Eigenvalues variation. II: Multidimensional problems. (English) Zbl 0807.34033

Dauge, Monique; Helffer, Bernard

J. Differ. Equations 104, No. 2, 263-297 (1993).

In part I [ibid. 243-262 (1993; Zbl 0784.34021)] the authors initiated the investigation of the variation of Neumann eigenvalues with respect to the domain; they studied Sturm-Liouville operators on intervals \([0,a]\) and proved a formula (F) \(\lambda'= c(q- \lambda)\) which linked the potential \(q\) and the derivative of the eigenvalues with \(a\).
In this paper the authors extend such a formula to higher dimensional domains: for instance, they give a one-parameter family of nested domains \(\Omega_ t\) in \(\mathbb{R}^ n\) which depends in an analytic way on that parameter \(t\). They consider the Neumann problem on \(\Omega_ t\) for a Schrödinger operator \(-\Delta+ V\). Then (F) extends to an equation of the form \[ \lambda'(t)= \int_{\partial\Omega_ t} (| Tv_ t|^ 2- \lambda| v_ t|^ 2)dv_ t, \] where \(T\) is an order one tangent differential operator, \(v_ t\) is the eigenvector and \(dv_ t\) is an absolutely continuous measure on \(\partial\Omega_ t\).
Unfortunately, they do not know anything in general about the first trace of \(v_ t\) along the whole boundary \(\partial\Omega_ t\); so in general, such a formula is not very informative about the variations of \(\lambda\). On the contrary, when the problem is spherically symmetric, for instance Coulomb potential or harmonic oscillator, the traces of \(v_ t\) are eigenfunctions of \(T\); in such a case, a separation of variables allows to reduce to a one-dimensional problem, which is in general a singular Sturm-Liouville problem. So the authors study Schrödinger operators with spherically symmetric potentials on balls with variable radii.
Then, by neighbour methods, they study the Neumann problem associated with the Laplace-Beltrami operator on spherical sectors and finally, by a different method, they study the evolution of the Neumann eigenvalues when a crack is propagating in a plate.

Reviewer: P.Bolley (Nantes)

MSC:

34B24	Sturm-Liouville theory
34L05	General spectral theory of ordinary differential operators
35P15	Estimates of eigenvalues in context of PDEs

Keywords:

Neumann problem; Schrödinger operator; singular Sturm-Liouville problem; spherically symmetric potentials; Laplace-Beltrami operator; Neumann eigenvalues; crack; plate