### 数学代写|数论作业代写number theory代考|Courant’s Nodal Domain Theorem

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## 数学代写|数论作业代写number theory代考|Courant’s Nodal Domain Theorem

Let $\phi$ be an eigenfunction of (1.1). The nodal set $\mathcal{Z}(\phi)$ of $\phi$ is defined as the closure of the set of (interior) zeros of $\phi$,
$$\mathcal{Z}(\phi):=\overline{{x \in \Omega \mid \phi(x)=0}}$$
A nodal domain of $\phi$ is a connectèd component of the set $\Omega \backslash \mathcal{Z}(\phi)$. Cäll $\beta_{0}(\phi)$ the number of nodal domains of $\phi$. We recall the following classical theorem $[12$, Chap. VI.6].

Theorem 1.1 (Courant 1923) Assume that the eigenvalues of (1.1) are listed in non-decreasing order, with multiplicities,
$$\mu_{1}<\mu_{2} \leq \mu_{3} \leq \cdots$$
Then, for any eigenfunction $\phi \in \mathcal{E}(\mu)$ of $(1.1)$, associated with the eigenvalue $\mu$,
$$\beta_{0}(\phi) \leq \kappa(\mu)$$
In particular, any $\phi \in \mathcal{E}\left(\mu_{k}\right)$ has a most $k$ nodal domains,
Courant’s theorem is a partial generalization, to higher dimensions, of a classical theorem of C. Sturm (1836). Indeed, in dimension 1, a $k$ th eigenfunction of the Sturm-Liouville operator $-\frac{d^{2}}{d x^{2}}+q(x)$ in $] a, b[$, with Dirichlet, Neumann, or mixed Dirichlet-Neumann boundary condition at ${a, b}$, has exactly $k$ nodal domains in ]$a, b[$. In dimension 2 (or higher), Courant’s theorem is not sharp. On the one hand, Stern (1925) proved that for the square with Dirichlet boundary condition, or for the 2-sphere, there exist eigenfunctions of arbitrarily high energy, with exactly two or

three nodal domains. On the other hand, Pleijel (1956) proved that, for any bounded domain in $\mathbb{R}^{2}$, there are only finitely many Dirichlet eigenvalues for which Courant’s theorem is sharp. We refer to $[7,24]$ for more details, and to [20] for Pleijel’s estimate under Neumann boundary condition.

Another remarkable theorem of Sturm states that any non trivial linear combination $u=\sum_{k=m}^{n} a_{j} u_{j}$ of eigenfunctions of the operator $-\frac{d^{2}}{d x^{2}}+q(x)$ has at most $(n-1)$ zeros (counted with multiplicities), and at least $(m-1)$ sign changes in the interval $] a, b[$, see [10].

A footnote in [12, p. 454], states that Courant’s theorem may be generalized as follows: Any linear combination of the first $n$ eigenfunctions divides the domain, by means of its nodes, into no more than $n$ subdomains. See the Göttingen dissertation of H. Herrmann, Beiträge zur Theorie der Eigenwerten und Eigenfunktionen, $1932 .$ For later reference, we introduce the following definition.

## 数学代写|数论作业代写number theory代考|Symmetries and Spectra

In this subsection, we analyze how symmetries influence the structure of the eigenvalues and eigenfunctions. The analysis is carried out for the equilateral rhombus, but the basic ideas work for the regular hexagon as well, and will be used in Sect. 3 .
In the sequel, we denote by the same letter $L$ a line in $\mathbb{R}^{2}$, and the mirror symmetry with respect to this line. We denote by $L^{}$ the action of the symmetry $L$ on functions, $L^{} \phi=\phi \circ L$.

A function $\phi$ is even (or invariant) with respect to $L$ if $L^{} \phi=\phi$. It is odd (or anti-invariant) with respect to $L$ if $L^{} \phi=-\phi$. In the former case, the line $L$ is an anti-nodal line for $\phi$, i.e., the normal derivative $v_{L} \cdot \phi$ is zero along $L$, where $v_{L}$ denotes a unit field normal to $L$ along $L$. In the latter case, the line $L$ is a nodal line for $\phi$, i.e., $\phi$ vanishes along $L$.

