Sierpinski地毯上Laplacian算子的谱与特征函数的同伦
Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet
摘要：
考虑平面（或更一般的任何欧几里得空间）中的一族有界域$Ω_{t}$，它们解析地依赖于参数$t$，并考虑每个域上的普通诺依曼拉普拉斯算子$Δ_{t}$。然后，我们可以将所有本征函数组织成连续族$u_{t}^{(j)}$，其中本征值$λ_{t}^{(j)}$也随$t$连续变化，尽管本征值的相对大小在$λ_{t}^{(j)}=λ_{t}^{(k)}$的交叉处随$t$变化。我们称这些族为本征函数同伦。我们研究两个明确的例子。第一个例子中，$Ω_{0}$等于一个正方形，$Ω_{1}$等于一个圆;在这两种情况下，本征函数都是显式已知的，所以我们的同伦连接了这两个显式族。在第二个例子中，我们从一个正方形开始近似谢尔宾斯基地毯，并不断删除不同大小的子正方形。(Data全文可查阅www.math.cornell.edu/cnsmh82）
英文摘要：
Consider a family of bounded domains $Ω_{t}$ in the plane (or more generally any Euclidean space) that depend analytically on the parameter $t$, and consider the ordinary Neumann Laplacian $Δ_{t}$ on each of them. Then we can organize all the eigenfunctions into continuous families $u_{t}^{(j)}$ with eigenvalues $λ_{t}^{(j)}$ also varying continuously with $t$, although the relative sizes of the eigenvalues will change with $t$ at crossings where $λ_{t}^{(j)}=λ_{t}^{(k)}$. We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has $Ω_{0}$ equal to a square and $Ω_{1}$ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at www.math.cornell.edu/~smh82)
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