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Spectra of a class of non-symmetric operators in Hilbert spaces with applications to singular differential operators

Part of: Ordinary differential operators Operator theory: Ordinary differential operators

Published online by Cambridge University Press:  15 February 2019

Huaqing Sun  and
Bing Xie
Huaqing Sun
Affiliation:
Department of Mathematics, Shandong University at Weihai Weihai, Shandong264209, P. R. China ([email protected]; [email protected])
Bing Xie
Affiliation:
Department of Mathematics, Shandong University at Weihai Weihai, Shandong264209, P. R. China ([email protected]; [email protected])
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Abstract

This paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

Keywords

𝒥-symmetric operatordefect indexeigenvalueessential spectrumsingular Hamiltonian system

MSC classification

Primary: 34L05: General spectral theory
Secondary: 47E05: Ordinary differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1769 - 1790
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Anderson, T. G. and Hinton, D. B.. Relative boundedness and compactness theory for second-order differential operators. J. Inequal. Appl. 1 (1997), 375–400.Google Scholar
2Behncke, H. and Hinton, D.. Spectral theory of Hamiltonian systems with almost constant coefficients. J. Diff. Equ. 250 (2011), 1408–1426.CrossRefGoogle Scholar
3Behncke, H. and Nyamwala, F. O.. Spectral analysis of higher-order differential operators with unbounded coefficients. Math. Nachr. 285 (2012), 56–73.CrossRefGoogle Scholar
4Brown, B. M. and Marletta, M.. Spectral inclusion and spectral exactness for singular non-self-adjoint Sturm-Liouville problems. Proc. R. Soc. Lond. A 457 (2001), 117–139.CrossRefGoogle Scholar
5Brown, B. M. and Marletta, M.. Spectral inclusion and spectral exactness for singular non-self-adjoint Hamiltonian systems. Proc. R. Soc. Lond. A 459 (2003), 1987–2009.CrossRefGoogle Scholar
6Brown, B. M., McCormack, D. K. R., Evans, W. D. and Plum, M.. On the spectrum of second-order differential operators with complex coefficients. Proc. R. Soc. Lond. A 455 (1999), 1235–1257.CrossRefGoogle Scholar
7Cascaval, R. and Gesztesy, F.. J-self-adjointness of a class of Dirac-type operators. J. Math. Anal. Appl. 294 (2004), 113–121.CrossRefGoogle Scholar
8Clark, S. and Gesztesy, F.. On self-adjoint and J-self-adjoint Dirac type operators: a case study. Contemp. Math. 412 (2006), 103–140.CrossRefGoogle Scholar
9Ding, W.. The spectrum of J-self-adjoint extensions of J-symmetric operators. J. Zhaoqing University 28 (2007), 11–13 (in Chinese).Google Scholar
10Dunford, N. and Schwartz, J. T.. Linear operators (II) (New York: Wiley-Interscience, 1963).Google Scholar
11Eastham, M. S. P.. The asymptotic solution of linear differential systems (Oxford: Clarendon Press, 1989).Google Scholar
12Edmunds, D. E. and Evans, W. D.. Spectral theory and differential operators (Oxford: Clarendon Press, 1987).Google Scholar
13Evans, W. D.. On the J-self-adjointness of Schrödinger operators with a singular complex potential. J. London Math. Soc. 20 (1979), 495–508.CrossRefGoogle Scholar
14Everitt, W. N. and Markus, L.. Boundary value problems and symplectic algebra for ordinary differential and quasidifferential operators, Mathematical Surveys and Monographs Vol. 61 (Providence, RI: Amer. Math. Soc., 1999).Google Scholar
15Galindo, A.. On the existence of J-self-adjoint extensions of J-symmetric operators with adjoint. Comm. Pure Appl. Math. 15 (1962), 423–425.CrossRefGoogle Scholar
16Garcia, S. R. and Wogen, W. R.. Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), 6065–6077.CrossRefGoogle Scholar
17Garcia, S. R. and Poore, D. E.. On the closure of the complex symmetric operators: Compact operators and weighted shifts. J. Funct. Anal. 264 (2013), 691–712.CrossRefGoogle Scholar
18Glazman, I. M.. An analogue of the extension theory of Herimitian operators and a non-symmetric one-dimensional boundary-value problem on a half-axis. Dokl. Akad. Nauk. SSSR 115 (1957), 214–216.Google Scholar
19Halvorsen, S. G.. Counterexamples in the spectral theory of singular Sturm-Liouville operators. In Ordinary and partial differential equations (eds. Sleeman, B. D. and Michael, I. M.). Lecture Notes in Math. 415, 373–382 (Berlin: Springer, 1974).CrossRefGoogle Scholar