Let $\mathcal{R} h_{e}$ be the interior of the equilateral rhombus with sides of length 1 , and verticês $\left.\left(-\frac{\sqrt{3}}{2}, 0\right),\left(0,-\frac{1}{2}\right),\left(\frac{\sqrt{3}}{2}\right), 0\right)$ ând $\left(0, \frac{1}{2}\right)$. Câll $D$ ând $M$ its diagononâls (rêsp̄. the longer one and the shorter one). The diagonal $M$ divides the rhombus into two equilateral triangles. The diagonals $D$ and $M$ divide the rhombus into four hemiequilateral triangles. In the sequel, we use the generic notation $\mathcal{T}{e}$ (resp. $\mathcal{T}{h}$ ) for any of the equilateral triangles (resp. hemiequilateral triangles) into which the rhombus decomposes, see Fig. 1 .
For $L \in{D, M}$, define the sets
$$\left{\begin{array}{l} \mathcal{S}{L,+}=\left{\phi \in L^{2}\left(\mathcal{R} h{e}\right) \mid L^{} \phi=+\phi\right. \ \mathcal{S}{L,-}=\left{\phi \in L^{2}\left(\mathcal{R} h{e}\right) \mid L^{} \phi=-\phi\right. \end{array}\right}$$

## 数学代写|数论作业代写number theory代考|Riemann-Schwarz Reflection Principle

In this subsection, we recall the “Riemann-Schwarz reflection principle” which we will use repeatedly in the sequel.

Consider the decomposition $\mathcal{R} h_{e}=\mathcal{T}{e, 1} \bigsqcup \mathcal{T}{e, 2}$, with $M\left(\mathcal{T}{e, 1}\right)=\mathcal{T}{e, 2}$. Choose a boundary condition $\mathfrak{a} \in{\mathfrak{d}, \mathrm{n}}$ on $\partial \mathcal{R} h_{e}$. Given an eigenvalue $\lambda$ of $-\Delta$ for $\left(\mathcal{R} h_{e}, \mathfrak{a}\right)$,

and $\sigma \in{+,-}$, consider the subspace $\mathcal{E}(\lambda) \cap \mathcal{S}{M, \sigma}$ of eigenfunctions $\phi \in \mathcal{E}(\lambda)$ such that $M^{} \phi=\sigma \phi$. If $0 \neq \phi \in \mathcal{E}(\lambda) \cap \mathcal{S}{M, \sigma}$, then $\phi \mid \mathcal{T}{e, 1}$ is an eigenfunction of $-\Delta$ for $\left(\mathcal{T}{e, 1}, \mathfrak{a} \mathfrak{b}\right)$,
with $\mathfrak{b}=\mathfrak{n}$ if $\sigma=+$, and $\mathfrak{b}=\mathfrak{d}$ if $\sigma=-$, associated with the same eigenvalue $\lambda$.
Conversely, let $\psi$ be an eigenfunction of $\left(\mathcal{T}{e, 1}, \mathfrak{a} \mathfrak{a} \mathfrak{b}\right)$, with eigenvalue $\mu{m}\left(\mathcal{T}{e, 1}, \mathfrak{a} \mathfrak{a} \mathfrak{b}\right)$, for some $m \geq 1$. Define the function $\breve{\psi}$ on $\mathcal{R} h{e}$ such that $\breve{\psi} \mid \mathcal{T}{e, 1}=\psi$ and $\breve{\psi} \mid \mathcal{T}{e, 2}=$
and $\sigma \in{+,-}$, consider the subspace $\mathcal{E}(\lambda) \cap \mathcal{S}{M, \sigma}$ of eigenfunctions $\phi \in \mathcal{E}(\lambda)$ such that $M^{} \phi=\sigma \phi$.
If $0 \neq \phi \in \mathcal{E}(\lambda) \cap \mathcal{S}{M, \sigma}$, then $\phi \mid \mathcal{T}{e, 1}$ is an eigenfunction of $-\Delta$ for $\left(\mathcal{T}{e, 1}, \mathfrak{a a b}\right)$,
with $\mathfrak{b}=\mathfrak{n}$ if $\sigma=+$, and $\mathfrak{b}=\mathfrak{d}$ if $\sigma=-$, associated with the same eigenvalue $\lambda$.
Conversely, let $\psi$ be an eigenfunction of $\left(\mathcal{T}{e, 1}, \mathfrak{a} \mathfrak{a} b\right)$, with eigenvalue $\mu{m}\left(\mathcal{T}{e, 1}\right.$, a a $\left.\mathfrak{b}\right)$, for some $m \geq 1$. Define the function $\breve{\psi}$ on $\mathcal{R} h{e}$ such that $\breve{\psi} \mid \mathcal{T}{e, 1}=\psi$ and $\breve{\psi} \mid \mathcal{T}{e, 2}=$
$\sigma \psi \circ M$. This means that $\breve{\psi}$ is obtained by extending $\psi$ across $M$ to $\mathcal{T}{e, 2}$ by sym- metry, in such a way that $M^{} \breve{\psi}=\sigma \breve{\psi}$. It is easy to see that the function $\bar{\psi}$ is an eigenfunction of $-\Delta$ for $\left(\mathcal{R} h{e}, \mathfrak{a}\right)$ (in particular it is smooth in a neighborhood of
$M)$, with eigenvalue $\mu_{m}\left(\mathcal{T}{e, 1}, \mathfrak{a a b}\right)$, so that $\breve{\psi} \in \mathcal{E}\left(\mu{m}\right) \cap \mathcal{S}{M, \sigma}$. The above considerations prove the first two assertions in the following proposi- tion. The proof of the third and fourth assertions is similar, using the symmetries $D$ and $M$, and the decomposition of $\mathcal{R} h{e}$ into hemiequilateral triangles $\mathcal{T}{h, j}, 1 \leq j \leq 4$. $\sigma \psi \circ M$. This means that $\psi$ is obtained by extending $\psi$ across $M$ to $\mathcal{T}{e, 2}$ by sym-
metry, in such a way that $M^{} \psi=\sigma \breve{\psi}$. It is easy to see that the function $\dot{\psi}$ is an
eigenfunction of $-\Delta$ for $\left(\mathcal{R} h_{e}\right.$, a) (in particular it is smooth in a neighborhood of
$M)$, with eigenvalue $\mu_{m}\left(\mathcal{T}{e, 1}, \mathfrak{a} a \mathfrak{b}\right)$, so that $\breve{\psi} \in \mathcal{E}\left(\mu{m}\right) \cap \mathcal{S}{M, \sigma}$. The above considerations prove the first two assertions in the following proposition. The proof of the third and fourth assertions is similar, using the symmetries $D$ and $M$, and the decomposition of $\mathcal{R} h{e}$ into hemiequilateral triangles $\mathcal{T}_{h, j}, 1 \leq j \leq 4$.

