In numerical computation in quantum mechanics, we found something surprising. Let the Hamiltonian be

$$ H = H_0 + lambda H_1, $$

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

It is common for an eigenstate to vanish $ lambda $ approaches $ lambda_c $ (say from the positive side). In general, however, it disappears smoothly in the sense that the eigenstate (in physical terms, a bound state) is becoming more and more extended $ lambda rightarrow lambda_c ^ + $,

To be more quantitative, consider the integral

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

from where $ B $ is any ball with finite radius at the origin. In general we have

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

But in our case it is like this

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

I wonder if this is known to the mathematicians.