I have the following exercise from Gustafson's time dependence methods and finite difference first edition.
Exercise2.7.1; what explicit method could be used for the Schrödinger type equation $u_t=iu_{xx}$? Derive the stability condition.
I am trying to follow section 2.7 of higher order equations on page 74. So it seems that for this case we have $a=i, p=2$ , but in this section on the book they only discuss for the case that $a$ is real, but here it's imaginary. Does it make a difference in this question?
Thanks in advance.
Edit: As requested below I am copying the section from the textbook.
In this section, we briefly discuss differnetial equations of the form: $$(2.7.1a)\frac{\partial u}{\partial t} = a \frac{\partial^p u}{\partial x^p}, p\ge 1, -\infty <x<\infty , t\ge 0$$ $$(2.7.1b)u(x,0)=f(x),-\infty < x<\infty,$$ where $a$ is a complex number and $f$ is $2\pi$-periodic. In Fourier-space, Eq. (2.7.1) becomes: $$(2.7.2a)\frac{\partial \hat{u}(\omega,t)}{\partial t}=a(i\omega)^p \hat{u}(\omega,t), $$ $$(2.7.2b)\hat{u}(\omega, 0) = \hat{f}(\omega),$$ that is, $$\hat{u}(\omega, t)=\exp(a(i\omega)^p t) \hat{f}(\omega),$$ with $$(2.7.3)| \hat{u}(\omega, t)| = |\exp(\Re[a(i\omega)^p]t\hat{f}(\omega)|.$$ For the problem to be well posed, it's sufficient that the condition $$(2.7.4) \Re[a(i\omega)^p]\le 0 $$ be fulfilled for all $\omega$. This ensures that the solution will satisfy the estimate $$(2.7.5)\| u(\cdot , t)\| \le \| f(\cdot)\|.$$ Since $\omega$ is real and can be positive or negative, we obtain the condition: $$(2.7.6a)sign(\Re a)=(-1)^{p/2+1}, \ \ \text{if $\Re a \ne 0$ and $p$ is even},$$ $$\Im a =0, \ \ \ \text{if $p$ is odd}$$ The most natural centred difference approximation is given by: $$(2.7.7)\frac{\partial^p}{\partial x^p} \to Q_p = (D_+ D_-)^{p/2}, \ \text{ $p$ is even,}$$ $$Q_p = D_0(D_+ D_-)^{(p-1)/2}, \ \ \ \text{$p$ is odd},$$ which, in Fourier space, yields: $$(2.7.8)(i\omega)^p\to \hat{Q}_p = (-\frac{4}{h^2}\sin^2(\omega h/2))^{p/2} , \text{$p$ is even,}$$ $$\frac{i}{h}\sin(\omega h)(\frac{-4}{h^2}\sin^2(\omega h/2)^{(p-1)/2} , \text{ $p$ is odd.}$$ If $a$ is real the Euler method can always be used if $p$ is even. Then the Leap-Frog scheme can always be used if $p$ is odd. This follows directly from the calculations made in Sections 2.2 and 2.5. For the Euler method, we have $$(2.7.9) \hat{Q}=1+ka\hat{Q}_p,$$ where $\hat{Q}_p$ is defined in Eq. (2.7.8). If $a$ is real, stability requires that the condition: $$(2.7.10)(-1)^{p/2-1}\cdot \frac{4^{p/2}ak}{h^p}\le 2 , \text{$p$ is even,}$$ be satisfied. For $p\ge 4$, the time step restriction is so severe that the method cannot be used in any realisitic computation. Similarly, for the leap-frog scheme, we obtain a condition of the form:$$(2.7.11)\frac{k}{h^p}\le \text{constant}, \text{$p$ is odd}.$$
Anyway, it doesn't seem they specify in this section which scheme would work for $a$ imaginary which is the case in this exercise.
I appreciate any help, cheers!