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A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions. (English) Zbl 1251.34100
The authors study the generally non-self-adjoint Schrödinger operators $$H^P$$ and $$H^{AP}$$ in the Hilbert space $$L^2([0,\pi];dx)$$ associated with the differential expression $$L:=-\frac{d^2}{dx^2}+V(x)$$, $$x\in [0,\pi]$$ and complex-valued potential $$V$$ satisfying $$V\in L^2([0,\pi];\,dx)$$, with periodic and antiperiodic boundary conditions. They give necessary and sufficient conditions in terms of spectral data for the above operators to possess a Riesz basis of root vectors (i.e. eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). The case of a Schauder basis for the above periodic and antiperiodic Schrödinger operators is also investigated.

##### MSC:
 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 47B40 Spectral operators, decomposable operators, well-bounded operators, etc. 47A10 Spectrum, resolvent 34L05 General spectral theory of ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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##### References:
 [1] Akhiezer, N.I.; Glazman, I.M., Theory of linear operators in Hilbert space, vol. I, (1981), Pitman Boston · Zbl 0467.47001 [2] Batchenko, V.; Gesztesy, F., On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV potentials, J. anal. math., 95, 333-387, (2005) · Zbl 1088.34071 [3] Birnir, B., Complex hillʼs equation and the complex periodic Korteweg-de Vries equations, Comm. pure appl. math., 39, 1-49, (1986) · Zbl 0592.47004 [4] Birnir, B., Singularities of the complex Korteweg-de Vries flows, Comm. pure appl. math., 39, 283-305, (1986) · Zbl 0605.35076 [5] Birnir, B., An example of blow-up, for the complex KdV equation and existence beyond blow-up, SIAM J. appl. math., 47, 710-725, (1987) · Zbl 0647.35076 [6] Christiansen, T., Isophasal, isopolar, and isospectral Schrödinger operators and elementary complex analysis, Amer. J. math., 130, 49-58, (2008) · Zbl 1140.35005 [7] Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1985), Krieger Malabar · Zbl 0042.32602 [8] P.A. Deift, An unpublished manuscript on the Hill equation with the potential $$K e^{- 2 i x}$$. [9] Dernek, N.; Veliev, O.A., On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator, Israel J. math., 145, 113-123, (2005) · Zbl 1073.34094 [10] Djakov, P.; Mityagin, B., Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps, J. funct. anal., 195, 89-128, (2002) · Zbl 1037.34080 [11] Djakov, P.; Mityagin, B., Spectral gaps of the periodic Schrödinger operator when its potential is an entire function, Adv. in appl. math., 31, 562-596, (2003) · Zbl 1047.34100 [12] Djakov, P.; Mityagin, B., Spectral triangles of Schrödinger operators with complex potentials, Selecta math., 9, 495-528, (2003) · Zbl 1088.34072 [13] Djakov, P.; Mityagin, B., Instability zones of 1D periodic Schrödinger and Dirac operators, Russian math. surveys, 61, 663-766, (2006) · Zbl 1128.47041 [14] Djakov, P.; Mityagin, B., Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Dokl. math., 83, 5-7, (2011) · Zbl 1242.34148 [15] Djakov, P.; Mityagin, B., Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. ann., 351, 509-540, (2011) · Zbl 1367.47050 [16] Djakov, P.; Mityagin, B., Criteria for existence of Riesz bases consisting of root functions of Hill and 1d Dirac operators, (2011), preprint [17] Dunford, N.; Schwartz, J.T., Linear operators, part III: spectral operators, (1988), Wiley-Interscience New York · Zbl 0635.47003 [18] Eastham, M.S.P., The spectral theory of periodic differential equations, (1973), Scottish Academic Press Edinburgh/London · Zbl 0287.34016 [19] Eberhard, W.; Freiling, G., Stone-reguläre eigenwertprobleme, Math. Z., 160, 139-161, (1978) · Zbl 0363.34021 [20] Eberhard, W.; Freiling, G.; Zettl, A., Sturm-Liouville problems with singular non-selfadjoint boundary conditions, Math. nachr., 278, 1509-1523, (2005) · Zbl 1090.34021 [21] Efendiev, R.F., The characterization problem for one class of second order operator pencil with complex periodic coefficients, Mosc. math. J., 