Question: Can the equation ${\psi}_{{{t}}}-{i}{\psi}_{{{x}{x}}}={e}^{{{x}}}{\psi}$ be solved with a canonical Fourier transform? If it requires a Fokas transform or inverse scattering transform, how would one apply them to generate an analytic solution?

Thoughts: I take Schrödinger's equation ${i}ℏ{\psi}_{{{t}}}{\left({x},{t}\right)}={\left[-\frac{{ℏ^{{2}}}}{{{2}{m}}}\partial_{{{x}{x}}}+{V}{\left({x},{t}\right)}\right]}{\psi}{\left({x},{t}\right)}$ with a potential ${V}{\left({x}\right)}=e^x$ defined on a one-dimensional model of a crystal lattice. The final PDE becomes

$$ \displaystyle {i}ℏ{\psi}_{{{t}}}{\left({x},{t}\right)}+\frac{{ℏ^{{2}}}}{{{2}{m}}}{\psi_{xx}}{\left({x},{t}\right)}={V}{\left({x},{t}\right)}{\psi}{\left({x},{t}\right)}\nonumber $$

Pardon me if the following simplification is egregious, but my PDEs textbook advocates for setting all coefficients to ${1}$.

$$ \displaystyle {\psi}_{{{t}}}-{i}{\psi}_{{{x}{x}}}=-i{e}^{{{x}}}{\psi}\nonumber $$

With the simplified formula, I attempt to produce an analytic Fourier solution; however, the spatial nonlinearity is breaking the validity of my approach.

$$ \displaystyle \frac{{{w}'{\left({t}\right)}}}{{{w}{\left({t}\right)}}}={i}\frac{{{v}{''}{\left({x}\right)}}}{{{v}{\left({x}\right)}}}-i{e}^{{{x}}}={\lambda}\in{\mathbb{{{C}}}}\nonumber $$

Finally, I have the system of equations ${w}'{\left({t}\right)}-{w}{\left({t}\right)}{{\lambda}}=0$ and ${v}{''}{\left({x}\right)}+{}{\left(i\lambda-{e}^{{{x}}}\right)}{v}{\left({x}\right)}={0}$. I admit setting $\frac{v(x)}{v(x)}=1$ is slightly careless as it assumes the limit approaching $0$ collapses to $1$, and does not account for any restrictions in domain. I had considered factoring the operator for the spatial equation with the hope of generating a solution in terms of the natural exponential that could produce a symmetric basis for an analytic Fourier transform. Unfortunately, that too appears to be an unhelpful endeavor.

$$ \text{Wrong!} \hspace 1em \displaystyle {\left(\partial_{{{x}}}+{i}\sqrt{{{}{\left({i\lambda}-{e}^{{{x}}}\right)}}}\right)}\underbrace{{{\left(\partial_{{{x}}}-{i}\sqrt{{{}{\left(i\lambda-{e}^{{{x}}}\right)}}}\right)}}v(x)}_{{{u}{\left({x}\right)}}} \ne 0\nonumber $$

By solving the comparatively simpler equation, ${v}{''}{\left({x}\right)}+{e}^{{{x}}}{v}{\left({x}\right)}$ with a similar method, knowing that $\frac{\partial}{\partial x} \left( ie^{\frac{x}{2}}\right)=-\frac{\partial}{\partial x}\left(ie^{\frac{-x}{2}}\right)$.

$$\left(\partial_x + ie^{\frac{x}{2}}\right)\underbrace{\left(\partial_x - ie^{\frac{x}{2}}\right)v(x)}_{u(x)}=0$$ I integrated a linear ODE to solve for ${u}{\left({x}\right)}$ and substituted ${u}{\left({x}\right)}$ as the inhomogeneous term in the second equation for ${v}{\left({x}\right)}$. Using the variation of parameters on the fundamental set of solutions created by the homogeneous equation, I produced a final solution for a spatial component ${v}{\left({x}\right)}$. I won't type the whole formula; however, it contained an ungainly term ${\int_{{{0}}}^{{{x}}}}{e}^{{-{2}{i}{e}^{{\frac{{{s}}}{{{2}}}}}}}{d}{s}$. Technically, I could still express ${v}{\left({x}\right)}$ as a formula of sines and cosines; however, the ${L}^{{{2}}}$-theory of Fourier series makes no mention of such a basis.

I believe that my question is fairly obvious given the rudimentary construction necessary to propose the equation, and I apologize in advance if the answer is equally obvious.

  • 1
    $\begingroup$ if you make a change of variables in your spatial equation $z={{e}^{x}}$ then you get an something that looks a lot like a Bessel function. Pretty sure you can then get these to pop out of it which if so, there's your classical orthogonal set and you're possibly looking at Hankel transforms (?) (polar version of Fourier). $\endgroup$ Apr 30, 2021 at 1:51
  • $\begingroup$ Would the appropriate formula be $\psi_{t}=i\left(\psi_{zz}z^2 + \psi_{z}z - z \psi\right)$? Then, wave functions that solve Bessel's differential equation would serve as steady-state solutions correct? Is there any corresponding PDE that would allow me to solve for arbitrary values of time. This problem is closely related to the Toda Lattice and inverse scattering transform. $\endgroup$
    – Talmsmen
    Apr 30, 2021 at 2:15
  • 1
    $\begingroup$ Not entirely sure what you mean here. However note that you have separated your PDE into two ODEs. the spatial ODE i think has solutions in terms of Bessel functions. Take those solutions and multiply them against the solutions for the temporal ODE to regain your time-dependence. Sturm-Liouville theory here is your friend by the way (assuming separation of variables is a valid approach to the problem at hand). $\endgroup$ Apr 30, 2021 at 2:58
  • $\begingroup$ Thank you, I'll read into the theory and respond in a few days. Do you mind if I contact you by chat? $\endgroup$
    – Talmsmen
    Apr 30, 2021 at 3:26


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