## 数学代写|数论作业代写number theory代考|Courant’s Nodal Domain Theorem

μ1<μ2≤μ3≤⋯

b0(φ)≤ķ(μ)

Courant 定理是 C. Sturm (1836) 的经典定理向更高维度的部分推广。实际上，在维度 1 中，aķSturm-Liouville 算子的特征函数−d2dX2+q(X)在]一个,b[, Dirichlet、Neumann 或混合 Dirichlet-Neumann 边界条件为一个,b, 正好ķ]中的节点域一个,b[. 在维度 2（或更高维度）中，Courant 定理并不尖锐。一方面，Stern (1925) 证明，对于具有 Dirichlet 边界条件的正方形，或者对于 2 球体，存在任意高能量的特征函数，恰好有两个或

Sturm 的另一个显着定理指出，任何非平凡的线性组合在=∑ķ=米n一个j在j算子的特征函数−d2dX2+q(X)最多有(n−1)零（以重数计算），并且至少(米−1)区间内的符号变化]一个,b[，见[10]。

[12, p. 中的脚注。454]，指出 Courant 定理可以概括如下：neigenfunctions 通过其节点将域划分为不超过n子域。参见 H. Herrmann 的 Göttingen 论文，Contributions to the theory of eigenvalues and eigenfunctions，1932.为了以后的参考，我们引入以下定义。

## 数学代写|数论作业代写number theory代考|Symmetries and Spectra

$$\left{\begin{array}{l} \mathcal{S}{L,+}=\left{\phi \in L^{2}\left(\mathcal{R} h{ e}\right) \mid L^{} \phi=+\phi\right. \ \mathcal{S}{L,-}=\left{\phi \in L^{2}\left(\mathcal{R} h{e}\right) \mid L^{} \phi=-\ φ\对。 \end{数组}\right}$$

## 数学代写|数论作业代写number theory代考|Riemann-Schwarz Reflection Principle

σψ∘米. 这意味着ψ˘通过扩展获得ψ穿过米至吨和,2通过对称，以这样的方式米ψ˘=σψ˘. 很容易看出函数ψ¯是一个特征函数−Δ为了(RH和,一个)（特别是在附近很光滑

，以这样的方式米ψ=σψ˘. 很容易看出函数ψ˙是一个

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