How can I recover professionally from being a crank? Three years ago I got kicked out of graduate school in physics. I was suffering from mental illness, which led me to believe I made a huge discovery in physics. But my advisor kept saying that what I was doing is nonsense and that I should focus on something else. I just ignored him and kept working on huge discovery, convinced I was going to win a Nobel prize.
I started sending many emails to many physics people (I had a list of thousand email addresses I found online) hoping someone would realize my amazing discovery. I got kicked from school which made me send more emails (almost one a day near the end), and when I got negative responses I sent an email to everyone on the list insulting them and saying I deserved to be in their place.
I got psychiatric help, and eventually doctors found a brain tumor, which they removed. I am now taking drugs, and doing much better. I now realize how insane my behavior was.
I want to re-apply to grad school, but I’m scared people will recognize my name or contact my university. My undergraduate grades were very good. Should I mention my illness (with proof) when applying? Should I send apology emails to the people I’ve contacted?
A few suggestions:

If it were me, I would be very forthcoming about the illness. Of course, no one is entitled to your private medical information, so you will have to decide what you're comfortable with. But &quot;an undiagnosed brain tumor caused me to act erratically&quot; is a very convincing explanation, and does not reflect poorly on you.
Consider reaching out to your previous advisors / colleagues. Even those who were furious at your previous behavior would probably accept that an undiagnosed brain tumor caused you to exhibit poor judgment. It's a &quot;Dr. Jekyll and Mr. Hyde&quot; situation.
Along similar lines, consider contacting the university, and the specific school within the university, to discuss whether the expulsion can be expunged or converted to some other classification. Most universities are really understanding if you speak to the right people.
When you apply, I would absolutely explain what happened. I would give a shorter version of what you wrote above -- even if they don't recognize your name, they will see that you were expelled, so it's better to tell your story. And you did a good job telling the story above, so I trust that you can write an explanation that is similarly clear and compelling.
There will likely be no need to provide medical proof. You could (perhaps even should) offer to provide proof, but I suspect most admissions committees would not request it. Among other reasons, it will likely be clear from your tone that you have recovered.
I don't think it's necessary to send an apology to your entire mailing list. But I don't think doing so would be inappropriate either. If you do so, be concise: no need for more than a sentence or two.