# Finding approximate eigenfunctions solutions with small eigenvalues

This question is about an appendix to chapter 7 of Aspects of Symmetry Erice lectures by Sidney Coleman. We have a SE for a 1-dimensional simple harmonic oscillator with $$\omega = 1$$, describing the bottom of a well of a symmetric double potential well, with vanishing minima centered at $$\pm a$$

$$-\frac{1}{2} \hbar \, \partial_x^2 \psi \, +\frac{1}{2}(x-a)^2\psi = E\psi$$

for the ground state, we know $$E=\frac{\hbar}{2}$$ and the even solution $$\psi_1 = e^{-(x-a)^2/2\hbar}$$. Because of linearity and reflection symmetry, we know there is another solution, $$\phi_1$$, odd and increasing for which we don't know an analytic form, but for small $$\hbar$$ its asymptotic approximation is $$\phi_1 = e^{(x-a)^2/2\hbar}$$. Here he notes that with the normalizations chosen for $$\psi_1$$ and $$\phi_1$$, the Wronskian of the two solutions is $$\phi_1 \partial_x \psi_1 - \psi_1 \partial_x \phi_1 = 2/\hbar$$

Now, we define a small dimensionless quantity $$\epsilon$$ to represent the first order correction to the ground state energy level, by $$E=\hbar \, \big(\frac{1}{2} + \epsilon\big)$$. Starting from $$\psi_1$$ and $$\phi_1$$, solutions to the SE for $$\epsilon=0$$, we want to find an approximate solution $$\psi$$ for small $$\epsilon$$, neglecting $$\epsilon^2$$ terms. Here Coleman says that this can be achieved by a standard method, namely turning the above SE into an integral equation and iterating once. He immediately writes this solution to be $$\psi = \psi_1 - \epsilon \int_{x}^{\infty} \mathrm{d}x' \, \psi_1(x')\big[\psi_1(x')\phi_1(x)-\phi_1(x')\psi_1(x)\big]$$

How did he derive this? What is this "standard method of integrating and iterating once" he's talking about? I substituted this solution $$\psi$$ back into the SE and after computing the derivatives, using the Wronskian written above and the fact that we know $$\psi_1$$,$$\phi_1$$ to be solutions I was able to verify by inspection that it is indeed the case that this $$\psi$$ solves the SE, except for terms of order $$\epsilon^2$$ which we neglect. But I would not be able to derive this $$\psi$$ by scratch. How did he do it?