I need to analyze this expectation with respect to $n$-dimensional random vectors $\mathbf{x} \in \mathbb{C}^n$.
$$E=\int_\mathbf{x} e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2} p_\mathbf{X}(\mathbf{x}) d\mathbf{x}$$
where $V$ is a fixed $n\times n$ unitary matrix and $\mathbf{a}$ is a fixed $n\times 1$ vector. Note that $V$ is not positive definite.
It is clear that $E$ must be a function of $\mathbf{a}$ and $V$.
1) Is there any non trivial distribution $p_\mathbf{X}(\mathbf{x})$ that can facilite the calculation of $E$ ? I have tried the uniform distribution over a fixed $(n-1)$-sphere without success.
2) Is there any inequality that can yield upper bound $E$ by a function of $\mathbf{a}$ and $V$. I mean something like $E = f(\mathbf{a},V) < g(\mathbf{a},V)$ without having $f(\mathbf{a},V)$ ?
I appreciate any hints and references.