Finite difference method boundary value problem

Question

Here is a boundary problem \begin{align} -\partial_x^2 \tilde{u}(x) + q(x)\tilde{u}(x) & = f(x), \; x \in (0,1)\\ \partial_x \tilde{u}(0) &= \alpha \\ \tilde{u}(1) &= \beta \end{align} with Neumann boundary conditions and $q(x) \geq Q^* > 0$. Now we can construct a finite difference method with a differential operator $$ \mathcal{L}(\tilde{u})(x) = -\partial_x^2 \tilde{u}(x) + q(x)\tilde{u}(x) $$ on a mesh $\Omega_h = \{x_i\}_{i=0}^N$ with $x_i = ih$ and $i \in [N]$. For clarity let $v_i := v_h(x_i)$ and the mesh dependent norm $$ ||v_h||_{\infty,h} := \max_{[N]}|v_i|. $$ Using the second difference we obtain $$ \mathcal{L}_h(v_h)_i = -\frac{v_{i+1} - 2v_i + v_{i-1}}{h^2} + q(x_i)v_i, \; i=1,...,N-1 $$ and $$ \mathcal{L}_h(v_h)_N = \beta. $$ For the Neumann boundary condition we create a ghost point $x_{-1}$ and obtain $$ \mathcal{L}_h(v_h)_0 = \frac 2h \alpha - 2\frac{v_1 - v_0}{h^2} + q(x_0)v_0 $$ where $\alpha = \frac{v_1 - v_{-1}}{2h}$. However now I'm having a hard time proving the stability estimate $$ ||v_h||_{\infty, h} \leq C ||\mathcal{L}_h(v_h)||_{\infty, h} $$ for the Neumann part. This leads me to believe my FDM method is incorrect?