Modular forms of varying weight. I
Let H be the upper half plane, \(\Gamma\) the full modular group and \(v_ r\) the 2r-th power of the multiplier system of the Dedekind eta-function. Let, for \(r\in {\mathbb{R}}\), \(L^ 2(\Gamma \setminus H,r+2\ell,r)\) be the Hilbert space of functions on H, square integrable with respect to \(y^{-2} dx dy\) on \(\Gamma\) \(\setminus H\) and transforming under \(\Gamma\) with weight \(r+2\ell\) and multiplier system \(v_ r\). In \(L^ 2(\Gamma \setminus H,r+2\ell,r)\) the differential operator \(L_{r+2\ell}=-y^ 2(\partial^ 2/\partial x^ 2+\partial^ 2/\partial y^ 2)+(r+2\ell)\partial /\partial x\) acts essentially selfadjointly; the spectrum of \(L_{r+2\ell}\) is discrete if \(r\not\in 12 {\mathbb{Z}}\). One may ask how the spectrum of \(L_{r+2\ell}\) changes with varying r. In this note it is shown that for \(r\in (0,12)\) the space \(L^ 2(\Gamma \setminus H,r,r)\) has an orthonormal basis \(\{\Phi_ j(r):\) \(j\geq 1\}\) with \(\Phi_ j(r)\) and the corresponding eigenvalue \(\mu_ j(r)\) holomorphic in r on a neighborhood \(W_ j\) of (0,12) in \({\mathbb{C}}\). If the Fourier coefficient \(\int^{1}_{0}e^{-2\pi ixr/12} \Phi_ j(r,iy+x) dx\) vanishes for all \(r\in (0,12)\) and all \(y>0\), then \(\mu_ j\) has a holomorphic extension to a neighbourhood of (-12,12). Furthermore, auxiliary results are given to be used in another note to obtain the meromorphic continuation of Eisenstein and Poincaré series in weight and parameter s jointly (where \(1/4-s^ 2\) is the eigenvalue of \(L_ r)\). - The proofs heavily use results from analytic perturbation theory of linear operators. Essential is also the concept of pseudo- Laplacian introduced by \textit{Y. Colin de Verdière} [Ann. Inst. Fourier 32, No.3, 275-286 (1982; Zbl 0489.58034) and ibid. 33, No.2, 87-113 (1983; Zbl 0496.58016)].
scientific article; zbMATH DE number 3880805