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It was proved by [Sel] tha t the Laplacian A(F) for congruence subgroups F of the modular group Fz has an infinite sequence of embedded eigenvalues {Ai} satisfying a Weyl law #{~i~<A}~(rFf/47r)A for A--+cx~. Here I l l is the area of the fundamental domain F of the group F, and the eigenvalues Ai are counted according to multiplicity. The same holds true for the Laplacian A(F; X), where X is a character on F and A(F; X) is associated with a congruence subgroup F1 of F. It is an important question whether this is a characteristic of congruence groups or it may hold also for some non-congruence subgroups of Fz. To investigate this problem Phillips and Sarnak studied per turbat ion theory for Laplacians A(F) with regular per turbat ions derived from modular forms of

Spectral theory of Laplacians for Hecke groups with primitive character