多维柱体在远距离扰动下的多重共振
On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders
摘要：
我们考虑一个一般的椭圆算子在一个无限的多维圆柱与几个遥远的扰动，这个运营商是通过“胶合”几个单扰动运营商$\mathcal{H}^{(k)}$，$k=1,\ldots,n$，在大的距离。每个算子$\mathcal{H}^{(k)}$的系数在圆柱的出口处是周期性的;这些周期部分在不同出口处的结构可以不同。本文考虑具有多个远距离扰动的算子的本质谱中的点$λ_0 \in\mathds{R}$，并假定该点不在中间算子$\mathcal{H}^{(k)}$，$k=2,\ldots,n-1$的本质谱中，而是$\mathcal{H}^{(k)}$，$k=1,\ldots,n$中至少一个的本征值.在这样的假设下，我们证明了具有多个远场扰动的算子在$λ_0$附近具有多个共振态。我们发现这些共振的渐近性中的首项，其结果是指数小的。我们还推测，所作的假设选择了唯一的情况下，当遥远的扰动在附近的$λ_0$产生许多共振。也就是说，由于$λ_0$在算子$\mathcal{H}^{(k)}$、$k=2,\ldots,n-1$中的至少一个的本质谱中，我们确实期望在$λ_0$附近出现无限多个共振。
英文摘要：
We consider a general elliptic operator in an infinite multi-dimensional cylinder with several distant perturbations; this operator is obtained by ``gluing'' several single perturbation operators $\mathcal{H}^{(k)}$, $k=1,\ldots,n$, at large distances. The coefficients of each operator $\mathcal{H}^{(k)}$ are periodic in the outlets of the cylinder; the structure of these periodic parts at different outlets can be different. We consider a point $λ_0 \in\mathds{R}$ in the essential spectrum of the operator with several distant perturbations and assume that this point is not in the essential spectra of middle operators $\mathcal{H}^{(k)}$, $k=2,\ldots,n-1$, but is an eigenvalue of at least one of $\mathcal{H}^{(k)}$, $k=1,\ldots,n$. Under such assumption we show that the operator with several distant perturbations possesses finitely many resonances in the vicinity of $λ_0$. We find the leading terms in asymptotics for these resonances, which turn out to be exponentially small. We also conjecture that the made assumption selects the only case, when the distant perturbations produce finitely many resonances in the vicinity of $λ_0$. Namely, as $λ_0$ is in the essential spectrum of at least one of operators $\mathcal{H}^{(k)}$, $k=2,\ldots,n-1$, we do expect that infinitely many resonances emerge in the vicinity of $λ_0$.
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