论文标题
在二维有吸引力的高斯杂质的情况下
The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity
论文作者
论文摘要
在本说明中，我们考虑了一个量子机械粒子在$ x $方向上受抛物线孔约束的无限薄层内移动的量子粒子，此外，在存在以有吸引力的高斯电位建模的杂质的情况下。我们研究了与模型相关的Birman-Schwinger操作员，假设该层内存在高斯杂质，并证明了这种积分操作员是Hilbert-Schmidt，它允许使用修改的Fredholm裁决物来计算Indrurity产生的结合状态。此外，我们考虑了高斯电位在$ x $方向上脱离$δ$的情况，而高斯在$ y $方向上的潜力。我们构建了相应的自我伴侣汉密尔顿人，并证明这是一系列具有适当缩放高斯电位的相应哈密顿量的序列的限制。展出了所有涉及的所有汉密尔顿人的基态能量的令人满意的界限。
In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the $x$-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a $δ$-potential in the $x$-direction and a Gaussian potential in the $y$-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited.
