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Zbl 1237.47046
Qi, Jianggang; Chen, Shaozhu
(Qi, Jiang-gang; Chen, Shao-zhu)
On an open problem of Weidmann: essential spectra and square-integrable solutions.
(English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 2, 417-430 (2011). ISSN 0308-2105; ISSN 1473-7124/e

Authors' abstract: In [Spectral theory of ordinary differential operators. Lecture Notes in Mathematics 1258. Berlin etc.: Springer-Verlag (1987; Zbl 0647.47052)], {\it J. Weidmann} proved that, for a symmetric differential operator $\tau$ and a real $\lambda$, if there exist fewer square-integrable solutions of $(\tau - \lambda )y = 0$ than needed and if there is a self-adjoint extension of $\tau$ such that $\lambda$ is not its eigenvalue, then $\lambda$ belongs to the essential spectrum of $\tau$. However, he posed as open problem whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.
[Vassilis G. Papanicolaou (Athena)]
MSC 2000:
*47E05 Ordinary differential operators
34B20 Weyl theory and its generalizations
34L05 General spectral theory for ODE

Keywords: $n$-th order ordinary differential operators; square-integrable solutions; deficiency indices; essential spectrum

Citations: Zbl 0647.47052

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