Dependence of eigenvalue of Sturm-Liouville operators on the real coupled boundary condition. (English) Zbl 07867878
Summary: In this paper, we discuss the continuous dependence of eigenvalue of Sturm-Liouville operators on the real coupled boundary condition by using of implicit function theorem. A geometric structure on \(\mathrm{SL}(2, \mathbb{R})\) containing real coupled boundary conditions is firstly clarified, that is, the smooth embedding submanifold. Under this structure, we verify the continuous differentiability of the \(n\)-th eigenvalue with regard to the boundary condition and explicitly present the expression for its differential. Moreover, a sufficient condition for recognizing double eigenvalues is given.
MSC:
| 34B24 | Sturm-Liouville theory |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 47J07 | Abstract inverse mapping and implicit function theorems involving nonlinear operators |
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