I have a self-adjoint operator $h$ acting in the Hilbert space $L^2(R^3)$ that has the following eigenvalues $$ h \varphi_s= \varepsilon_s \varphi_s\;,\quad s=1,2,... , $$ where $\varphi_s$ are orthonormal square integrable eigenfunctions: $$ \langle \varphi_s,\varphi_{s'}\rangle = \int_{R^3} \varphi_s^*(\vec r) \varphi_{s'}(\vec r) \; \text d^3 r = \delta_{ss'} \;. $$ The first $N$ eigenvalues are negative, $\varepsilon_i<0, 1\leq i\leq N$, whereas all other eigenvalues are positive, $\varepsilon_a>0$ for $a>N$.
I will also make use of the position operator (multiplication operator) $X$ acting like, $$ (X f)(\vec r) = \vec r f(\vec r)\;,\quad \forall f\in L^2(R^3)\;. $$
My question is, how can I interpret the operators
$$ A(\vec G , \vec G', \tau) = \text e^{-\text i \vec G X} \, \text e^{-h\tau} \,\text e^{-\text i \vec G' X} $$ or $$ B(\vec G , \vec G', t) = \text e^{-\text i \vec G X} \,\text e^{\text i h t} \,\text e^{-\text i \vec G' X}\,. $$ where $\vec G,\vec G'\in R^3$ and $\tau,t \in R^+$?
In particular I am interested in the matrix elements $$\langle \varphi_i | A(\vec G , \vec G', \tau)|\varphi_j\rangle$$ or $$\langle \varphi_i | B(\vec G , \vec G', t)|\varphi_j\rangle$$ with $1<i,j<N$. So I am only interested in how they act in the subspace where $h$ has negative eigentvalues.
Can I think of $A$ and $B$ as rotations of $\text e^{-h \tau}$ and $\text e^{\text i h t}$ ? And is there any strategy to simplify/decompose/approximate $A$ or $B$ since I am only interested in their action in the described subspace? Or can you recommend any literature?
Actually I am only interested in this operator which will turn out to be more convenient regarding to the question.
PS: I am a physicist, so please don't overestimeate my mathematical vocabulary.
PPS: The operator $h$ is the Hartree-Fock operator, acting like $$ (hf)(\vec r) = -\frac{1}{2}(\vec \nabla^2f)(\vec r) + \int_{R^3} \sum_{i=1}^N\frac{|\varphi_i(\vec r')|^2}{|\vec r - \vec r'|} f(\vec r) \,\text d^3 r' - \int_{R^3} \sum_{i=1}^N\frac{\varphi_i^*(\vec r')f(\vec r')}{|\vec r - \vec r'|}\varphi_i(\vec r) \,\text d^3 r' $$ where $\varphi_i$ are its eigenvectors as defined above.