7, 55-65, (2007), 166 · Zbl 1131.34010 [22] Freiling, G., Zur vollständigkeit des systems der eigenfunktionen und hauptfunktionen irregulärer operatorbüschel, Math. Z., 188, 55-68, (1984) · Zbl 0538.47004 [23] Freiling, G.; Rykhlov, V., On a general class of Birkhoff-regular eigenvalue problems, Differential integral equations, 8, 2157-2176, (1995) · Zbl 0837.34081 [24] Gasymov, M.G., Spectral analysis of a class of second-order non-self-adjoint differential operators, Funct. anal. appl., 14, 11-15, (1980) · Zbl 0574.34012 [25] Gasymov, M.G., Spectral analysis of a class of ordinary differential operators with periodic coefficients, Sov. math. dokl., 21, 718-721, (1980) · Zbl 0527.34023 [26] Gelʼfand, I.M., Expansion in characteristic functions of an equation with periodic coefficients, Dokl. akad. nauk, 73, 1117-1120, (1950), (in Russian) [27] Gesztesy, F.; Makarov, K.A., (modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels revisited, Integral equations operator theory, Integral equations operator theory, Integral equations operator theory, 48, 561-602, (2004), and the corrected electronic only version in [28] Gesztesy, F.; Tkachenko, V., When is a non-self-adjoint Hill operator a spectral operator of scalar type?, C. R. acad. sci. Paris, ser. I, 343, 239-242, (2006) · Zbl 1108.34064 [29] Gesztesy, F.; Tkachenko, V., A criterion for Hill operators to be spectral operators of scalar type, J. anal. math., 107, 287-353, (2009) · Zbl 1193.47037 [30] Gesztesy, F.; Weikard, R., Floquet theory revisited, (), 67-84 · Zbl 0946.47031 [31] Gesztesy, F.; Weikard, R., Picard potentials and hillʼs equation on a torus, Acta math., 176, 73-107, (1996) · Zbl 0927.37040 [32] Gesztesy, F.; Weikard, R., A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta math., 181, 63-108, (1998) · Zbl 0955.34073 [33] Gesztesy, F.; Weikard, R., Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach, Bull. amer. math. soc., 35, 271-317, (1998) · Zbl 0909.34073 [34] Gohberg, I.; Goldberg, S.; Kaashoek, M.A., Classes of linear operators, vol. I, Oper. theory adv. appl., vol. 49, (1990), Birkhäuser Basel · Zbl 0745.47002 [35] Gohberg, I.C.; Krein, M.G., The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. math. soc. transl. ser. 2, 13, 185-264, (1960) · Zbl 0089.32201 [36] Gohberg, I.C.; Krein, M.G., Introduction to the theory of linear nonselfadjoint operators, Transl. math. monogr., vol. 18, (1969), Amer. Math. Soc. Providence, RI · Zbl 0181.13504 [37] Golʼdberg, A.A.; Levin, B.Ya.; Ostrovskii, I.V., Entire and meromorphic functions, (), 67-98 [38] Guillemin, V.; Uribe, A., Hardy functions and the inverse spectral method, Comm. partial differential equations, 8, 1455-1474, (1983) · Zbl 0567.35073 [39] Ince, E.L., Ordinary differential equations, (1956), Dover New York · Zbl 0063.02971 [40] Kashin, B.S.; Saakyan, A.A., Orthogonal series, Transl. math. monogr., vol. 75, (1989), Amer. Math. Soc. Providence, RI · Zbl 0668.42011 [41] Kerimov, N.B.; Mamedov, Kh.R., On the Riesz basis property of root functions of some regular boundary value problems, Math. notes, 64, 483-487, (1998) · Zbl 0924.34072 [42] Keselʼman, G.M., On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. vyssh. uchebn. zaved. mat., 39, 2, 82-93, (1964), (in Russian) [43] Kirac, A.A., Riesz basis property of the root functions of non-selfadjoint operators with regular boundary conditions, Int. J. math. anal. (ruse), 3, 1101-1109, (2009) · Zbl 1204.34113 [44] Kotani, S., Generalized Floquet theory for stationary Schrödinger operators in one dimension, Chaos solitons fractals, 8, 1817-1854, (1997) · Zbl 0936.34074 [45] Kurbanov, V.M., A theorem on equivalent bases for differential operators, Dokl. math., 73, 11-14, (2006) · Zbl 1155.34363 [46] Levin, B.Ya., Distribution of zeros of entire functions, Transl. math. monogr., vol. 5, (1980), Amer. Math. Soc. Providence, RI · Zbl 0152.06703 [47] Levin, B.Ya., Lectures on entire functions, Transl. math. monogr., vol. 150, (1996), Amer. Math. Soc. Providence, RI, in collaboration with Yu. Lyubarskii, M. Sodin, and V. Tkachenko · Zbl 