## Sur les fonctions propres de l’opérateur de Laplace-Beltrami dont le carré est fonction propre. (On eigenfunctions of the Laplace-Beltrami operator whose square is an eigenfunction).(French)Zbl 0646.58028

Harmonic analysis, symmetric spaces and probability theory, Cortona/Italy 1984, Symp. Math. 29, 111-129 (1987).
[For the entire collection see Zbl 0633.00012.
Let D be the Laplace-Beltrami operator on the n-dimensional Lobatchevsky space. The authors prove that all normalized real solutions F of the system: $$DF=0$$; $$D(F^ 2)=2(n-1)^ 2F^ 2$$ are obtained by taking for F the hyperbolic Poisson kernel.
Reviewer: B.Helffer

### MSC:

 58J50 Spectral problems; spectral geometry; scattering theory on manifolds

### Keywords:

Laplace-Beltrami operator; hyperbolic Poisson kernel

Zbl 0633.00012