QUANTUM ERGODICITY FOR COMPACT QUOTIENTS OF IN THE BENJAMINI–SCHRAMM LIMIT

We study the limiting behavior of Maass forms on sequences of large-volume compact quotients of $\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ , $d\ge 3$ , whose spectral parameter stays in a fixed window. We prove a form of quantum ergodicity in this level aspect which extends results of Le Masson and Sahlsten to the higher rank case.

MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .

Asymptotic Behavior of Solutions of the Dirac System with an Integrable Potential

We consider the Dirac system on the interval [0, 1] with a spectral parameter $$\mu \in {\mathbb {C}}$$ μ ∈ C and a complex-valued potential with entries from $$L_p[0,1]$$ L p [ 0 , 1 ] , where $$1\le p$$ 1 ≤ p . We study the asymptotic behavior of its solutions in a strip $$|\mathrm{Im}\,\mu |\le d$$ | Im μ | ≤ d for $$\mu \rightarrow \infty $$ μ → ∞ . These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.

A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.