## sp.spectral theory – Sudden occurrence of an eigenvalue of a self-aligned operator \$ H = H_0 + lambda H_1 \$

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.