Estimates on spectral gaps for quantum waveguide Schrödinger operators

该项目的目的是研究描述量子波导和所谓的泄漏量子线的薛定谔算子的光谱特性。对这类模型的研究兴趣来自于介观物理。量子波导描述了电子在半导体结构中移动，产生可以模拟为硬壁的势垒。漏量子线描述了电子在两种不同半导体材料之间的运动，引起有限的势跳跃，描述量子波导的算符由三维欧氏空间中的管或二维欧氏空间中的带的拉普拉斯算子给出。已知即使对于无限长的管/带，非平凡曲率也可以诱导束缚态。在这种情况下，一个自然的问题出现了：一个可以给连续特征值之间的距离估计。像往常一样，频谱间隙的下限比上限更难推导。该项目的目的是推导出这样的下界，特别是对于低位本征值。对于模拟泄漏量子线的薛定谔算子，本征值分裂的下界也可以提出同样的问题。该算子由对应于动能的负拉普拉斯算子和作为势项的奇异相互作用组成。奇异势是吸引的，并且支撑在一个界面流形上。在温和的假设下，这个哈密顿量有束缚态。该项目的目的是再次推导出连续特征值距离的下限。

The aim of the project is to study spectral properties of Schrödinger operators which describe quantum waveguides and so-called leaky quantum wires. The strong interest of studying this kind of models comes from mesoscopic physics. Quantum waveguides describe an electron moving in a semiconductor structure producing potential barriers which can be modelled as hard walls. Leaky quantum wires describe the electron movement between two different semiconductor materials giving rise to a finite potential jump.Operators describing quantum waveguides are given by the Laplacian restricted to a tube in three dimensional Euclidean space or a strip in two dimensional Euclidean space. It is known that even for infinitely long tubes/strips non-trivial curvature can induce bound states. In this situation a natural question arises: can one give estimates on the distance between consecutive eigenvalues. As usual, lower bounds on the spectral gaps are more challenging to derive than upper ones. The aim of the project is to derive such lower bounds, in particular for the low lying eigenvalues.The same question on lower bounds for eigenvalue splittings can be posed for a Schrödinger operator which models leaky quantum wires. The operator consists of the negative Laplacian corresponding to the kinetic energy and a singular interaction as the potential term. The singular potential is attractive and supported on an interface manifold. Under mild assumptions this Hamiltonian has bound states. The aim of the project is again to derive lower bounds for the distance of consecutive eigenvalues.