0856.30001 [48] Lyubič, U.I.; Macaev, V.I., On the spectral theory of linear operators in Banach spaces, Sov. math. dokl., 1, 184-186, (1960) [49] Lyubič, Ju.I.; Macaev, V.I., Operators with separable spectrum, Mat. sb. (N.S.), 56, 98, 433-468, (1962), (in Russian) [50] Makin, A., On spectral decompositions corresponding to non-self-adjoint Sturm-Liouville operators, Dokl. math., 73, 15-18, (2006) · Zbl 1155.34364 [51] Makin, A., Convergence of expansions in the root functions of periodic boundary value problems, Dokl. math., 73, 71-76, (2006) · Zbl 1155.34365 [52] Makin, A., On periodic boundary value problem for the Sturm-Liouville operator · Zbl 1033.34037 [53] Makin, A., On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. equ., 42, 1717-1728, (2006) · Zbl 1128.34329 [54] Malamud, M.M., On the completeness of the root vector system of the Sturm-Liouville operator with general boundary conditions, Dokl. math., 77, 2, 175-178, (2008) · Zbl 1163.34395 [55] Malamud, M.M., On the completeness of the system of root vectors of the Sturm-Liouville operator with general boundary conditions, Funct. anal. appl., 42, 198-204, (2008) · Zbl 1167.34393 [56] Mamedov, Kh.R., On the basis property in $$L_p(0, 1)$$ of the root functions o f a class non-self adjoint Sturm-Liouville operators, European J. pure appl. math., 3, 831-838, (2010) · Zbl 1216.34089 [57] Mamedov, Kh.R.; Menken, H., On the basisness in $$L_2(0, 1)$$ of the root functions in not strongly regular boundary value problems, European J. pure appl. math., 1, 2, 51-60, (2008) · Zbl 1161.34057 [58] Marchenko, V.A., Sturm-Liouville operators and applications, (1986), Birkhäuser Basel [59] Meiman, N.N., The theory of one-dimensional Schrödinger operators with a periodic potential, J. math. phys., 18, 834-848, (1977) [60] Menken, H.; Mamedov, Kh.R., Basis property in $$L_p(0, 1)$$ of the root functions corresponding to a boundary-value problem, J. appl. funct. anal., 5, 351-356, (2010) · Zbl 1205.34112 [61] Mihaĭlov, V.P., Riesz bases in $$L_2(0, 1)$$, Sov. math. dokl., 3, 851-855, (1962) · Zbl 0133.37602 [62] Minkin, A., Resolvent growth and Birkhoff-regularity, J. math. anal. appl., 323, 387-402, (2006) · Zbl 1114.34064 [63] Minkin, A., Regularity of dissipative differential operators, (January 2010) [64] Naimark, M.A., Linear differential operators, part I, (1967), Ungar New York · Zbl 0219.34001 [65] Pastur, L.A.; Tkachenko, V.A., Spectral theory of Schrödinger operators with periodic complex-valued potentials, Funct. anal. appl., 22, 156-158, (1988) · Zbl 0717.34096 [66] Pastur, L.A.; Tkachenko, V.A., An inverse problem for a class of one-dimensional Schrödinger operators with a complex periodic potential, Math. USSR izv., 37, 611-629, (1991) · Zbl 0739.34022 [67] Pastur, L.A.; Tkachenko, V.A., Geometry of the spectrum of the one-dimensional Schrödinger equation with a periodic complex-valued potential, Math. notes, 50, 1045-1050, (1991) · Zbl 0781.34054 [68] Reed, M.; Simon, B., Methods of modern mathematical physics. IV: analysis of operators, (1978), Academic Press New York · Zbl 0401.47001 [69] Rofe-Beketov, F.S., The spectrum of non-selfadjoint differential operators with periodic coefficients, Sov. math. dokl., 4, 1563-1566, (1963) · Zbl 0199.14002 [70] Sansuc, J.-J.; Tkachenko, V., Spectral parametrization of non-selfadjoint hillʼs operators, J. differential equations, 125, 366-384, (1996) · Zbl 0844.34088 [71] Sansuc, J.-J.; Tkachenko, V., Spectral properties of non-selfadjoint hillʼs operators with smooth potentials, (), 371-385 · Zbl 0844.34087 [72] Sansuc, J.