# Perturbation theory with degenerate Zero Modes?

If we we know that the one dimensional Hamiltonian $\hat{H}=-\partial_{t}^2+V\left(\varphi\right)$ has two fold degenerate zero modes $\varphi_{1,2}\left(t\right)$ (i.e. has zero eigenvalue), if we deform it simply by some parameter $\mu$ so that $\hat{H}_{\mu}=\hat{H}-\mu$, and we know that it will not lift this degeneracy, then the degenerate stationary eigenfunctions of $\hat{H}_{\mu}$ on $\left[-L,+L\right]$ up to first order in $\mu$ are:$$\psi_{1,2}\left(t\right)=\varphi_{1,2}\left(t\right)\left\{ 1+\mu\intop_{-L}^{t}d\tau\left[\varphi_{2}\left(t\right)\varphi_{1}\left(\tau\right)-\varphi_{1}\left(t\right)\varphi_{2}\left(\tau\right)\right]\right\}$$ I already spent too much time trying to reproduce this using usual approaches in QM as in usual and degenerate perturbation theory, but in vain, any ideas on this?

• What are L and t here? – noah May 9 '17 at 1:28
• $t$ is just time as writing in the Hamilton after Wick rotation, and $L$ is the a domain over which the system is defined. – TMS May 9 '17 at 8:54
• Then it seems very odd to me that an integration limit for a time variable is a length. – noah May 9 '17 at 8:57
• Yes same situation is for me, the source of this is pg. 24 in laces.web.cern.ch/laces/LACES10/notes/instlargen.pdf – TMS May 9 '17 at 9:00
• Just in case will face this, I contacted the author of the paper and he confirmed that there was errors since it is actually a draft. – TMS May 22 '17 at 15:22