Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the Laplacian on toroidal surfaces
Suppose p,q,s are real, continuous, periodic functions with period \(L>0\); in addition, assume that p,s are positive and \(p'\) is continuous. Hill's equation and associated periodic eigenvalue problem is: \[ (p(x)y'(x))'+(\lambda s(x)-q(x))y(x)=0;\quad y(0)=y(L);\quad y'(0)=y'(L). \] It is well known that the eigenvalues \(\{\lambda_ n\}\) satisfy \(\lambda_ 0<\lambda_ 1\leq \lambda_ 2<\lambda_ 3\leq \lambda_ 4<...<\lambda_{2n-1}\leq \lambda_{2n}<\lambda_{2n+1}...\); [see \textit{M. S. P. Eastham}, The spectral theory of periodic differential equations (1973; Zbl 0287.34016)]. The authors give bounds for these eigenvalues based on the Sturm comparison theorem and a result of \textit{O. Haupt} [Math. Ann. 76, 67-104 (1914)]. More specifically, they show that \[ \min_{x\in [0,L]}B_ n(x)\leq \lambda_{2n-1}\leq \lambda_{2n}\leq \max_{x\in [0,L]}B_ n(x), \] where \(B_ n\) is the estimating function defined by \[ B_ n(x)=\frac{p(x)}{q(x)}((\frac{2\pi}{L})^ 2 n^ 2+\frac{q(x)}{p(x)}-\frac{S(P^*)}{2}), \] \(P^*\) is any primitive of p and where \(S(P^*)\) is the Schwarz derivative of \(P^*\). They follow this result with some refinements and apply their results to estimate the eigenvalues of the Laplacian on a surface of revolution homeomorphic to the torus. Many authors have given either upper bounds or lower bounds for the eigenvalues; for example, under certain restrictions on q, Eastham gives the upper bound \[ \lambda_{2n}\leq \frac{\sup (p(x))}{\inf (s(x))}(\frac{2n\pi}{L})^ 2. \] It is interesting and impressive that these authors have been able to give both global upper and lower bounds for these eigenvalues. It appears that their estimates are very good for certain Laplacian problems. This reviewer would like to see further work and a follow-up paper by the authors in order to see further comparisons made between their estimates and known numerical techniques to estimate eigenvalues (i.e. from the NAG library) of various periodic boundary value problems associated with Hill's equation. I am always impressed with analytic bounds but I still would like to see how they compare to numerical approximations.
scientific article; zbMATH DE number 4212545
scientific article