What are solutions to(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\psi''(x) = (a_0 + a_1 x)\psi(x)

[/tex]

?

First idea I've had was that I could try some kind of perturbation with respect to the [itex]a_1[/itex] variable, so that

[tex]

\psi(x) = A_1e^{\sqrt{a_0}x} + A_2e^{-\sqrt{a_0}x} + \psi_1(x)

[/tex]

would be an attempt. But I couldn't find anything useful for [itex]\psi_1[/itex] when [itex]a_1\neq 0[/itex], so it got stuck there.

Second idea was to use Fourier transform to transform the problem into this form

[tex]

\phi'(x) = (b_0 + b_1x^2)\phi(x)

[/tex]

Unfortunately the solutions

[tex]

\phi(x) = Be^{b_0x + \frac{1}{3}b_1x^3}

[/tex]

don't have converging Fourier transforms, so that doesn't help either.

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# Wave function on slope potential

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