# How to analyse the smallest eigenvalue of this linear ODE?

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.

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.