Fourier transform, Fourier integral

$\renewcommand{\Re}{\operatorname{Re}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$

Eigenvalues of Laplacian in the disk and a ball


  1. From wave equation to Helmholtz equation
  2. Separation of variables in polar coordinates
  3. Separation of variables in spherical coordinates
  4. Separation of variables in cylindrical coordinates

From wave equation to Helmholtz equation

Consider wave equation \begin{equation} u_{tt}-c^2 \Delta u =0. \label{equ-LE.1} \end{equation} Separating $t$ from spatial variables $(x,y,z)$ we get as usual $T(t)=e^{i\omega t}$ and $v= v(x,y,z)$ satisfying \begin{equation} (\Delta +\lambda )v =0 \label{equ-LE.2} \end{equation} with the corresponding boundary condition and with $\lambda =k^2=\omega^2 c^2$. Equation (\ref{equ-LE.2}) is called Helmholtz equation.

Separation of variables in polar coordinates

Using $\Delta v= v_{rr}+r^{-1}v_r + r^{-2}v_{\theta\theta}$ and separating $r$ and $\theta$: $v=R(r)\Theta (\theta)$ we arrive to \begin{equation*} (R'' + r^{-1}R'+k^2R) \Theta + r^{-2} R\Theta ''=0 \end{equation*} which can be rewritten as \begin{equation*} \frac{r^2R'' + rR'+r^2k^2R}{R} + \frac{\Theta ''}{\Theta}=0 \end{equation*} and then $\frac{\Theta ''}{\Theta}=-\mu$ and since we consider a disk we must assume that $\Theta$ is $(2\pi)$-periodic and therefore $\Theta =e^{\pm im\theta}$ and $\mu =-m^2$, $m=0,1,2,\ldots$. Then \begin{equation} r^2R'' + rR'+(r^2k^2 - m^2) R=0. \label{equ-LE.3} \end{equation} In this equation parameter $k$ is superficial and we can make it $1$. Indeed, scaling $x= k r$ (it is not an original Cartesian coordinate) we observe that equation becomes \begin{equation} x^2R'' + xR'+(x^2 - m^2) R=0. \label{equ-LE.4} \end{equation} This is Bessel equation and its solutions (bounded at $0$--as our domain is a disk $\mathcal{D}=\{r < a\}$) are Bessel functions $J_m (x)$ -- see Bessel function $J_m$.

Plugging $x=kr$ we get $R(r)=J_m (kr)$ and then

  1. Under Dirichlet boundary condition $u|_{r=a}=0$ we have $R(a)=0$ which means \begin{equation} J_m(ka)=0 \iff k_{mn}= a^{-1} s_{mn} \label{equ-LE.5} \end{equation} where $s_{mn}$ is $n$-th $0$ of $J_m(x)$.

  2. Under Neumann boundary condition $u|_{r=a}=0$ we have $R'(a)=0$ which means \begin{equation} J'_m(ka)=0 \iff k_{mn}= a^{-1} s'_{mn} \label{equ-LE.6} \end{equation} where $s'_{mn}$ is $n$-th $0$ of $J'_m(x)$.

Separation of variables in spherical coordinates

Using $\Delta v= v_{\rho\rho}+2\rho^{-1}v_r + \rho^{-2}\Lambda v$ where $\Lambda$ is a Laplace-Beltrami operator on the sphere and separating $\rho$ and $(\phi,\theta)$: $v=P(\rho)Y(\phi,\theta)$ we arrive to \begin{equation*} (P'' + 2\rho^{-1}P'+k^2P) Y + \rho{-2} R\Lambda Y=0 \end{equation*} which can be rewritten as \begin{equation*} \frac{P'' + 2\rho^{-1}P'+k^2P}{P} + \frac{\Lambda Y}{Y}=0 \end{equation*} and then $\frac{\Lambda Y}{Y}=-\mu$ and since we consider a ball we must assume that $Y$ is a spherical harmonic and $\mu =-l(l+1)$, $l=0,1,2,\ldots$. Then \begin{equation} \rho ^2P'' + \rho P'+(\rho ^2k^2 - l(l+1)^2) P=0. \label{equ-LE.7} \end{equation} In this equation parameter $k$ is superficial and we can make it $1$. Indeed, scaling $x= k \rho$ (it is not an original Cartesian coordinate) we observe that equation becomes \begin{equation} x^2P'' + 2xP'+(x^2 - l(l+1)^2) P=0. \label{equ-LE.8} \end{equation} One can rewrite this equation as (\ref{equ-LE.4}) with $R=x^{\frac{1}{2}} P$ and $m=(l+\frac{1}{2}$ and therefore we arrive to Bessel functions with half-integer parameter $m$: $R= J_{l+\frac{1}{2}}(x)$ (we are looking for its solutions bounded at $0$--as our domain is a ball $\mathcal{B}=\{\rho < a\}$) are Bessel functions $J_m (x)$.

However in contrast to Bessel functions with integer parameter $l$ solutions to (\ref{equ-LE.8}) known as spherical Bessel functions $j_l(x)$ are elementary functions--see spherical Bessel functions $j_l(x)$ .

Plugging $x=k\rho$ we get $P(\rho)=j_l (k\rho)$ and then

  1. Under Dirichlet boundary condition $u|_{\rho=a}=0$ we have $P(a)=0$ which means \begin{equation} j_l(ka)=0 \iff k_{ln}= a^{-1} s_{mn} \label{equ-LE.9} \end{equation} where $s_{ln}$ is $n$-th $0$ of $j_l(x)$.

  2. Under Neumann boundary condition $u|_{\rho=a}=0$ we have $P'(a)=0$ which means \begin{equation} j'_l(ka)=0 \iff k_{ln}= a^{-1} s'_{ln} \label{equ-LE.10} \end{equation} where $s'_{ln}$ is $n$-th $0$ of $j'_l(x)$.

Separation of variables in cylindrical coordinates

Consider cylinder $\{ r< a, 0< z < b\}$ and Laplace equation in it with Dirichlet boundary conditions on both lids: $u|_{z=0}=u_{z=b}=0$. Separating variable $z$ from other spatial variables $u(x,y,z)=v(x,y)Z(x)$ we arrive to $Z''+\mu Z=0$, $(\Delta _{x,y} +\mu )v=0$ and therefore $\mu = m^2\pi^2 b^{-2}$, $Z=\sin (m\pi z/b)$ and $(\Delta _{x,y}-k^2)v =0$ with $k= m\pi b^{-1}$. Continuing further we arrive to \begin{equation} r^2R'' + rR'-(r^2k^2 + m^2) R=0. \label{equ-LE.11} \end{equation} In this equation parameter $k$ is superficial and we can make it $1$. Indeed, scaling $x= k r$ (it is not an original Cartesian coordinate) we observe that equation becomes \begin{equation} x^2R'' + xR'-(x^2 + m^2) R=0. \label{equ-LE.12} \end{equation} This equation could be reduced to the original Bessel equation by replacing $x$ by $ix$ (thus going into complex domain).