This question defines two non-commuting self-adjoint projection operators $A$ and $B$ on a separable Hilbert space and asks about the spectral decomposition of their sum, $A+B$.
Consider the Hilbert space of complex-valued functions $\psi(x)$ of a single real variable $x$ with the usual inner product $\langle\psi,\phi\rangle = \int dx\ \psi^*(x)\phi(x)$. Consider these two projection operators $A$ and $B$:
$A$ projects onto the positive half-space in the original ($x$) representation.
$B$ projects onto the positive half-space in the Fourier-transformed representation.
Explicitly, define $A$ by $$ A\psi(x)=\begin{cases} \psi(x)&\text{ if }x > 0 \\ 0 &\text{ otherwise}. \end{cases} $$ Define $B$ by $$ B\psi(x) =\int_{0}^\infty dp\ \int_{-\infty}^\infty dy\ e^{ip(x-y)}\psi(y). $$ Question: Does the self-adjoint operator $A+B$ have a purely discrete spectrum?
I suspect the answer is no (I suspect the spectrum is purely continuous), but this is just a gut feeling. I'm looking for insights about how to deduce at least qualitative information about the spectrum of this particular operator.