 
Summary: A note on the generalized Dumbbell problem
Colette ANN '
E
Abstract.  This note is devoted to the calculation of the
asymptotics of the small eigenvalues and corresponding eigenfunc
tions for the Laplace operator with Neumann boundary conditions
on a domain obtained by adding several thin channels between given
bounded domains.
This note will employ a useful lemma on quadratic forms to improve, with a simple
proof, a recent result of Jimbo and Morita. This lemma was introduced by Helffer and
Sj¨ostrand to study the tunnelling effect, then by Colin de Verdi`ere in his work about stable
multiplicity and also by myself to study the growth of multiplicity by adding handles (see
also an illustration of this techniques in [CCdV]). One can find the following weak version
in [A].
Lemma.  Let (q; D) be a closed nonnegative quadratic form in the Hilbert space
(H; ! ; ?). Define the associated norm kfk 2
1 = kfk 2 + q(f), and the spectral projector
\Pi I for any interval I =]ff; fi[ for which the boundary does not meet the spectrum (q will
denote also the symmetric bilinear associated form). Then
(i) there exists a constant C ? 0, which depends on I, such that, if f 2 D and – 2 I
