Consider a finite rank complex bundle $E$ over $S^1$ with connection $\nabla$. Let $Q_0, Q_1 \in C^\infty(S^1, E)$ be pseudo-differential operators. $Q_0$ is defined by the symbol $\sigma_0(x, \xi) = i \xi$, and $Q_1$ by $\sigma_1(x,\xi) = 2H(\xi)-1$ where $H$ is the Heaviside step function. Both of these have discrete spectra.
Now take $Q_t$ to be defined by $\sigma_t = t \sigma_0 + (1-t) \sigma_1$. Does $Q_t$ have a discrete spectrum?
Less rigorously, is there a ``continuous" way to connect $Q_0$ and $Q_1$ through operators with discrete spectra? Also, are there well known theorems about when such operator have discrete spectra, or well known classes of operators with discrete spectra? (Compact operators do, but that is insufficient for my purposes). Apologies for being vague, I'm a physicist.