can be fulfilled even in “dirty” superconductors with $l\ll\xi$ because typically $\exp(-R/\xi)\ll1$. The effect of smearing due to the finite temperature is controlled by the parameter $T/\delta$.

To take into account the finite temperature effects and investigate the LDOS profiles in the whole range of energies and distances from the vortex core we solve numerically the BdG system (\[BdG\]) for a vortex in mesoscopic cylinder. We use the same method which was successfully applied to study the mesoscopic oscillations of vortex core levels[@MelnikovPrl2005], the spectrum of multiquanta vortices in mesoscopic superconductors[@MelnikovPrb2008] and the heat transport along vortex lines[@MelnikovPrb2007]. The resulting spectrum as a function of the angular momentum is shown in Fig. \[FigSpectrumDisc\] which clearly demonstrates the splitting of the anomalous CdGM branch at low energies caused by the boundary effects.

![\[FigSpectrumDisc\] (Color online) Quasiparticle spectrum vs $\mu$ for a singly quantized vortex in a disk of the radius $R=2\xi$. The CdGM spectrum is shown by the solid line. We choose here $k_F\xi=200$.](Spectrum){width="1.0\linewidth"}

For large energies $\varepsilon\sim\Delta_0$ the spectrum is not well described by the Eq.(\[SpectrumDisc\]).

Using the spectrum and the expression (\[DOS\]) for the LDOS we can find the local differential conductance as a function of voltage $V$: $$\frac{dI}{dV}=\left(\frac{dI}{dV}\right)_{N}
  