标题： 新的三维薛定谔方程精确解 显示英文标题

标题： New exact solutions of the 3D Schrödinger equation

摘要： 此前我们发现了一个具有正的规范不变 Weyl-Stratonovich 准概率密度函数的独特量子系统，该函数可以通过所谓的{\guillemotleft }二次漏斗{\guillemotright }势来定义 [Phys. Rev. A 110 02222 (2024)]。在这项工作中，我们基于微观和宏观系统的 -模型，构造了一类三维薛定谔方程对于双参数{\guillemotleft }二次漏斗{\guillemotright }势的精确解。找到了能量谱和本征函数集的显式表达式。利用标量势和矢量势的规范不变性，得到了电磁薛定谔方程的解，其中磁场以由奇异涡旋概率流场定义的{\guillemotleft }磁通弦{\guillemotright }的形式存在。详细研究了导致各种类型的涡旋和势概率电流场的本征函数叠加。量子系统的性质分析是在 Wigner-Vlasov 形式主义框架内进行的。 显示英文摘要

摘要： Previously we found a unique quantum system with a positive gauge-invariant Weyl-Stratonovich quasi-probability density function which can be defined by the so-called {\guillemotleft}quadratic funnel{\guillemotright} potential [Phys. Rev. A 110 02222 (2024)]. In this work we have constructed a class of exact solutions to the 3D Schr\"odinger equation for a two-parameter {\guillemotleft}quadratic funnel{\guillemotright} potential based on the -model of micro and macro systems. Explicit expressions for the energy spectrum and the set of eigenfunctions have been found. Using gauge invariance for scalar and vector potentials, a solution to the electromagnetic Schr\"odinger equation has been obtained, with a magnetic field in the form of a {\guillemotleft}Dirac string{\guillemotright} defined by a singular vortex probability flux field. Superpositions of eigenfunctions leading to various types of vortex and potential probability current fields have been investigated in detail. The analysis of the quantum system's properties has been carried out within the Wigner-Vlasov formalism.