Let $A$ and $B$ be self-adjoint operators on some Hilbert space and let their commutator be proportional to identity operator:
$AB - BA = \lambda I$
Let us also assume that $A$ has a discrete spectrum, i.e. a countable set $\{\psi_n\}$ of eigenvectors with eigenvalues $\{\lambda_n\}$:
$A\psi_n = \lambda_n\psi_n$
Does this necessarily imply that the operator $B$ also has discrete spectrum? I.e. can it be shown that there exists a countable set of eigenvectors $\{\phi_n\}$ and eigenvalues $\{\mu_n\}$, such that:
$B\phi_n = \mu_n\phi_n$
Here I use the words "spectrum" and "eigenvalues" interchangeably, i.e. as a physicist, not as a mathematician :)
The specific application I have in mind is the Extended Hilbert Phase Space (see my paper on this) where the operators of energy and time are introduced and their commutator is proportional to identity. Now, in many situations the energy has discrete spectrum, so my question can be rephrased as: does this imply that the time is also, likewise, discrete?