Spectral theory: sudden appearance of a self-adjoint operator value $ H = H_0 + lambda H_1 $

When doing some numerical calculations in quantum mechanics, we find something surprising for us. Let the Hamiltonian be

$$ H = H_0 + lambda H_1, $$

where both $ H_0 $ Y $ H_1 $ they are self-adjoint, and $ lambda $ It is some real parameter. The finding is that some eigenvalues ​​and eigenstates disappear (or appear) suddenly as $ lambda $ crosses some critical point $ lambda_c $.

It is common for a state of its own to disappear as $ lambda $ approaches $ lambda_c $ (Let's say, from the positive side). But, in general, it disappears smoothly, in the sense that the state itself (a state linked in physical terms) spreads more and more as $ lambda rightarrow lambda_c ^ + $.

For more quantitative, consider the integral.

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

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

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

But in our case, it is.

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

I wonder if this is a known fact for mathematicians.