The test_input_* files test only, that an input file included in the distro works (= some result is obtained, and the process converged). They do not test actual physical/mathematical correctness of the results (yet).
Yes, this test file is not finished. There are probably others that could test more than they test so far. I have usually written tests only when something went actually wrong :)
Yes, from now on, if something goes wrong, let's write a test for it.
So for schroedinger: you can use Test( TestInput ) to have test_input() method for free and add, for example, test_energies() and other methods to your Test class to verify correctness - those can use the solution already computed in test_input(), but you have to enforce a correct order of evaluation by using the 'tests' list (attribute of your Test class, see test_meshio.py).
Perfect, I'll do that.
Generally, I think it's a very good idea, when you (or we) implement something and you are sure that it works perfectly (which - at least for me - takes some time playing with the problem to be sure that what I get is correct), to write a test testing the correct result immediatelly. Because once we test it, we can forget about it, as it will always work.