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Suppose t h a t A > 0 and l e t 0 < X 5 A _ . Then t h e equation ( A + X ) u = 0 does not admit a p o s i t i v e s o l u t i o n i n M which i s G-multiplicative. + It i s w e l l known Cseg I n t r o d u c t i o n ) t h a t the equation ( A A)u = 0 admits Thus Theorem 4 . 2 shows t h a t Theorem 4 . 1 a p o s i t i v e s o l u r i o n on M i f X 5 A i s f a l s e i f A > 0. This will be f o r i n s t a n c e t h e case i f M h a s a negative s e c t i o n a l curvature (see McKean 171). Recently Brooks [41 has shown t h a t A = 0 i f and o n l y i f wl(M) i s a n amenable group.

For further details of the work presented here see 121. In the present context we define resonance energies as follows: Definition. e -+Ex tion $ satisfying $(O) = 0 and $ ( x ) as x + -. (Our convention cn the square root function is that it is positive on the positive real axis and has a cut along the negative real axis). Remark. [5,7] The condition IJJ(x)-eifi boundary condition. is called an outgoing wave or Gamow-Siegert Since our potential vanishes for x > a this is equi- valent to the condition $ ' (a) = i& $(a) so that our problem may be viewed as a (non-self-adjoint) Sturm-Liouville problem on the interval [Oral with eigenvalue parameter appearing in the boundary conditions.

Given a real valued function u in M one says the u is G-multiplicative if for any g E G: for all x u(gx> = y,u(x) f4 in 'a is a constant. When yg = 1 for all g in G one says that u is a G-invariant funstion. Similarly one defines the notion of a G-invariant differential on M. It is clear that a G-invariant function (differential) on M can be considered as a function (differential) on M. where Let now u be a positive G-multiplicative function of class C1 on M. Clearly dlogu is a G-invariant differential on M and thus can be considered as a closed differential on M.

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