# How to solve for the scattering solution of following Schrodinger equation?

Suppose you have non-relativistic fermions scattering off a delta function potential.

It is an easy job to solve $$H=-\partial_x^2+\epsilon \delta(x)$$ by starting with an eigenfunction of the form $$\psi(x)=(A e^{-ikx}+B e^{ikx})\theta(-x)+(C e^{-ikx}+D e^{ikx})\theta(x)$$ and looking at the continuity of the function at $$x=0$$ and discontinuity of the slope of the wavefunction at $$x=0$$. Then one can compute the S-matrix.

It is also easy to solve Hamiltonian describing two fermions with attraction/repulsion at the point of contact i.e. $$H=-\partial_{x_1}^2-\partial_{x_2}^2+g\delta(x_1-x_2)$$.

For example to solve above you start with the eigenstate of the form \begin{aligned} &\psi(x_1,x_2)=\theta\left(x_{2}-x_{1}\right)\left\{A \mathrm{e}^{\mathrm{i}\left(k_{1} x_{1}+k_{2} x_{2}\right)}+B \mathrm{e}^{\mathrm{i}\left(k_{2} x_{1}+k_{1} x_{2}\right)}\right\} \\ &+\theta\left(x_{1}-x_{2}\right)\left\{C \mathrm{e}^{\mathrm{i}\left(k_{1} x_{2}+k_{2} x_{1}\right)}+D\mathrm{e}^{\mathrm{i}\left(k_{2} x_{2}+k_{1} x_{1}\right)}\right\} \end{aligned}

and then demand the continuity of the eigenfunction at $$x_1=x^2$$ and the condition that the above function is eigenstate with eigenvalue $$k_1^2+k_2^2$$, we can compute the S-matrix relative the incoming and outgoing amplitudes.

Now, consider a problem mixing both of the cases. You have fermions scattering off the delta function potentials and also attracting/repelling each other at the point of contact i.e. take the Hamiltonian of the form $$$$H=-\partial_{x_1}^2-\partial_{x_2}^2+\epsilon(\delta(x_1)+\delta(x_2))+ g \delta(x_1-x_2).$$$$

Now, how would one solve this problem?

Let's take SE $$\left[-\partial_x^2+\epsilon\delta(x)\right]\psi(x)=E\psi(x)$$ The first boundary condition is $$\psi(+\eta)=\psi(-\eta) =\psi(0),$$where $$\eta$$ is infinitesimal. I.e., the wave function is continuous at $$x=0$$. The second boundary condition is obtained by integrating the SE: $$\int_{-\eta}^\eta dx\left[-\partial_x^2+\epsilon\delta(x)\right]\psi(x)=\int_{-\eta}^\eta dx E\psi(x)\\\Rightarrow -\partial_x\psi(x)|_{x=\eta}+\partial_x\psi(x)|_{x=-\eta}+\epsilon\psi(0)\approx2\eta\psi(0)\rightarrow 0\\ \Rightarrow \partial_x\psi(x)|_{x=\eta}-\partial_x\psi(x)|_{x=-\eta}=\epsilon\psi(0)$$ This also can be used to find the bound states in a delta-potential well. See, e.g., Intuition for the number of bound states to the double Dirac potential well