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