The spectral analysis of Jacobi (semi-infinite, tridiagonal) operators acting on $\ell^{2}(\mathbb{N})$ is deeply investigated. A crucial role is played by function $m$ which is usually known as Weyl $m$-function and which can be expressed as $$m(z)=\langle e_1, (J-z)^{-1}e_1\rangle $$ where $e_1$ is the first vector of the standard basis of $\ell^{2}(\mathbb{N})$, see. for example, G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, AMS, Rhode Island, 2000.
Function $m$ reflects many spectral properties of $J$ (this is due to $e_1$ is cyclic vector for $J$). For example, if $J$ is self-adjoint operator with discrete spectrum, then $m(z)$ is a meromorphic function whose singular points coincide with eigenvalues of $J$ (even with multiplicities) and vice versa.
One could concern other functions that could encode at least some spectral properties of self-adjoint Jacobi operators. For instance, consider function $$e(z)=\langle e_1, \exp(iJz)e_1\rangle.$$ Function $e$ is well defined and it is in fact the Fourier transform of the spectral measure of $J$ (sandwiched by vector $e_1$). My question is: Having the function $e$ at hand (as a complex function of complex variable), it is possible to say something about the spectrum of $J$? Has it been already studied somewhere? Many thanks.
Your claim about multiplicities seems flawed: The spectrum of an Jacobi operator always has multiplicity one (since $e_1$ is a cyclic vector) and the multiplicity of an isolated pole of $m(z)$ is also always one, but this last fact is due to self-adjointness and has nothing to do with the multiplicity of an eigenvalue.
Concerning your question: If your operator is bounded, you can read off the support of the spectral measure from the growth of $e(z)$ - this is essentially the Paley–Wiener theorem. The decay of $e(t)$ as $t\to\infty$ is related to the smoothness of the spectral measure. See the section on the RAGE Theorem in G. Teschl, Mathematical Methods in Quantum Mechanics, GSM 157, Amer. Math.Soc., Providence, 2014.