20Hao, X., Sun, J. and Zettl, A.. The spectrum of differential operators and square-integrable solutions. J. Funct. Anal. 262 (2012), 1630–1644.CrossRefGoogle Scholar
21Hartman, P. and Wintner, A.. An oscillation theorem for continuous spectra. Proc. Natl. Acad. Sci. 33 (1947), 376–379.CrossRefGoogle ScholarPubMed
22Hartman, P. and Wintner, A.. On the essential spectra of singular eigenvalue problems. Am. J. Math. 72 (1950), 545–552.CrossRefGoogle Scholar
23Hassi, S. and Kuzhel, S.. On J-self-adjoint operators with stable C-symmetries. Proc. Roy. Soc. Edinburgh 143A (2013), 141–167.CrossRefGoogle Scholar
24Knowles, I. W.. On the boundary conditions characterizing J-self-adjoint extensions of J-symmetric operators. J. Diff. Equ. 40 (1981), 193–216.CrossRefGoogle Scholar
25Krall, A. M.. M(λ) theory for singular Hamiltonian systems with one singular point. SIAM J. Math. Anal. 20 (1989a), 664–700.CrossRefGoogle Scholar
26Krall, A. M.. M(λ) theory for singular Hamiltonian systems with two singular points. SIAM J. Math. Anal. 20 (1989b), 701–715.CrossRefGoogle Scholar
27Kuzhel, S., Shapovalova, O. and Vavrykovych, L.. On J-self-adjoint extensions of the Phillips symmetric operators. Methods Funct. Anal. Topology 16 (2010), 333–348.Google Scholar
28Liu, J.. On J-self-adjoint extensions of J-symmetric operators. Acta Sci. Natur. Univ. Intramongol. 23 (1992), 312–316 (in Chinese).Google Scholar
29Mcleod, J. B.. Square-integrable solutions of a second-order differential equation with complex coefficients. Quart. J. Math. Oxford (3) 13 (1962), 129–133.CrossRefGoogle Scholar
30Meyer, K. R. and Hall, G. R.. Introduction to Hamiltonian dynamical systems and the N-body problem (New York: Springer-Verlag, 1992).CrossRefGoogle Scholar
31Mogilevskii, V.. Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators. J. Funct. Anal. 261 (2011), 1955–1968.CrossRefGoogle Scholar
32Qi, J. and Chen, S.. On an open problem of Weidmann: essential spectra and square-integrable solutions. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), 417–430.CrossRefGoogle Scholar
33Qi, J. and Chen, S.. Essential spectra of one-dimensional Dirac operators. Integr. Equ. Oper. Theory 74 (2012), 7–24.CrossRefGoogle Scholar
34Race, D.. On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh 85A (1980), 1–14.CrossRefGoogle Scholar
35Race, D.. On the essential spectra of linear 2n-th order differential operators with complex coefficients. Proc. Roy. Soc. Edinburgh 92A (1982), 65–75.CrossRefGoogle Scholar
36Race, D.. The theory of J-self-adjoint extensions of J-symmetric operators. J. Diff. Equ. 57 (1985), 258–274.CrossRefGoogle Scholar
37Remling, C.. Essential spectra and L 2-solutions of one dimensional Schrödinger operators. Proc. Am. Math. Soc. 124 (1996), 2097–2100.CrossRefGoogle Scholar
38Remling, C.. Spectral analysis of higher-order differential operators I: general properties of the M-functions. J. London Math. Soc. (2) 58 (1998), 367–380.CrossRefGoogle Scholar
39Remling, C.. Spectral analysis of higher-order differential operators II: fourth-order equations. J. London Math. Soc. (2) 59 (1999), 188–206.CrossRefGoogle Scholar
40Schmidt, K. M.. A remark on the essential spectra of Dirac systems. Bull. Lond. Math. Soc. 32 (2000), 63–70.CrossRefGoogle Scholar
41Shang, Z.. On J-self-adjoint extensions of J-symmetric ordinary differential operators. J. Diff. Equ. 73 (1988), 153–177.CrossRefGoogle Scholar
42Sims, A. R.. Secondary conditions for linear differential operators of the second-order. J. Math. Mech. 6 (1957), 247–285.Google Scholar
43Sun, H. and Qi, J.. On classification of second-order differential equations with complex coefficients. J. Math. Anal. Appl. 372 (2010), 585–597.CrossRefGoogle Scholar
44Sun, H. and Shi, Y.. On essential spectra of singular linear Hamiltonian systems. Linear Algebra Appl. 469 (2015), 204–229.CrossRefGoogle Scholar
45Sun, J., Wang, A. and Zettl, A.. Continuous spectrum and square-integrable solutions of differential operators with intermediate deficiency index. J. Funct. Anal. 255 (2008), 3229–3248.CrossRefGoogle Scholar
46Sun, H., Shi, Y. and Wen, J.. 𝒥-self-adjoint extensions of a class of Hamiltonian differential systems. Linear Algebra Appl. 462 (2014), 204–232.CrossRefGoogle Scholar
47Weidmann, J.. Linear operators in Hilbert spaces (New York: Springer-Verlag, 1980).CrossRefGoogle Scholar
48Weidmann, J.. Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258 (Berlin: Springer-Verlag, 1987).CrossRefGoogle Scholar