axis. The LDOS can be expressed through the eigenfunctions of the BdG equation (\[BdG\]) in the following form (see, e.g., Ref.): $$\label{DOS}
  N({\bf r},\varepsilon)=\sum_n
  |u_n({\bf{r}})|^2\delta(\varepsilon-\varepsilon_n)\,,$$ where $u_n({\bf{r}})$ is electron component of quasiparticle eigenfunction corresponding to an energy level $\varepsilon_n$ (we sum over both positive and negative values $\varepsilon_n$). The eigenfunction has to be normalized: $$\int\left(|u_n ({\bf r})|^2+|v_n ({\bf r})|^2\right)d^2{\bf r}=1\,.$$

In general case BdG equations are rather complicated. A simplification can be obtained if one considers a quasiclassical approximation, assuming that the wavelength of quasiparticles is much smaller than the superconducting coherence length (see, e.g., Ref. ). Within such an approximation, quasiparticles move along linear trajectories, i.e. straight lines along the direction of the quasiparticle momentum ${\bf
k}_F=k_F(\cos\theta_p,\sin\theta_p)$. Generally, the quasiclassical form of the wave function can be constructed as follows: $(u,v)=e^{i{\bf
k}_F{\bf r}}(\tilde U,\tilde V)$, where $(\tilde U({\bf r}),\tilde V({\bf
r}))$ is a slowly varying envelope function. Then the system (\[BdG\]) is reduced to the quasiclassical Andreev equations: $$\begin{aligned}
\label{quasiclass}
  \frac{\hbar\mathbf{k}_F}{m}\left(-i\hbar\nabla+
  \frac{e}{c}\mathbf{A}\right)\tilde U+\Delta\tilde V
  =\varepsilon\tilde U\,,
  \notag\\
  \frac{\hbar\mathbf{k}_F}{m}\left(i\hbar\nabla+
  \frac{e}{c}\mathbf{A}\right)\tilde V+\Delta^*\tilde U
  =\varepsilon\tilde V\,,\end{aligned}$$ which are defined at the linear trajectories determined by the direction of the quasiparticle momentum ${\bf k}_F$.

The quasiclassical approximation allows to develop a powerful method 