Is it possible to pass a PHD without full understanding of ones own thesis?

Of course, it would be good if someone around you can help you see what needs to be done to legitimize your use of "the other 5 equations". If you post to MathStackExchange, someone might be able to help, for example, if no one physically nearby can.

In any case, honesty and at least a nominal "dispassion" about things is the best approach, I think. Say what you assume, say what you can verify, say what previous sources say, etc. It's ok, I think, if your results are "relative" (=relative to assuming that certain sources are ok, plus assuming that certain extensions, which you've tried but failed (so far) to validate yourself first-hand, are ok...), as long as you are very clear on what your results are relative to.

Don't bluff... :)

Or do I just throw in the towel and save myself another 2 months of absolute torment?

Nobody except you would be able to decide. But, given what you write, failing could be an option. Academia is a hard sport, and your health might be more important than some theory. I see bad examples around me.

In case you decide to proceed, here are some thoughts:

• Ask an online service (such as Google scholar) for which papers cite the original framework. Some of the authors referencing the framework might be living. Write them e-mails.

• Apply Feynman's problem-solving algorithm. (It's not a joke, I'm very serious.) In less extravagant terms, improve your working style.

• Apply for an extension.

• ...

Bear in mind that, after all, the problem might be unsolvable with the methods at your disposal.

E.g., the mature Ostrogradsky once asked Paris mathematicians for help for solving a particular problem. They took quite some time, and finally responded "This task can be solved only by one person: the Russian professor Ostrogradsky. He lives in Petersburg. You should contact him.".

But one proper piece of advice: no torment; go home at 6pm whatever you do.

For a PhD you should demonstrate two things: that you have become an expert in your (very very small) chosen topic, and that you have learned how to do credible research in that topic. From your description you don't seem to have demonstrated either at this point, so I would find it concerning if you did get a PhD.

Kicking in the towel isn't the only option though.

• You say no-one in your department understands the material. But it is extremely unlikely that no-one in the world does. There are many theoretical physicists, and many mathematicians. Even if they are not familiar with the exact problem, there will be someone you can talk to who knows far more about eigenfunctions than you seem to.
• Get an extension. If you've been allowed to go substantially beyond the expertise of those supervising you where you don't understand and no-one else is supporting you, then someone has not really been taking their responsibilities seriously enough.
• Try and find a summer school on the mathematics you need.
• Explore the options for graduating with a lower degree.