Toronto Math Forum
APM3462018S => APM346Lectures => Topic started by: Jingxuan Zhang on February 27, 2018, 10:12:26 AM

In chap 5.2 repeatedly I find "we need to take a correct branch". For instance, for the Schrodinger equation with $t\gtrless 0$, what in fact is the reason of taking $\sqrt{i}=\pm e^{i\pi/4}$?

If we solve a crossbreed of Schr\"odinger and Heat
\begin{equation}
u_t =au_{xx}
\tag{1}
\end{equation}
with $a\in \mathbb{C}$ with the positive real part (which is wellposed in the direction of positive $t$) we get a formula with kernel $\frac{1}{\sqrt {4\pi a t}} e^{x^2/4at}$. This square root is positive for real $a>0$.
Let us look, what happens if $a=bi+\varepsilon$, $\varepsilon \to +0$ and $b>0$. We get
$\frac{1}{\sqrt {4\pi (bi+\varepsilon) t}} e^{x^2/4(bi+\varepsilon)t}$. But if $\varepsilon \to +0$ then this equation (1) becomes Schr\"odinger equation
\begin{equation}
u_t =ibu_{xx}
\tag{2}
\end{equation}
exponent factor tends to $e^{x^2/4b i t}=e^{ix^2/4b t}$, but since $bi +\varepsilon \to bi$ from the right, then its argument tends to $\pi/2$ from below, and argument of the square root tends to ${\pi/4}$, so we get
\begin{equation}
\frac{1}{\sqrt {4\pi bt}} e^{i\pi/4}e^{ix^2/4b t}
\tag{3}
\end{equation}
Recall that $b>0$, $t>0$.
The same is true for $b<0, t<0$. Indeed, changing $t\mapsto t$, $b\mapsto b$ preserves equation (2).
So (3) holds for $bt>0$. But taking complex conjugation and $b\to b$ also preserves (2). This operation with (3) brings
\begin{equation}
\frac{1}{\sqrt {4\pi bt}} e^{i\pi/4}e^{ix^2/4b t}.
\tag{4}
\end{equation}
So, correct formula (I suspect I skipped on the lecture) for convolution kernel is
\begin{equation}
\frac{1}{\sqrt {4\pi bt}} e^{\mp i\pi/4}e^{ix^2/4b t}\qquad \text{for} \ \ bt \gtrless 0.
\tag{5}
\end{equation}

Indeed, changing $t\mapsto t$, $b\mapsto b$ preserves equation (2).
So (3) holds for $bt>0$. But taking complex conjugation and $b\mapsto b$ also preserves (2). This operation with (3) brings
\begin{equation}
\frac{1}{\sqrt {4\pi bt}} e^{i\pi/4}e^{ix^2/4b t}.
\tag{4}
\end{equation}
Is this because (heuristically) since $u$ is realvalued
$$u_t=ibu_{xx}=\overline{i(b)u_{xx}} \implies u = \frac{1}{\sqrt {4\pi bt}} e^{i (\pi/4+x^2/4b t)} = \overline{\frac{1}{\sqrt {4\pi (b)t}} e^{i (\pi/4+x^2/4(b) t)}}=\frac{1}{\sqrt {4\pi (b)t}} e^{i (\pi/4+x^2/4bt)}?$$
I hesitated a lot before posting. This $b$ under square root is really poignant. How would you then justify your "taking complex conjugation and $b\mapsto b$ also preserves (2)"?

Just check equation (2)