论文标题
优化软量子环的最低特征值
Optimization of the lowest eigenvalue of a soft quantum ring
论文作者
论文摘要
我们考虑自我辅助二维schrödinger操作员$h_μ$与差分表达式相关的$-Δ-μ$，描述了一种粒子，暴露于封闭曲线胶带中支撑的$μ$的粒子，并具有固定的横向横向一维轮廓度量$μ___\ bot $。该运算符具有非空负离散频谱，我们为其最低特征值获得了两个优化结果。对于第一个，我们修复$μ_\ bot $，并最大程度地提高最低特征值，相对于曲线条的形状，第一个问题中的优化器证明是环。我们还将这一结果推广到情况，该情况涉及$H_μ$的额外扰动，其形式是由曲面条包围的域的特征函数的正倍数。其次，我们固定曲线条带的形状，并根据$μ__\ bot $的变化将最低特征值最小化，这是在总体配置文件量$α> 0 $的限制下是固定的。此问题中的优化器为$μ_\ bot $，由$α$的产品和最佳位置支持的dirac $δ$ function。
We consider the self-adjoint two-dimensional Schrödinger operator $H_μ$ associated with the differential expression $-Δ-μ$ describing a particle exposed to an attractive interaction given by a measure $μ$ supported in a closed curvilinear strip and having fixed transversal one-dimensional profile measure $μ_\bot$. This operator has nonempty negative discrete spectrum and we obtain two optimization results for its lowest eigenvalue. For the first one, we fix $μ_\bot$ and maximize the lowest eigenvalue with respect to shape of the curvilinear strip the optimizer in the first problem turns out to be the annulus. We also generalize this result to the situation which involves an additional perturbation of $H_μ$ in the form of a positive multiple of the characteristic function of the domain surrounded by the curvilinear strip. Secondly, we fix the shape of the curvilinear strip and minimize the lowest eigenvalue with respect to variation of $μ_\bot$, under the constraint that the total profile measure $α>0$ is fixed. The optimizer in this problem is $μ_\bot$ given by the product of $α$ and the Dirac $δ$-function supported at an optimal position.
