I am studying Vassili N. Kolokoltsov's paper "On the Mathematical Theory of Quantum Stochastic Filtering Equations for Mixed States" and need to understand the role of the control operator $C$ from the Mora-Rebolledo (MR) Hypothesis.
The MR Hypothesis (Section 2.1) introduces $C$ as a strictly positive self-adjoint operator with discrete spectrum $0 \leq \lambda_1 \leq \lambda_2 \leq \cdots$ and corresponding eigenbasis $\{e_m\}$. It requires:
- Domain Control: $Dom(C) \subset Dom(H) \cap Dom(L) \cap Dom(L^*L)$
- Operator Bounds: $\|Hx\|^2 \leq K\|x\|_C^2$, $\|L^*Lx\|^2 \leq K\|x\|_C^2$ for all $x \in Dom(C)$
- Generalized Dissipativity: $-2Re(Cx, iCP_mHx) - Re(Cx, CP_mL^*Lx) + \|CP_mLx\|^2 \leq \alpha(\|x\|_C^2 + \beta)$
Additionally, Hypothesis A provides more structure: there exists $l \in \mathbb{N}$ such that $L: \mathcal{H}_m \to \mathcal{H}_{m+l}$ and $H: \mathcal{H}_m \to \mathcal{H}_{m+l}$ for all $m$.
!!!!! What specific mathematical pathology in quantum filtering equations necessitates introducing $C$? The paper proves well-posedness for the linear Belavkin equation (Theorem 2.1): $d\chi(t) = -[iH\chi(t) + \frac{1}{2}L^*L\chi(t)]dt + L\chi(t)dY(t)$ Does $C$ primarily prevent solutions from escaping to "too singular" states in $Dom(C)^\perp$, or is it essential for controlling the stochastic integral $\int_0^t L\chi(s)dY(s)$ when $L$ is unbounded?
I'm trying to understand if $C$ addresses a physical obstruction in quantum stochastic evolution with unbounded operators, or if alternative approaches could circumvent its necessity.
