2019-08-012024-05-18https://scholars.lib.ntu.edu.tw/handle/123456789/699386摘要：特徵值問題 (eigenvalue problem) 一直是數學和應用數學領域的重要問題。傳統上，線性 偏微分方程 (如線性薛丁格方程) 以線性泛函分析的方法，建構變分學理論用來解決特徵值 問題。然而關於非線性偏微分方程 (如非線性薛丁格方程)之特徵值問題似乎沒有任何理論被 建立。本計畫將研究可飽和 (saturable) 與平方根 (square-root) 非線性薛丁格方程組之 特徵值問題，推導特徵值估計理論並用來估計基態 (ground state) 能量。預期將發展一套 方法可用來解決在L2 範數條件下能量極小化問題與其所對應之Lagrange 乘數估計。另外將 針對具有空間效應 (steric effect) 之擴散 (diffusion) 方程組研究其特徵值問題與估計 方法。<br> Abstract: Eigenvalue problem is important in the fields of mathematics and applied mathematics. Conventionally, one may use methods of linear functional analysis to study the eigenvalue problem of linear partial differential equations (including linear Schrodinger equations) and develop theorems of calculus of variation. However, it seems that there is no theorem for the eigenvalue problem of nonlinear partial differential equations (including nonlinear Schrodinger equations). In this project, we propose to study the eigenvalue problem of nonlinear Schrodinger systems with saturable and square root nonlinearities and derive the estimate of eigenvalue and ground state energy. It is expected to have a method for the energy minimization problems under the L2 norm constraints and the estimate of the associated Lagrange multipliers. Moreover, we propose to study the eigenvalue problem of a diffusion system with steric effects and develop a method for the eigenvalue estimate.