标题： Schrödinger 算符在牛顿-卡特兰几何中的高阶对称性 显示英文标题

标题： Higher Symmetries of the Schrödinger Operator in Newton-Cartan Geometry

摘要： 我们建立了牛顿-卡特几何的非相对论共形对称性与薛定谔方程之间的若干关系。 特别是，我们讨论了保持平坦牛顿-卡特时空共形不变的矢量场代数 $\mathfrak{sch}(d)$，并证明其曲率推广生成了协变薛定谔方程的对称群，该方程耦合到牛顿势和广义科里奥利力。 我们给出了牛顿-卡特框架下Killing张量和共形Schrödinger-Killing张量的内在定义，并讨论了它们分别与守恒量和薛定谔方程更高对称性的联系。 最后，我们研究了共形对称性在牛顿旋量理论中的作用，在该理论中，旋量空间上全纯矢量场的无限维代数对应于牛顿-卡特时空中代数 $\mathfrak{cnc}(3)$。 显示英文摘要

摘要： We establish several relationships between the non-relativistic conformal symmetries of Newton-Cartan geometry and the Schr\"odinger equation. In particular we discuss the algebra $\mathfrak{sch}(d)$ of vector fields conformally-preserving a flat Newton-Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schr\"odinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing tensors and conformal Schr\"odinger-Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schr\"odinger equation. Finally we consider the role of conformal symmetries in Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to the symmetry algebra $\mathfrak{cnc}(3)$ on the Newton-Cartan spacetime.