# 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.