Eigenvalues variation. I: Neumann problem for Sturm–Liouville operators. (English) Zbl 0784.34021

Dauge, Monique; Helffer, Bernard

J. Differ. Equations 104, No. 2, 243-262 (1993).

The authors investigate variations with respect to the domain, of eigenvalues for Neumann problems in one dimensions of the type \(D_ x(p(x)D_ x u(x))+ q(x)u(x)=f(x)\), \(0<x<a\), \(u'(0)=0\), \(u'(a)=0\). They prove formula (F) which relates the potential \(q\) and the first derivative of the eigenvalues \(\lambda\) with respect to the domain. Equation (F) is a differential equation which has the form: (F) \(\lambda'=c(q-\lambda)\) where \(c\) is a positive function. Together with asymptotics when the width of the domain goes to zero or to infinity, they derive from this formula precise information about the variation of the eigenvalues. In a second paper [ibid. 104, 244-278 (1993)], the authors study the same problems for operators which arise in mathematical physics.

Reviewer: P.Bolley (Nantes)

MSC:

34B24	Sturm-Liouville theory
34L05	General spectral theory of ordinary differential operators

Keywords:

Neumann problems; variation of the eigenvalues; operators; mathematical physics