TY - JOUR
T1 - The spectral decomposition of |θ|2
AU - Nelson, Paul D.
PY - 2021/8
Y1 - 2021/8
N2 - Let θ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for ⟨ | θ| 2, φ⟩ as φ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically | θ| 2∉ L 2. Applications to the subconvexity, quantum variance and 4-norm problems are indicated.
AB - Let θ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for ⟨ | θ| 2, φ⟩ as φ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically | θ| 2∉ L 2. Applications to the subconvexity, quantum variance and 4-norm problems are indicated.
KW - Half-integral weight modular forms
KW - Spectral decomposition
KW - Theta functions
UR - http://www.scopus.com/inward/record.url?scp=85097666539&partnerID=8YFLogxK
U2 - 10.1007/s00209-020-02665-8
DO - 10.1007/s00209-020-02665-8
M3 - Journal article
SN - 0025-5874
VL - 298
SP - 1425
EP - 1447
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 3-4
ER -