Non-real eigenvalues of indefinite Sturm-Liouville problems. (English) Zbl 1291.34052
Consider the regular indefinite Sturm-Liouville problem
\[
-(py')' + q y = \lambda w y\text{ on }[0,1]
\]
with separated boundary conditions and with real coefficients \(w, q, 1/p \in L^1[0,1]\) satisfying \(p(x) > 0, \, w(x) \neq 0\) a.e. If the weight function \(w\) changes its sign then the theory of definitizable operators in Krein spaces implies a relation between the number \(n\) of negative eigenvalues of the corresponding definite problem (with \(w\) replaced by \(|w|\)) and the number (\(\leq 2n\)) of non-real eigenvalues of the original problem. The present paper presents a refinement of this result improving some earlier estimates. In particular, two statements on a priori bounds for (the real and imaginary parts of) the non-real eigenvalues are obtained, involving the coefficients and the boundary conditions: one estimate assuming certain smoothness conditions on the weight function \(w\) and the other estimate assuming \(p=1\) and only a single sign change of \(w\). Furthermore, two existence statements are obtained, using the so-called eigencurve method: sufficient conditions for the existence of exactly \(2\) non-real eigenvalues and also for its complete absence (even if above we have \(n > 0\)).
Reviewer: Andreas Fleige (Dortmund)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 47B50 | Linear operators on spaces with an indefinite metric |
References:
| [1] | Atkinson, F. V.; Jabon, D., Indefinite Sturm-Liouville problems, (Kaper, H. G.; Kwong, M. K.; Zettle, A., Proceedings of the Focused Research Program on Spectral Theory and Boundary Value Problems, vol. I (1988), Argonne National Lab.: Argonne National Lab. Lemont, IL, USA), 31-45 |
| [2] | Behrndt, J.; Philipp, F.; Trunk, C., Bounds on the non-real spectrum of differential operators with indefinite weights (2012) |
| [3] | Behrndt, J.; Katatbeh, Q.; Trunk, C., Non-real eigenvalues of singular indefinite Sturm-Liouville operators, Proc. Amer. Math. Soc., 137, 3797-3806 (2009) · Zbl 1182.47036 |
| [4] | Binding, P.; Browne, P. J., Applications of two parameter spectral theory to symmetric generalised eigenvalue problems, Appl. Anal., 29, 107-142 (1988) · Zbl 0683.47011 |
| [5] | Binding, P.; Volkmer, H., Eigencurves for two-parameter Sturm-Liouville equations, SIAM Rev., 38, 27-48 (1996) · Zbl 0869.34020 |
| [6] | Čurgus, B.; Langer, H., A Krein space approach to symmetric ordinary differential operators with an indefinite weight functions, J. Differential Equations, 79, 31-61 (1989) · Zbl 0693.34020 |
| [7] | Fleckinger, J.; Mingarelli, A. B., On the eigenfunctions of non-definite elliptic operators, (Knowles, I. W.; Lewis, R. T., Differential Equations (1964), North-Holland: North-Holland The Netherlands) · Zbl 0559.35055 |
| [8] | Haupt, O., Über eine Methode zum Beweis von oszillations Theoremen, Math. Ann., 76, 67-104 (1915) · JFM 45.0494.01 |
| [9] | Mingarelli, A. B., Indefinite Sturm-Liouville problems, (Lecture Notes in Math., vol. 964 (1982), Springer: Springer Berlin, New York), 519-528 · Zbl 0503.34016 |
| [10] | Mingarelli, A. B., A survey of the regular weighted Sturm-Liouville problem - The non-definite case (June 2011) |
| [11] | Qi, J.; Chen, S., A priori bounds and existence of non-real eigenvalues of indefinite Sturm-Liouville problems, J. Spectr. Theory (2013), in press |
| [12] | Richardson, R. G.D., Theorems of oscillation for two linear differential equations of second order with two parameters, Trans. Amer. Math. Soc., 13, 22-34 (1912) · JFM 43.0400.03 |
| [13] | Richardson, R. G.D., Contributions to the study of oscillatory properties of the solutions of linear differential equations of the second order, Amer. J. Math., 40, 283-316 (1918) |
| [14] | Zettl, A., Sturm-Liouville Theory, Math. Surveys Monogr., vol. 121 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1074.34030 |
| [15] | Turyn, L., Sturm-Liouville problems with several parameters, J. Differential Equations, 38, 239-259 (1980) · Zbl 0421.34023 |
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