Calculating eigenvalues for operator Given relation $[a,a^\dagger]=I$. Operator $K$ is defined as $K=a^\dagger a+\lambda a^\dagger+\lambda^* a$. I need to find the eignevalues of operator $K$. How realtion that involves commutator could help me? Thanks for any suggestions.
 A: You may just complete the square:
$$ K  = (a^\dagger+\lambda^*)(a+\lambda) - \lambda\lambda^* $$
Expand the product and subtract the last term to see that you get the same three terms. One may define $b=a+\lambda$. Then 
$$ K = b^\dagger b - \lambda \lambda^* $$
and $[b,b^\dagger]={\bf 1}$, so these $b$ operators are isomorphic to $a$ and the spectrum of $K$ is the same as the spectrum of $a^\dagger a - \lambda\lambda^*$.
The spectrum of $a^\dagger a$ is $0,1,2,3,\dots$, by the usual raising-and-lowering operator solution of the harmonic oscillator (there is a state with the eigenvalue $0$ satisfying $a|0\rangle = 0$ and one may raise the eigenvalue by $1$ or an integer by acting with $a^\dagger$ on this ground state) so the spectrum of $K$ is $-\lambda \lambda^*, 1-\lambda \lambda^*, $ and so on.
In the language where $a\sim x+ip$, this subtraction of the linear terms from the quadratic $a^\dagger a$ is equivalent to simply moving the harmonic oscillator to a different mean value of $x$ and $p$ (the real and imaginary part of $\lambda$, up to some simple normalization factors).
