[论文解读] 2D Schrodinger Operator, (2+1) Systems and New Reductions. The 2D Burgers Hierarchy and Inverse Problem Data

P. G. Grinevich, А. Миронов+1 more|arXiv (Cornell University)|May 4, 2010

Spectral Theory in Mathematical Physics被引用 7

一句话总结

本文提出了基于二维Schrödinger算子的(2+1)维可积体系GKMMN系统，并将二维Burgers约化识别为一个关键的可积系统。通过在黎曼曲面上使用代数几何数据，建立了二维Schrödinger算子的新逆谱理论，证明了二维Burgers体系源于具有特定Baker-Akhiezer函数的可约黎曼曲面，并展示了其与自伴磁Pauli算子在谱数据下的相容性。

ABSTRACT

The Theory of (2+1) Systems based on 2D Schrodinger Operator was started by S.Manakov, B.Dubrovin, I.Krichever and S.Novikov in 1976. The Analog of Lax Pairs introduced by Manakov, has a form $L_t=[L,H]-fL$ ("The $L,H,f$-triples") where $L=\partial_x\partial_y+G\partial_y+S$ and $H,f$-some linear PDEs. Their Algebro-Geometric Solutions and therefore the full higher order hierarchies were constructed by B.Dubrovin, I.Krichever and S.Novikov. The Theory of 2D Inverse Spectral Problems for the Elliptic Operator $L$ with $x,y$ replaced by $z,\bar{z}$, was started by B.Dubrovin, I.Krichever and S.Novikov: The Inverse Spectral Problem Data are taken from the complex "Fermi-Curve" consisting of all Bloch-Floquet Eigenfunctions $Lψ=const$. Many interesting systems were found later. However, specific properties of the very first system, offered by Manakov for the verification of new method only, were not studied more than 10 years until B.Konopelchenko found in 1988 analogs of Backund Transformations for it. He pointed out on the "Burgers-Type Reduction". Indeed, the present authors quite recently found very interesting extensions, reductions and applications of that system both in the theory of nonlinear evolution systems (The Self-Adjoint and 2D Burgers Hierarhies were invented, and corresponding reductions of Inverse Problem Data found) and in the Spectral Theory of Important Physical Operators ("The Purely Magnetic 2D Pauli Operators"). We call this system GKMMN by the names of authors who studied it.

研究动机与目标

• 基于二维Schrödinger算子及其Lax三元组形式，发展一类新的(2+1)维可积系统。
• 识别并表征二维Burgers体系作为GKMMN系统的自然约化。
• 建立黎曼曲面上代数几何逆谱数据与非线性演化系统全体系之间的联系。
• 在自伴椭圆算子背景下，证明二维Burgers系统与自伴二维Pauli算子（用于自旋1/2粒子）的相容性。
• 阐明逆问题数据的谱意义，特别是Fermi曲线与Bloch-Floquet本征函数的作用。

提出的方法

• 本文采用Lax三元组形式，其中算子$ L = \partial_x\partial_y + G\partial_y + S $，$ H = \partial_x^2 + F\partial_y + A $，以及标量$f$，满足方程$L_t = [L,H] - fL$。
• 通过在具有两个无穷远点和极点除子$D$的黎曼曲面$\Gamma$上构造两点Baker-Akhiezer函数$\psi(P,x,y,t)$，实现代数几何解的构建。
• 通过设定$S = 0$，导出二维Burgers约化，得到线性系统$c_t - c_{xx} + c_{yy} = (U(x) + V(y))c$，该系统推广了1维Burgers方程。
• 将理论扩展至复化情形$x \to z, y \to \bar{z}$，得到$GKMMN$-II与$B_2$-II系统，且在特定谱条件下保持自伴性。
• 逆谱数据编码于复代数曲线$\Gamma$与Bloch-Floquet本征函数中，谱数据决定势能与磁场。
• 通过规范变换将一般系统约化为标准形式，并证明动力学与实数及自伴约化形式的相容性。

实验结果

研究问题

• RQ1 如何通过Lax三元组形式利用二维Schrödinger算子生成(2+1)维可积体系？
• RQ2 两点Baker-Akhiezer函数在构造GKMMN系统及其约化解中的作用是什么？
• RQ3 二维Burgers体系如何作为GKMMN系统的约化而出现？其谱解释是什么？
• RQ4 在自旋1/2粒子的Pauli哈密顿量背景下，二维Schrödinger算子在何种条件下为自伴算子？
• RQ5 能否从复代数曲线与Bloch-Floquet本征函数完全重构二维Schrödinger算子的逆谱数据？

主要发现

• GKMMN系统被推导为(2+1)维非线性演化系统，其方程为$G_t = G_{xx} - G_{yy} + (F^2)_x - (G^2)_x - A_x + 2S_y$与$S_t = -S_{xx} + S_{yy} + 2(GS)_x - 2(FS)_y$，并满足约束$F_x = 2G_y$，$A_y = 2S_x$，以及$f = 2G_x - F_y$。
• 当$S = 0$时，二维Burgers系统$B_2$作为约化出现，得到线性方程$c_t - c_{xx} + c_{yy} = (U(x) + V(y))c$，该方程推广了1维Burgers方程。
• 二维Burgers体系通过可约黎曼曲面$\Gamma = \Gamma' \cup \Gamma''$与除子$D = D' + D''$构造，其中$\psi$为满足在交点$Q_s$处匹配条件的Baker-Akhiezer函数。
• 在复变量$z, \bar{z}$下的$GKMMN$-II系统中，当磁场与势能为实数时，自伴情形在时间$t \in \mathbb{R}$下保持相容，得到系统$c_t - 4c_{xy} = 8a_y c$与$S_t + 4S_{xy} = 8[S\Phi_{xy} - S_x\Phi_y - S_y\Phi_x]$。
• 当$S = 0$时，二维Burgers系统$B_2$-II约化为$c_t - 4c_{xy} = T(x,y,t)c$，其中$\Delta T = 0$，磁场为$B = -\frac{1}{2}\Delta(\log c) \in \mathbb{R}$，将该体系与纯磁二维Pauli算子联系起来。
• 二维Schrödinger算子的逆谱数据编码于复代数曲线$\Gamma$与Bloch-Floquet本征函数中，且推测非平凡自伴算子的Bloch谱集中，$\Gamma$的Zariski开部分仅在平凡情况下存在。