溶解尖点形式：高阶费米的黄金法则
Dissolving cusp forms: Higher order Fermi's Golden Rules
摘要：
对于双曲曲面，拉普拉斯算子的嵌入本征值是不稳定的，并且倾向于成为共振。菲利普斯-萨尔纳克确定了充分溶解条件，并在费米黄金法则中得到了优雅的表达。我们证明了更高的近似公式，并获得了充要条件，溶解尖点形式的本征函数$u_j$的共振。在字符品种的扰动框架中，我们将结果与$L$ -系列$L(u_j\otimes F^n, s)$的特殊值联系起来。这是$u_j$与$F(z)^n$的Rankin-Selberg卷积，其中$F(z)$是权重为2的尖点形式的反导数。在一个例子中，我们表明，上述条件迫使嵌入的本征值成为一个穿孔附近的变形空间的共振。
英文摘要：
For a hyperbolic surface embedded eigenvalues of the Laplace operator are unstable and tend to become resonances. A sufficient dissolving condition was identified by Phillips-Sarnak and is elegantly expressed in Fermi's Golden Rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin-Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$ is the antiderivative of a weight 2 cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.
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