Schrödinger operators with complex potentials

Schrödinger方程描述了量子力学粒子的运动。方程的空间部分由一个所谓的Schrödinger算子控制，一个偏微分算子可以看作是经典哈密顿算子的量子化。这个算子的特征值（或者更一般地说，谱）是系统的可能能量。能量守恒要求Schrödinger算子是自伴随的（“对称/厄米”）；特别地，在这种情况下，势必须是实值。然而，许多有趣的物理现象（共振、能量耗散等）是用Schrödinger复值势算子来模拟的。在数学上，复势提出了重大的挑战，并且该理论比只处理实值势的经典对应物要落后得多。尽管近年来取得了迅速的进展，但仍需要更多的（反）例子。这个项目的目的是构建明确的（反）复杂势的例子，导致从经典理论的角度来看是“意想不到的”光谱行为。候选人将有机会参加LMS联合研究小组“非自伴随谱理论的挑战”的研讨会。

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".