Comparison of Maaß Wave Forms

In this note we will discuss various ways to deduce the equality of two cusp forms. Primarily we will work with non-holomorphic cusp forms of weight zero on the classical modular group, although some generalizations will be mentioned. Theorems of this type are often also referred to as multiplicity one theorems. They have been studied by many individuals and many interesting results have been established. Our approach has two goals. We would like to establish theorems that require only approximate amounts of information. The “information” about the two cusp forms we will examine is the set of Fourier coefficients, and the Laplace eigenvalues. The reason that we want to use only approximate information is that, unlike the holomorphic case, Maaß cusp forms have no integrality conditions on their Fourier coefficients, so saying that two numbers are equal is a rather strong statement. The second goal is to have, in some of our theorems, results that lead to bounds on the multiplicity of the Laplace spectrum.