The Maslov Index and the Spectra of Second Order Elliptic Operators
The Maslov Index and the Spectra of Second Order Elliptic Operators
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DOI：
10.1016/j.aim.2018.02.027
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发表时间：
2016-10
期刊：
影响因子：
--
通讯作者：
Y. Latushkin;Selim Sukhtaiev
中科院分区：
文献类型：
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作者：
Y. Latushkin;Selim Sukhtaiev
我们考虑有界 Lipschitz 域 Ω 上的二阶椭圆微分算子。首先，我们在它们的自伴扩张之间建立自然的一一对应关系，定义域包含在 H 1 (Ω) 中，拉格朗日平面包含在 H 1/2 (∂ Ω)× H− 1/2 (∂ Ω) 中。其次，我们推导了一个将此类算子的单参数族谱流与马斯洛夫指数联系起来的公式，马斯洛夫指数是计算 H 1/2 (∂ Ω)× H− 1/2 (∂ Ω) 中拉格朗日平面路径共轭点有符号数的拓扑不变量。此外，我们根据几类二阶算子的马斯洛夫指数计算莫尔斯指数，即负特征值的数量：周期单元 Q⊂ R n 上的 θ→-周期薛定谔算子、具有 Robin 型边界条件的椭圆算子以及星形域上薛定谔算子的抽象自伴扩张。我们的工作建立在 B. Booß-Bavnbek、K. Furutani 和 C. Zhu 最近开发的技术的基础上，并通过将二阶算子的自伴扩展与一阶 Sobolev 空间 H 1 (Ω) 中的域结合起来，扩展了谱流公式的有效性范围。此外，我们还概括了 G. Cox、J. Deng、C. Jones、J. Marzuola、A. Sukhtayev 和作者最近获得的有关 Maslov 指数和 Morse 指数之间关系的结果。最后，我们描述并研究了抽象边界三元组理论与抽象对称算子自伴扩张的拉格朗日描述之间的联系。
We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H 1 (Ω), and Lagrangian planes in H 1/2 (∂ Ω)× H− 1/2 (∂ Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H 1/2 (∂ Ω)× H− 1/2 (∂ Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ→-periodic Schrödinger operators on a period cell Q⊂ R n, the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on star-shaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H 1 (Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators.