Applications of a one-dimensional Sobolev inequality to eigenvalue problems
The paper deals with upper and lower bounds for the eigenvalues of regular Sturm-Liouville boundary eigenvalue problems \(- y'' + qy = \lambda y\) where \(q \in L_s (a,b)\), \(1 \leq s \leq \infty\), with boundary conditions \(y(a) = y(b) = 0\) or \(y(a) = y'(b) = 0\). The case of a Sturm Liouville problem of this type with a positive weight function \(k \in L_s (a,b)\) is also considered. To this end a one-dimensional Sobolev type inequality is used, supplemented with a Prüfer transformation argument. In particular, a shorter proof for a known result on the maximization or minimization of an eigenvalue \(\lambda_n\) subject to a norm constraint on \(q\) is given. The inequalities obtained for the eigenvalues are applied to estimate the number of eigenvalues in \((\infty, \lambda]\), the gaps between eigenvalues and to derive Lyapunov and de La Vallée Poussin type inequalities.
upper and lower bounds for the eigenvalues
Sturm-Liouville boundary eigenvalue problems
scientific article; zbMATH DE number 856933