I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities

\begin{align} -\mathrm{i} u'(x) +f^*(x) v(x) &= \lambda u(x) \\ f(x)u(x) + \mathrm{i} v'(x) &= \lambda v(x) \end{align}

where $f(x)$ is a complex-valued function mainly varying around $x=0$ where its norm decreases from $1$ to $1-\delta$ and goes back and its phase varies from $0$ to $\phi$ and $\delta\leq1,\phi\leq2\pi$. $f^*(x)$ is its complex conjugate. For example, we can have two forms of $f(x)$ (using tanh or Cauchy distribution) of the similar profile $$ f(x)= \left(1-\delta\frac{\tanh(x/a+1)-\tanh(x/a-1)}{2\tanh{1}} \right) \exp{ \left[\mathrm{i}\phi\frac{\tanh(x/a)}{2} \right]}\\ f(x)= \left(1-\delta\frac{a^2}{x^2+a^2} \right) \exp{\left[\mathrm{i}\phi\frac{\tanh(x/a)}{2} \right]} $$ where $a$ controls the width of the region in which $f(x)$ varies quickly.enter image description here

I tried reducing it to a 2nd-order ODE. But it messes up the eigenstructure $$v''-\frac{f'}{f}v'+(\lambda^2-|f|^2-\mathrm{i}\frac{f'}{f}\lambda)v=0.$$ I have no idea if any of the two cases can be solved analytically. If possible, it would be the best.

It is known that the system will have a few (at least one) discrete real eigenvalues in $(-1,1)$ if $\delta,\phi$ are not too small and the eigenfunction is more or less localized around $x=0$. Outside $(-1,1)$, there will be a continuous spectrum.

I am interested in the eigenvalue $\lambda_0$ closest to $0$ only. If one cannot solve the system, is it possible to (roughly) understand how $\lambda_0(a,\delta,\phi)$ behaves to some extent? Perhaps some variation trend or even more. E.g., $\lambda_0$ monotonically increases with $\delta$ or something like this.



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