Schrödinger operators with complex potentials

薛定谔方程描述了量子力学粒子的运动。方程的空间部分由所谓的薛定谔算子控制，薛定谔算子是一种偏微分算子，可以被视为经典哈密顿的量子化。这个算符的本征值(或者更一般地说，谱)是系统的可能能量。能量守恒要求薛定谔算符是自伴的(“对称/厄米特”)；特别是在这种情况下，势必须是实值的。然而，许多有趣的物理现象(共振、能量耗散等)是由具有复值位势的薛定谔算子模拟的。从数学上讲，复势是一个巨大的挑战，而且该理论比只处理实值势的经典对应理论要落后得多。尽管近年来取得了快速的进展，但仍然需要更多的(反)例子。这个项目的目的是构造显式的(反)例子，即从经典理论的观点来看，导致光谱行为的复势的显式(反)例子。候选人将有机会参加我们的LMS联合研究小组的工作坊“非自伴光谱理论中的挑战”。

The Schrödinger equation describes the motion of quantum mechanical particles. The spatial part of the equation is governed by a so-called Schrödinger operator, a partial differential operator that can be viewed as a quantization of the classical Hamiltonian. The eigenvalues (or more generally, the spectrum) of this operator are the possible energies of the system. Conservation of energy requires that the Schrödinger operator is self-adjoint ("symmetric/hermitian"); in particular, the potential must be real-valued in this case. However, many interesting physical phenomena (resonances, dissipation of energy etc.) are modelled by Schrödinger operators with complex-valued potentials. Mathematically, complex potentials pose a significant challenge, and the theory is much less developed than its classical counterpart dealing only with real-valued potentials. Even though rapid progress has been made in recent years, there is a need for more (counter)-examples.The aim of this project is to construct explicit (counter)-examples of complex potentials leading to spectral behaviour that is "unexpected" from the point of view of the classical theory. The candidate will have the opportunity to participate in workshops of our LMS Joint Research Group "Challenges in Non-Self-Adjoint Spectral Theory".