Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series

Abstract

The problem of quantum unique ergodicity (QUE) of weight $\frac{1}{2}$ Eisenstein series for ${\Gamma }_0\left(4\right)$ leads to the study of certain double Dirichlet series involving ${GL}_2$ automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.