双曲曲面的谱收敛与量子遍历性
Spectral convergence of hyperbolic surfaces and quantum ergodicity
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中文总结 AI 辅助
该研究将Plancherel收敛推广到有限面积双曲曲面，建立其量子遍历性定理，涵盖退化情形，为退化双曲曲面的量子遍历性提供自然谱框架。
中文摘要 AI 辅助
我们将局部紧群紧商群研究中引入的Plancherel收敛概念推广到有限面积双曲曲面，并建立Plancherel序列的量子遍历性定理。我们证明，对于核一致有界且传播一致有界的积分算子，在固定紧区间内的特征值对应的期望值，平均而言会渐近等分布。我们的结果涵盖了Le Masson与Sahlsten研究的乘法算子情形，对应传播界为零且核支撑在对角线上的退化情形。此外，我们的结果允许 systole（最短闭测地线长度）收缩而无一致下界，其允许的退化受Plancherel收敛约束。这表明Plancherel收敛是退化双曲曲面上量子遍历性的自然谱框架。
英文摘要
We extend the notion of Plancherel convergence, introduced in the study of compact quotients of locally compact groups, to finite-area hyperbolic surfaces and establish a quantum ergodicity theorem for Plancherel sequences. We show that the expectation values of integral operators whose kernels are uniformly bounded and have uniformly bounded propagation become asymptotically equidistributed, on average, for eigenvalues in a fixed compact interval. Our result recovers the case of multiplication operators studied by Le Masson and Sahlsten, corresponding to the degenerate case of propagation bound zero and kernels supported on the diagonal. Moreover, our result allows the systole to shrink without a uniform lower bound, with the allowed degeneration constrained by Plancherel convergence. This identifies Plancherel convergence as a natural spectral framework for quantum ergodicity on degenerating hyperbolic surfaces.
