= Stability of monomial ladder-operator perturbations
{title2=$\Delta E_0^{(2)}=-\lambda^2\hbar\omega(r-1)!$}
The perturbation $\lambda\hbar\omega(a^r+(a^\dagger)^r)$ couples the oscillator vacuum only to $|r\rangle$ at first order. Its squared coupling is $(\hbar\omega)^2r!$ and the <energy> denominator is $-r\hbar\omega$, giving the displayed formal second-order coefficient. For $r>2$ and nonzero real $\lambda$, coherent-state expectations contain $R^2+2\lambda R^r\cos r\theta$; choosing its phase negative proves the <Hamiltonian> is unbounded below. The formal vacuum branch must therefore not be called an exact ground state. For $r=2$ boundedness requires $|\lambda|<1/2$, while the displaced $r=1$ oscillator is stable for every real strength.
Back to article page
