×

When is a non-self-adjoint Hill operator a spectral operator of scalar type? (English) Zbl 1108.34064

The paper is concerned with a nonselfadjoint Hill operator \(H = - d / d x^{2} + V\) in \(L^{2} (\mathbb{R})\), where \(V\) is \(\pi\)-periodic and \(V \in L^{2} [0,\pi].\) The authors give necessary and sufficient conditions for \(H\) to be a spectral operator of scalar type. These conditions are formulated in terms of the fundamental solutions \(\theta (z,x), \phi (z,x)\) of the equation
\[ -y'' + V (x) y = z y \]
with the boundary conditions
\[ \theta (z,0) = \phi ' (z,0) = 1,\quad \theta ' (z,0) = \phi (z,0) = 0. \]
Thus, \(H\) is a spectral operator of scalar type if and only if the estimates
\[ \biggl| \frac{\phi (\lambda,\pi)}{\Delta_{+}^{\bullet} (\lambda)} \biggl| \leq c, \;\biggl| \frac{\theta ' (\lambda,\pi)}{(\mid \lambda \mid + 1) \;\Delta_{+}^{\bullet}(\lambda)} \biggl| \leq c, \;\biggl| \frac{\Delta_{-} (\lambda)}{(\sqrt{\mid \lambda \mid} + 1) \Delta_{+}^{\bullet}(\lambda)} \biggl| \leq c \]
hold for all \(\lambda \in \sigma (H)\) with a finite positive constant \(c,\) where \(\Delta_{\pm} (z) = [\theta (z,\pi) \pm \phi ' (z,\pi)] / 2, \;z \in C, \bullet \) denotes the derivative with respect to \(z\). Also, \(H\) is a spectral operator of scalar type if and only if the following conditions are satisfied:
(i) The function \((\Delta_{+} (z)^{2} - 1 - \Delta_{-} (z)^{2}) / \phi (z,\pi) \Delta_{+}^{\bullet} (z)\) is analytic for \(z\) in an open neighborhood of \(\sigma (H);\)
(ii) The inequalities \[ \biggl| \frac{\phi (\lambda,\pi)}{\Delta_{+}^{\bullet} (\lambda)} \biggl| \leq c, \;\;\biggl| \frac{\Delta_{-} (\lambda)}{(\sqrt{\mid \lambda \mid} + 1) \Delta_{+}^{\bullet}(\lambda)} \biggl| \leq c \] are satisfied for all \(\lambda \in \sigma (H)\) (\(c\) is a finite positive constant).
Another criterion involves notions connected with the method of direct integral decompositions according to which the Hill operator \(H\) is decomposed in a family of operators \(H (t), t \in [0,2\pi]\), defined in \(L^{2} [0,\pi]\) by the differential expression \(-d^{2} / d x^{2} + q (x)\) restricted to \(x \in [0,\pi]\) and the boundary conditions \(y (\pi) = e^{it} y (0)\) and \(y ' (\pi) = e^{it} y ' (0)\).

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Dunford, N.; Schwartz, J. T., Linear Operators, Part III: Spectral Operators (1988), Wiley-Interscience: Wiley-Interscience New York · Zbl 0635.47003
[2] Gel’fand, I. M., Expansion in characteristic functions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR, 73, 1117-1120 (1950), (in Russian)
[3] F. Gesztesy, V. Tkachenko, A criterion for Hill operators to be spectral operators of scalar type, in preparation; F. Gesztesy, V. Tkachenko, A criterion for Hill operators to be spectral operators of scalar type, in preparation
[4] Gesztesy, F.; Weikard, R., Floquet theory revisited, (Knowles, I., Differential Equations and Mathematical Physics (1995), International Press: International Press Boston), 67-84 · Zbl 0946.47031
[5] McGarvey, D., Operators commuting with translations by one, J. Math. Anal. Appl.. J. Math. Anal. Appl., J. Math. Anal. Appl.. J. Math. Anal. Appl.. J. Math. Anal. Appl., J. Math. Anal. Appl., J. Math. Anal. Appl., 12, 187-234 (1965), Part III · Zbl 0188.21201
[6] Meiman, N. N., The theory of one-dimensional Schrödinger operators with a periodic potential, J. Math. Phys., 18, 834-848 (1977)
[7] 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
[8] Sansuc, J.-J.; Tkachenko, V., Spectral parametrization of non-selfadjoint Hill’s operators, J. Differential Equations, 125, 366-384 (1996) · Zbl 0844.34088
[9] 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
[10] Titchmarsh, E. C., Eigenfunction problems with periodic potentials, Proc. Roy. Soc. London A, 203, 501-514 (1950) · Zbl 0040.04701
[11] 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
[12] Tkachenko, V. A., Spectra of non-selfadjoint Hill’s operators and a class of Riemann surfaces, Ann. Math., 143, 181-231 (1996) · Zbl 0856.34087
[13] Veliev, O. A., Spectrum and spectral singularities of differential operators with complex-valued periodic coefficients, Differential Equations, 19, 983-989 (1983) · Zbl 0539.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.