# Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction
## 2 Theoretical Model for the Point Interaction  
The present analysis follows the model developed by [PERSON] [19]. The two-dimensional Helmholtz equation for the domain \(\Omega\) with boundary shape \(\delta\Omega\) is given by  
\[\Big{(}\
abla_{2D}^{2}+k_{n}^{2}\Big{)}\psi_{n}(\mathbf{r})=0, \tag{1}\]  
where  
\[\
abla_{2D}^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{ \partial y^{2}} \tag{2}\]  
\(\mathbf{r}=(x,y)\) and \(k_{n}\) and \(\psi_{n}\), with the indices of \(n=1,2,3\cdots\), are the eigenvalues and eigenfunctions, respectively. The issue of quantum billiards is associated with the Dirichlet  
Figure 1: Experimental setups of (**a**) classical and (**b**) modern Chladni plates. boundary conditions with \(\psi_{n}=0\). Considering a d-function potential at \(\mathbf{r}_{s}\) inside the domain, the Helmholtz equation in Equation (1) can be expressed as  
\[\Big{(}\
abla_{2D}^{2}+\alpha\delta(\mathbf{r}-\mathbf{r}_{s})+k^{2}\Big{)} \Psi(\mathbf{r};k)=0, \tag{3}\]  
where the parameter \(\alpha\) is a real number and its absolute value is directly related to the coupling strength. Although the singular potential is not an ordinary function, Equation (3) can be physically analyzed using a truncated basis. The truncation of a basis certainly leads the effective potential to be equivalent to a scatterer with small finite size accompanied with Gibbs-like oscillations. Nevertheless, the finiteness of the scatterer size should be applicable to realistic systems from a physical perspective. In terms of a truncated basis \(\{\,\psi_{n}(\mathbf{r})\,\}\) (\(n=1,2,3\cdots\)), the functions \(\delta(\mathbf{r}-\mathbf{r}_{s})\) and \(\Psi(\mathbf{r})\) in Equation (3) can be approximately expressed as  
\[\delta(\mathbf{r}-\mathbf{r}_{s}) =\sum_{n=0}^{N}\ \psi_{n}^{*}(\mathbf{r}_{s})\ \psi_{n}(\mathbf{r}), \tag{4}\] \[\Psi(\mathbf{r};k) =\sum_{n=0}^{N}\ a_{n}(k)\ \psi_{n}(\mathbf{r}), \tag{5}\]  
Substituting Equations (4) and (5) into Equation (3) and using Equation (1), the eigenfunction of Equation (3) can be derived as  
\[\Psi(\mathbf{r};k)=-\alpha\ \Psi(\mathbf{r}_{s};k)\ G(\mathbf{r},\mathbf{r}_{s} ;k), \tag{6}\]  
where \(G(\mathbf{r},\mathbf{r}_{s};k)\) is the Green function given by  
\[G(\mathbf{r},\mathbf{r}_{s};k)=\sum_{n=0}^{N}\ \frac{\psi_{n}^{*}(\mathbf{r}_{s} )\ \psi_{n}(\mathbf{r})}{k^{2}-k_{n}^{2}}. \tag{7}\]  
The identification \(\delta(\mathbf{r}-\mathbf{r}_{s})\Psi(\mathbf{r};k)=\delta(\mathbf{r}- \mathbf{r}_{s})\Psi(\mathbf{r}_{s};k)\) was used in deriving Equation (6). Setting \(\mathbf{r}=\mathbf{r}_{s}\) in both sides of Equation (6), the transcendental equation for solving the eigenvalues \(k\) is given by \(1+\alpha\ \xi(k)=0\), where \(\xi(k)=G(\mathbf{r}_{s},\mathbf{r}_{s};k)\) is a meromorphic function. It has been discussed [30; 31] that a d-function potential in the Helmholtz equation is problematic in the limit of \(N\rightarrow\infty\) for a complete basis. The renormalization process from the self-adjoint extension theory [30; 31] is employed to treat short-range singularities in a proper manner. Based on the renormalization process, the eigenvalues are determined by  
\[\alpha_{b}^{-1}=\sum_{n=0}^{\infty}\ |\ \psi_{n}(\mathbf{r}_{s})\ |^{2}\Big{(} \frac{1}{k^{2}-k_{n}^{2}}+\frac{k_{n}^{2}}{k_{n}^{4}+1}\Big{)}\, \tag{8}\]  
where \(\alpha_{b}\) is called the renormalized coupling constant. It has been fully discussed [30] that \(\alpha_{b}\) can be related to the physical coupling constant \(\alpha\) defined in Equation (3) with a truncated basis. We use a square-shape quantum billiard to illustrate the relationship between the coupling parameter \(\alpha\) and the spectrum of eigenvalues. For a free square-shape billiard with the region in \(0\leq x,y\leq L\), the eigenfunctions are given by  
\[\psi_{n,m}(\mathbf{r})=\frac{2}{L}\sin\Big{(}\frac{n\pi}{L}x\Big{)}\sin\Big{(} \frac{m\pi}{L}y\Big{)}, \tag{9}\]  
where \(n=1\),\(2\),\(3\cdots\) and \(m=1\),\(2\),\(3\cdots\).