One convenient way to do analysis on the symmetric Fock space is to use its isomorphism to
the Bargmann (reproducuing Kernel Hilbert) space (sometimes called the Bargmann-Fock pace)
of analytic functions on $\mathbb C^s$ (with respect to the Gaussian measure) defined in the classical paper:
Bargman V. On a Hilbert space of analytic functions and associated integral transform I,
Pure Appl. Math. 14(1961), 187-214.
An introduction to the Bargmann space may be found in chapter 4 of the
book by Uri Neretin
On the Bargmann space the creation and anihilation operators
are just the multiplication $a_j = z_j$ and the derivation $a^*_j = d/dZ_j$
and consequently, the theory of several complex variables can be used for the analysis on this space,
for example the trace of (a trace class) operator can be represented as an integral on its symbol.
Remark: The isomorphism between the symmetric Fock and Bargmann spaces is not proved in the Book. It can be found for example in the references of the following article:
Regarding the question about $a(f)+a^*(f)$, it is proportional to the position operator of quantum mechanics.
This is an unbounded operator, its spectrum is the whole real line, but it does not have
eigenfunvectors within the Fock space (Loosly speaking, they are Dirac delta functions), however one can find a series of vectors which approximate arbitrarily closely its eigenvectors. Using the corresponding projectors, one can approximate the spectral decomposition of this operator.
The case of the momentum operator $i(a(f)-a^*(f))$ is used more frequently, a possible choice of the approximate eigenvectors is by means of wave packets.
Let me try to answer this question. I apologise if my notation is slightly different, since I will work in some more generality, since the equivariance properties of the creation and annihilation operators are actually more transparent, I believe, relative to the the general linear group instead of the orthogonal group. Also the fact that this is the harmonic oscillator is a red herring. In the case of the harmonic oscillator, we have to introduce more structure, reducing the group of symmetries.
It is also in this case, where the grading to be discussed below coincides (up to a choice of scale) with the energy of the system.
Let $E$ be an $n$-dimensional real vector space and let $E^*$ denote its dual. Then on $H = E \oplus E^* \oplus \mathbb{R}K$ one defines a Lie algebra by the following relations:
$$ [x,y]= 0 = [\alpha,\beta] \qquad [x,\alpha] = \alpha(x) K = - [\alpha,x] \qquad [K,*]=0$$
for all $x,y \in E$ and $\alpha,\beta \in E^*$. This is called the Heisenberg Lie algebra of $E$, denoted $\mathfrak{h}$.
The automorphism group of $\mathfrak{h}$ is the group $\operatorname{Sp}(E\oplus E^*)$ of linear transformations of $E\oplus E^*$ which preserve the symplectic inner product defined by the dual pairing:
$$\omega\left( (x,\alpha), (y,\beta) \right) = -\alpha(y) + \beta(x).$$
Let $\mathfrak{a} < \mathfrak{h}$ denote the abelian subalgebra with underlying vector space $E \oplus \mathbb{R}K$. One can induce a $\mathfrak{h}$-module from an irreducible (one-dimensional) $\mathfrak{a}$-module as follows. Let $W_k$ denote the one-dimensional vector space on which $E$ acts trivially and $K$ acts by multiplication with a constant $k$. Then letting $U$ be the universal enveloping algebra functor, we have that
$$ V_k = U\mathfrak{h} \otimes_{U\mathfrak{a}} W_k$$
is an $\mathfrak{h}$-module. The Poincaré-Birkhoff-Witt theorem implies that $V_k$ is isomorphic as a vector space to the symmetric algebra of $E^*$, which we may (as we are over $\mathbb{R}$) identify with polynomial functions on $E$.
The subgroup of $\operatorname{Sp}(E\oplus E^*)$ which acts on $V_k$ is the general linear group $\operatorname{GL}(E)$ and hence $V_k$ becomes a $\operatorname{GL}(E)$-module. In fact, $V_k$ is graded (by the grading in the symmetric algebra of $E^*$ or equivalently the degree of the polynomial):
$$V_k = \bigoplus_{p\geq 0} V_k^{(p)}$$
and each $V_k^{(p)}$ is a finite-dimensional $\operatorname{GL}(E)$-module isomorphic to $\operatorname{Sym}^p E^*$.
Every vector $x \in E$ defines an annhilation operator: $A(x): V_k^{(p)} \to V_k^{(p-1)}$ via the contraction map
$$E \otimes \operatorname{Sym}^p E^* \to \operatorname{Sym}^{p-1} E^*$$
whereas every $\alpha \in E^*$ defines a creation operator: $C(\alpha): V_k^{(p)} \to V_k^{(p+1)}$ by the natural symmetrization map
$$E^* \otimes \operatorname{Sym}^p E^* \to \operatorname{Sym}^{p+1} E^*.$$
Both of these maps are $\operatorname{GL}(E)$-equivariant, and this is perhaps the most invariant statement I can think of concerning the creation and annihilation operators.
Best Answer
An explicit construction of generalized ladder operators $A^\pm=\mp d/dx+W(x)$ exists if the Hamiltonian can be factorized as $$H=-\frac{d^2}{dx^2}+V(x)=A^+ A^- +E_0,$$ with $E_0$ the lowest eigenvalues of $H$. The function $W(x)$ satisfies the Ricatti equation, $$W(x)^2-W'(x)=V(x)-E_0.$$ A class of "shape-invariant" potentials that can be treated in this way is discussed in Generalized Ladder Operators for Shape-invariant Potentials (2001).
Note that typically one also wants the ladder operators to satisfy a commutation relation of the form $$B^+ (x)B^- (x)-B^- (x)B^+ (x)=B_0$$ with $B_0$ independent of $x$. How to transform $A^\pm$ into $B^\pm$ satisfying this property is also discussed in this paper.