相关论文: Non-escape of mass for arithmetic quantum limits o…

We work toward the arithmetic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on hyperbolic $4$-manifolds. We show that limits of such forms can only scar on totally geodesic $3$-submanifolds, and in fact…

We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients $\Gamma\backslash (\mathbb{H}^{(2)})^r \times (\mathbb{H}^{(3)})^s$. An argument by induction on dimension of the orbit…

We eliminate the possibility of "escape of mass" for Hecke-Maass forms of large eigenvalue for the modular group. Combined with the work of Lindenstrauss, this establishes the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak for…

We prove the quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak for the sequence of Pitale lifts, which are Hecke-Maass forms on a congruence quotient of $\mathbb{H}^4$ constructed as lifts from half-integral weight forms…

We establish two new variants of arithmetic quantum ergodicity. The first is for self-dual $\mathrm{GL}_2$ Hecke-Maass newforms over $\mathbb{Q}$ as the level and Laplace eigenvalue vary jointly. The second is a nonsplit analogue wherein…

We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2(…

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the…

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $\Gamma \backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of…

We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over…

We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the…

We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic…

We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke…

We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density $1$ subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound…

It is a folklore result in arithmetic quantum chaos that quantum unique ergodicity on the modular surface with an effective rate of convergence follows from subconvex bounds for certain triple product $L$-functions. The physical space…

Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple…

We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical phase-space…

This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result.

Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $\Gamma\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms…

We prove a quantum ergodicity theorem for sequences of closed hyperbolic surfaces converging to the Poincar\'e disc in the Benjamini-Schramm sense. Assuming a uniform lower bound on the injectivity radius and a spectral gap, we establish…

This short note proves that a Laplacian cannot be quantum uniquely ergodic if it possesses a quasimode of order zero which (i) has a singular limit, and (ii) is a linear combination of a uniformly bounded number of eigenfunctions (modulo an…