In cartesian coordinates, the Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\qquad(1)$$ If it’s converted to spherical coordinates, we get $$\nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 \frac{\partial}{\partial r}\right)+\frac{1}{r^2 sin\theta}\frac{\partial}{\partial \theta}\left(sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^2 sin^2 \theta}\frac{\partial^2}{\partial \phi^2}\qquad(2)$$ I am following the derivation (i.e. the method of conversion from cartesian to spherical) in “Quantum physics of atoms, molecules, solids, nuclei […]

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^1)=\{n^2 : n\ \in \mathbb{N}^*\}$, the trace (It this relevant for the question?) as distribution is simply $$w(t)=\sum_{k \geq 1} […]

In my previous question I asked about evaluating the following integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy, $$ and it turned out that the answer is $(4\pi)^2$ for a unit sphere. Now what about the case where $x$ and $y$ are not on the surface of the […]

It is well known that for Dirichlet problem for Laplace equation on balls or half-space, we could use the green function to construct a solution based on the boundary data. For instance, one could find a nice proof in Evans PDE book, chapter 2.2, it is called the Poisson’s formula. Now, it comes to my […]

I am looking to find the radial part of Laplace’s operator in three dimensions. I looked up Laplace’s operator in spherical coordinates and from there I guess the radial part is: $\frac{\partial^2}{\partial r^2} + \frac{2\: \partial}{r\:\partial r}$. Is this correct and is there a better, quicker way to find the radial part?

I am trying to identify for which $\lambda$ so that I can obtain the Maximum principle, that is, $$\sup_{\overline{\Omega}}u\leq \sup_{\partial \Omega}u$$ for equation $-\triangle u = \lambda u$, where the domain $\Omega$ is open bounded. I tried to mimic the prove in Evans book. Indeed, by assuming that $\lambda\leq 0$, I have $$-\triangle u \leq […]

Consider an operator $H=-\Delta +U(x)$ on $L^2(\mathbb R)$ for a function $U(x): \mathbb R \to \mathbb R$ that tends to $+\infty$ as $|x|$ grows. These kinds of operators appear all over non-relativistic quantum mechanics as Hamiltonians. A statement that I have read is that such an operator has a discrete spectrum, and it was presented […]

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ u(x)=\begin{cases} \dfrac{1}{(2-d)\Omega_d}|x|^{2-d}\text{ for } d=1,3,4,5,\ldots\\ \dfrac{1}{2\pi}\log|x| \ \ \ \ \ \ \ \ \ \ \ \ \text{for } d=2, \end{cases} $$ where $\Omega_d$ is the $d$-dimensional solid angle […]

I would like to find the spectrum of a circle. It seems that there are no boundary conditions, but I’m not quite certain. How could I find the spectrum of a circle over the Laplacian operator?

I know, from a recent enlightening answers received here, that, if we define the distribution represented by Dirac’s $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is contained in a compact subset of $\mathbb{R}^3$, then$$\nabla^2\left(\frac{1}{\|\boldsymbol{x}-\boldsymbol{x}_0\|}\right)=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$$where the Laplacian obviously is to be intended in the sense of the derivatives of distributions. […]

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