Inequalities for sums of reciprocals of eigenvalues
The study of inequalities of eigenvalues of differential operators was stimulated by \textit{M. S. Ashbaugh's} and \textit{R. D. Benguria's} [Ann. Math., II. Ser. 135, 601-628 (1992; Zbl 0757.35052)] proof of the Payne- Polya-Weinberger conjecture. In this paper, the authors prove a very general and interesting inequality for the sums of reciprocals of eigenvalues involving the Rayleigh quotient \(R(u)= A(u, u)/B(u, u)\). The \(A\) and \(B\) are Hermitian sesquilinear functionals on a vector space \({\mathbf V}\) over \({\mathbf R}\) or \({\mathbf C}\), with \(B\) positive definite, and \(A\) positive semidefinite with kernel \({\mathbf K}\). Let the Rayleigh quotient \(R(u)\) generate a nondecreasing sequence of positive eigenvalues \(\{\lambda_ n\}\) and associate \(B\)-orthonormal eigenfunctions \(\{u_ i\}_{1\leq i\leq j}\) as \[ \lambda_ n= \{R(u): B(v, k)= 0,\;\forall k\in{\mathbf K},\;B(v, u_ i)= 0,\;1\leq i\leq n- 1\}. \] The main result states that if \(\{w_ 1,\dots, w_ n\}\) is any set of nonzero vectors in \({\mathbf V}\) that are pairwise \(A\)-orthogonal \(B\)-orthogonal to \({\mathbf K}\) and the eigenvectors \(\{u_ i\}_{1\leq i\leq j}\), then \[ \sum^{i= j+ n}_{i= j+1} \lambda^{-1}_ i\geq \sum^{i= n}_{i= 1} [R(w_ i)]^{-1}. \] Three applications of the theorem are given. The first considers the Neuman eigenvalue problem \(\Delta u= -\lambda u\), in \(\Omega\), \(\partial u/\partial n= 0\), on \(\partial\Omega\) for the Laplace operator on a bounded smooth domain \(\Omega\subset{\mathbf R}^ n\). The second involves the conormal derivative eigenvalue problem for a self-adjoint elliptic operator \(- \nabla\cdot (a\nabla u)= \lambda\rho u\), in \(\Omega\), \((a\nabla u)\cdot n= 0\), on \(\partial\Omega\), with \(\rho(x)> 0\). The third is the Neuman problem for the Schrödinger operator \(-\Delta u+ \rho u= \lambda u\), in \(\Omega\), \(\partial u/\partial n= 0\), on \(\partial\Omega\).
inequalities of eigenvalues of differential operators
Laplace operator on a bounded smooth domain
Neuman problem for the Schrödinger operator
scientific article; zbMATH DE number 566484
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