## Spectral theory – Sudden emergence of a self-worth of a self-adjoint operator \$ H = H_0 + lambda H_1 \$

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.