By performing numerical calculations in quantum mechanics, we found something surprising. Whether the Hamiltonian is

$$ H = H_0 + lambda H_1, $$

or both $ H_0 $ and $ H_1 $ are self-adjoint, and $ lambda $ is a real parameter. The conclusion is that some eigenvalues and eigenstates disappear (or appear) suddenly when $ lambda $ crosses a critical point $ lambda_c $.

It is common that a clean state disappears $ lambda $ approaches $ lambda_c $ (say on the positive side) But in general, it disappears progressively, in that the eigenstate (a state bound in physical terms) widens more and more. $ lambda rightarrow lambda_c ^ + $.

For more quantitative, consider the integral ($ f $ be clean)

$$ I = int_B d tau | f | ^ 2, $$

or $ B $ is an arbitrary ball of finite radius centered at the origin. In general, we have

$$ lim _ { lambda rightarrow lambda_c ^ +} I = 0. $$

But in our case, it's

$$ lim _ { lambda rightarrow lambda_c ^ +} I = c> 0. $$

I wonder if this is a fact known to mathematicians.