Eclipse won't start / stay on

Recently my husband's Eclipse started giving him trouble by not staying on,
like it would crank but then go right back off. Then the next day it would start
just fine and run like usual. And then randomly not stay on again. Then the
other day it just died on him while driving. We had to have it towed and at the
garage the mechanic cranked it and it ran without issues again. They
couldn't really find what could be causing this. We are stationed in Germany
at the moment so it's German mechanics looking at our US car so that's not
helping. They tried to test drive it a few times and it worked every time so we
went to pick it up and ...yep, it wouldn't stay on. So they said they would look
some more...The error codes are P0135, P1400, P0403 and P0340.
The battery was old so they replaced it.
Any ideas on what's going on?