-J.; Tkachenko, V., Characterization of the periodic and antiperiodic spectra of nonselfadjoint hillʼs operators, (), 216-224 · Zbl 0884.34090 [73] Serov, M.I., Certain properties of the spectrum of a non-selfadjoint differential operator of the second order, Sov. math. dokl., 1, 190-192, (1960) · Zbl 0106.05902 [74] Shin, K.C., On half-line spectra for a class of non-self-adjoint Hill operators, Math. nachr., 261-262, 171-175, (2003) · Zbl 1044.34045 [75] Shin, K.C., Trace formulas for non-self-adjoint Schrödinger operators and some applications, J. math. anal. appl., 299, 19-39, (2004) · Zbl 1070.34116 [76] Shin, K.C., On the shape of spectra for non-self-adjoint periodic Schrödinger operators, J. phys. A, 37, 8287-8291, (2004) · Zbl 1064.81029 [77] Shiryaev, E.A.; Shkalikov, A.A., Regular and completely regular differential operators, Math. notes, 81, 566-570, (2007) · Zbl 1156.34075 [78] Shkalikov, A.A., The completeness of the eigenfunctions and associated functions of an ordinary differential operator with irregular-separated boundary conditions, Funct. anal. appl., 10, 305-316, (1976) · Zbl 0354.34023 [79] Shkalikov, A.A., On the basis problem of the eigenfunctions of an ordinary differential operator, Russian math. surveys, 34, 5, 249-250, (1979) · Zbl 0471.34014 [80] Shkalikov, A.A., The basis problem of the eigenfunctions of ordinary differential operators with integral boundary conditions, Moscow univ. math. bull., 37, 6, 10-20, (1982) · Zbl 0565.34020 [81] Shkalikov, A.A.; Veliev, O.A., On the Riesz basis property of eigen- and associated functions of periodic and anti-periodic Sturm-Liouville problems, Math. notes, 85, 647-660, (2009) · Zbl 1190.34111 [82] Titchmarsh, E.C., Eigenfunction expansions associated with second-order differential equations, part II, (1958), Oxford University Press Oxford · Zbl 0097.27601 [83] Tkachenko, V.A., Spectral analysis of the one-dimensional Schrödinger operator with periodic complex-valued potential, Sov. math. dokl., 5, 413-415, (1964) · Zbl 0188.46103 [84] Tkachenko, V.A., Spectral analysis of a nonselfadjoint Hill operator, Sov. math. dokl., 45, 78-82, (1992) · Zbl 0791.34061 [85] Tkachenko, V.A., Discriminants and generic spectra of nonselfadjoint hillʼs operators, Adv. sov. math., 19, 41-71, (1994) · Zbl 0815.34018 [86] Tkachenko, V.A., Spectra of non-selfadjoint hillʼs operators and a class of Riemann surfaces, Ann. math., 143, 181-231, (1996) · Zbl 0856.34087 [87] Tkachenko, V., Characterization of Hill operators with analytic potentials, Integral equations operator theory, 41, 360-380, (2001) · Zbl 0994.34015 [88] Tkachenko, V., Non-selfadjoint Sturm-Liouville operators with multiple spectra, (), 403-414 · Zbl 1036.34010 [89] Veliev, O.A., The one-dimensional Schrödinger operator with a periodic complex-valued potential, Sov. math. dokl., 21, 291-295, (1980) · Zbl 0449.34015 [90] Veliev, O.A., Spectrum and spectral singularities of differential operators with complex-valued periodic coefficients, Differ. equ., 19, 983-989, (1983) · Zbl 0539.47029 [91] Veliev, O.A., Spectral expansions related to non-self-adjoint differential operators with periodic coefficients, Differ. equ., 22, 1403-1408, (1986) · Zbl 0665.34025 [92] Veliev, O.A., Spectral expansion for a nonselfadjoint periodic differential operator, Russ. J. math. phys., 13, 101-110, (2006) · Zbl 1130.34060 [93] Veliev, O.A., On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. math., 176, 195-207, (2010) · Zbl 1204.34117 [94] Veliev, O.A.; Toppamuk Duman, M., The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. math. anal. appl., 265, 76-90, (2002) · Zbl 1003.34075 [95] Veliev, O.A.; Kirac, A.A., On the nonself-adjoint differential operators with the quasiperiodic boundary conditions, Int. math. forum, 2, 1703-1715, (2007) · Zbl 1132.47037 [96] Walker, P.W., A nonspectral Birkhoff-regular differential operator, Proc. amer. math. soc., 66, 187-188, (1977) · Zbl 0367.34019 [97] Weikard, R., On hillʼs equation with a singular complex-valued potential, Proc. London math. soc., 76, 603-633, (1998) · Zbl 0905.34004 [98] Weikard, R., On a theorem of hochstadt, Math. ann., 311, 95-105, (1998) · Zbl 0930.34076 [99] Young, R.M., An introduction to nonharmonic Fourier series, (2001), Academic Press San Diego · Zbl 0981.42001 [100] Zheludev, V.A., Perturbations of the spectrum of the schroedinger operator with a complex periodic potential, (), 25-41 [101] Zygmund, A., Trigonometric series, (1990), Cambridge University Press Cambridge, vols. I & II combined · JFM 58.0280.01
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