论文标题
Schrödinger操作员在无限缸和其他产品上的共鸣
Resonances for Schrödinger operators on infinite cylinders and other products
论文作者
论文摘要
我们研究了schrödinger运营商在无限产品上的共鸣$ x = \ mathbb {r}^d \ times \ times \ mathbb {s}^1 $，其中$ d $是奇数，$ \ mathbb {s}^1 $是单位圈，并且潜在的$ v \ in l^\ in l^\ in^\ in^\ infty_c（x）$。 This paper shows that at high energy, resonances of the Schrödinger operator $-Δ+V$ on $X=\mathbb{R}^d\times \mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-Δ+V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle.这些共振反过来是根据$ \ mathbb {r}^d $在有限集中的$ \ mathbb {r}^d $上给出的共鸣的。如果电势光滑，我们将获得共振的改进定位，特别是在简单的，排名相应的散射的一个极点上，在$ \ mathbb {r}^d $上分辨出来。在这种情况下，我们获得了相应高能共振位置的领先顺序校正。除了有关共振位置的直接结果外，我们还表明，在高能远离共振的情况下，模型运营商$-Δ+v_0 $的分解$ x $ x $近似于$ x $上的$-δ+v $。如果$ d = 1 $，在某些情况下，这意味着波动方程溶液的渐近扩展存在。再次，对于$ d = 1 $的特殊情况，我们获得了所有实现潜力之间零电位的共振刚度结果。
We study the resonances of Schrödinger operators on the infinite product $X=\mathbb{R}^d\times \mathbb{S}^1$, where $d$ is odd, $\mathbb{S}^1$ is the unit circle, and the potential $V\in L^\infty_c(X)$. This paper shows that at high energy, resonances of the Schrödinger operator $-Δ+V$ on $X=\mathbb{R}^d\times \mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-Δ+V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schrödinger operator on $\mathbb{R}^d$ which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank one poles of the corresponding scattering resolvent on $\mathbb{R}^d$. In that case, we obtain the leading order correction for the location of the corresponding high energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator $-Δ+V_0$ on $X$ approximates that of $-Δ+V$ on $X$. If $d=1$, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of $d=1$, we obtain a resonant rigidity type result for the zero potential among all real-valued potentials.
