2018/05/01 by Xijun Hu, Hu, Xijun, Lei Liu +5
Mathematics · Physics and Astronomy · #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.1805.00253
It is natural to consider continuous dependence of the n-th eigenvalue on d-dimensional (d≥2) Sturm-Liouville problems after the results on 1-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the n-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the n-th eigenvalue. This renders a precise description of the jump phenomena of the n-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into 2d+1 layers and show that the n-th eigenvalue is continuously dependent on Sturm-Liouville equations and on boundary conditions when restricted into each layer. In addition, we prove that the analytic and geometric multiplicities of an eigenvalue are equal. Finally, we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions.