \text{ as } n\in2\mathbb{N}\ (\text{or } n\in1+2\mathbb{N}), \quad n\to\infty. $$ For such potentials we prove that $\lambda_n^+-\lambda_n^-\sim\pm 2\sqrt{V(-2n)V(2n)}$ and $$ \mu_n-\frac12(\lambda_n^++\lambda_n^-)\sim-\frac12(V(-2n)+V(2n)). $$ References: 32 entries.

Bibliographie

[1] Anahtarci B., Djakov P., “Refined asymptotics of the spectral gap for the Mathieu operator”, J. Math. Anal. Appl., 396 (2012), 243–255 | DOI | MR | Zbl

[2] Avron J., Simon B., “The asymptotics of the gap in the Mathieu equation”, Ann. Phys., 134 (1981), 76–84 | DOI | MR | Zbl

[3] Dernek N., Veliev O., “On the Riesz basisness of the root functions of the nonself-adjoint Sturm–Liouville operator”, Israel J. Math., 145 (2005), 113–123 | DOI | MR | Zbl

[4] Djakov P., Mityagin B., “Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps”, J. Funct. Anal., 195:1 (2002), 89–128 | DOI | MR | Zbl

[5] Djakov P., Mityagin B., “Spectral gaps of the periodic Schrödinger operator when its potential is an entire function”, Adv. in Appl. Math., 31:3 (2003), 562–596 | DOI | MR | Zbl

[6] Djakov P., Mityagin B., “Spectral triangles of Schrödinger operators with complex potentials”, Selecta Math. (N.S.), 9:4 (2003), 495–528 | DOI | MR | Zbl

[7] Dzhakov P. B., Mityagin B. S., “Zony neustoichivosti odnomernykh periodicheskikh operatorov Shredingera i Diraka”, UMN, 61:4(370) (2006), 77–182 | DOI | MR | Zbl

[8] Djakov P., Mityagin B., “Asymptotics of instability zones of the Hill operator with a two term potential”, J. Funct. Anal., 242:1 (2007), 157–194 | DOI | MR | Zbl

[9] Djakov P., Mityagin B., “Fourier method for one-dimensional Schrödinger operators with singular periodic potentials”, Oper. Theory Adv. Appl., 203, Birkhäuser-Verlag, Basel, 2010, 195–236 | MR

[10] Djakov P., Mityagin B., “Spectral gaps of Schrödinger operators with periodic singular potentials”, Dyn. Partial Differ. Equ., 6:2 (2009), 95–165 | DOI | MR | Zbl

[11] Dzhakov P. B., Mityagin B. S., “Skhodimost spektralnykh razlozhenii operatorov Khilla s trigonometricheskimi mnogochlenami kak potentsialy”, Doklady RAN, 436:1 (2011), 11–13 | Zbl

[12] Djakov P., Mityagin B., “Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials”, Math. Ann., 351:3 (2011), 509–540 | DOI | MR | Zbl

[13] Djakov P., Mityagin B., “Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators”, J. Funct. Anal., 263:8 (2012), 2300–2332 1106.5774 | DOI | MR | Zbl

[14] Djakov P., Mityagin B., “Riesz bases consisting of root functions of 1D Dirac operators”, Proc. AMS, 141:4 (2013), 1361–1375 | DOI | MR | Zbl

[15] Djakov P., Mityagin B., “Divergence of spectral decompositions of Hill operators with two exponential term potentials”, J. Funct. Anal., 265:4 (2013), 660–685 | DOI | MR | Zbl

[16] Djakov P., Mityagin B., Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators, arXiv: 1309.1751

[17] Dunford N., Schwartz J., Linear Operators, v. III, Spectral Operators, Wiley, New York, 1971 | MR | Zbl

[18] Gesztesy F., Tkachenko V., “A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions”, J. Diff. Equations, 253 (2012), 400–437 | DOI | MR | Zbl

[19] Hryniv R. O., Mykytyuk Ya. V., “1-D Schrödinger operators with periodic singular potentials”, Methods Funct. Anal. Topology, 7:4 (2001), 31–42 | MR | Zbl

[20] Kappeler T., Mityagin B., “Gap estimates of the spectrum of Hill's equation and action variables for KdV”, Trans. AMS, 351:2 (1999), 619–646 | DOI | MR | Zbl

[21] Kappeler T., Mityagin B., “Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator”, SIAM J. Math. Anal., 33:1 (2001), 113–152 | DOI | MR | Zbl

[22] Kerimov N. B., Mamedov Kh. R., “O bazisnosti Rissa kornevykh funktsii nekotorykh regulyarnykh kraevykh zadach”, Matem. zametki, 64:4 (1998), 558–563 | DOI | MR | Zbl

[23] Keselman G. M., “O bezuslovnoi skhodimosti razlozhenii po sobstvennym funktsiyam nekotorykh differentsialnykh operatorov”, Izv. vuzov. Matematika, 39:2 (1964), 82–93 | MR | Zbl

[24] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970 | MR | Zbl

[25] Makin A. S., “O skhodimosti razlozhenii po kornevym funktsiyam periodicheskoi kraevoi zadachi”, Doklady RAN, 406:4 (2006), 452–457 | MR | Zbl

[26] Marchenko V. A., Operatory Shturma–Liuvilya i ikh prilozheniya, Naukova dumka, Kiev, 1975 | MR

[27] Mikhailov V. P., “O bazisakh Rissa v $L^2[0,1]$”, DAN SSSR, 144:5 (1962), 981–984 | MR

[28] Mityagin B. S., “Skhodimost razlozhenii po sobstvennym funktsiyam operatora Diraka”, Doklady RAN, 393:4 (2003), 456–459 | MR

[29] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969

[30] Polya G., Szegö G., Problems and Theorems in Analysis, v. I, Classics in Mathematics, Springer-Verlag, Berlin, 1998

[31] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Tr. MMO, 64, 2003, 159–212 | Zbl

[32] Veliev O. A., Shkalikov A. A., “O bazisnosti Rissa sobstvennykh i prisoedinennykh funktsii periodicheskoi i antiperiodicheskoi zadach Shturma–Liuvillya”, Matem. zametki, 85:5 (2009), 671–686 | DOI | MR | Zbl