On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients. (English) Zbl 07870832
Summary: In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator \(L\) generated by the differential expression of odd order \(n\) with the \(m\times m\) periodic matrix coefficients, where \(n>1.\) We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set \(\left[ (2\pi N)^n,\infty \right) \cup (-\infty,(-2\pi N)^n]\) belongs to at least \(m\) bands, where \(N\) is the smallest integer satisfying \(N\ge \pi^{-2}M+1\) and \(M\) is the sum of the norms of the coefficients. Moreover, we prove that if \(M\le \pi^22^{-n+1/2}\), then each point of the real line belong to at least \(m\) bands.
MSC:
| 34L05 | General spectral theory of ordinary differential operators |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 47E05 | General theory of ordinary differential operators |
References:
| [1] | Danford, N., Schwartz, J.T.: Linear Operators, Part II: Spectral Theory, New York. Wiley-Interscience, USA (1988) |
| [2] | Gelfand, IM, Expansions in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73, 1117-1120, 1950 · Zbl 0037.34505 |
| [3] | Kato, T., Perturbation Theory for Linear Operators, 1980, Berlin: Springer, Berlin · Zbl 0435.47001 |
| [4] | Levitan, BM, Inverse Sturm-Liouville Problems, 2018, Boston: De Gruyter, Boston |
| [5] | McGarvey, DC, Differential operators with periodic coefficients in \(L_p(-\infty,\infty )\), J. Math. Anal. Appl., 11, 564-596, 1965 · Zbl 0188.21103 · doi:10.1016/0022-247X(65)90105-8 |
| [6] | Naimark, MA, Linear Differential Operators, 1967, London: George G. Harap & Company, London · Zbl 0219.34001 |
| [7] | Rofe-Beketov, FS, The spectrum of non-self-adjoint differential operators with periodic coefficients, Soviet Math. Dokl., 4, 1563-1566, 1963 · Zbl 0199.14002 |
| [8] | Veliev, OA, On the bands of the Schrodinger operator with a matrix potential, Mathematische Nachrichten., 2023 · Zbl 1541.34103 · doi:10.1002/mana.202100481 |
| [9] | Veliev, OA, On the spectrum of the differential operators of even order with periodic matrix coefficients, Lett. Math. Phys., 113, 53, 2023 · Zbl 1521.34078 · doi:10.1007/s11005-023-01679-7 |
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