A relatively well-known example of (continuous) Schrödinger operator with empty spectrum is the complex Airy operator on the line, i.e., the operator acting on $L^{2}(\mathbb{R})$ given by the differential expression $$ -\frac{d^{2}}{dx^{2}}+i x, $$ where $i$ is the imaginary unit; see for example Sec. 3.1 in this paper.
The question is whether there is a discrete counterpart of this phenomenon. More precisely, I would like to know an example (if any) of the operator $T_{v}:Dom\,T_{v}\subset\ell^{2}(\mathbb{Z})\to\ell^{2}(\mathbb{Z})$ with the maximal domain $Dom\,T_{v}=\{u\in\ell^{2}(\mathbb{Z}) \mid T_{v}u\in\ell^{2}(\mathbb{Z})\}$ and acting as $$ (T_{v}u)_{n}:=u_{n-1}+v_{n}u_{n}+u_{n+1}, \quad n\in\mathbb{Z}, $$ where $v=\{v_{n}\}_{n\in\mathbb{Z}}\subset\mathbb{C}$, such that the spectrum of $T_{v}$ is empty. The operator $T_{v}$ is refered to as the discrete Schrödinger operator with potential sequence $v$. Of course, such an example (if any) must have $v$ an unbounded and non-real sequence.
Remark: The discrete analogue of the complex Airy operator is the discrete Schrödinger operator $T_{v}$ with $v_{n}= in$. This is not an operator with empty spectrum. Actually, the spectrum of this operator can be computed fully explicitly and equals $i\mathbb{Z}$.
