Fundamental solution of the stationary Dirac equation with a linear potential

Abstract

We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier transforms of parabolic cylinder functions. This solution describes the flux of quasiparticles in graphene emitted by a pointlike source of electrons that are partially converted into holes (antiparticles). Using our explicit formula, we calculate its semiclassical asymptotic behavior in the hole region.

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Acknowledgments

The author is grateful to A. A. Davydov, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov, and M. Rouleux, whose help in various forms was very useful in the work on this paper.

Funding

This research was supported in part by the Russian Foundation for Basic Research and the Japanese Society for the Advancement of Science in the framework of scientific project No. 19-51-50005.

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Appendix

We show that a solution of Eq. (1) that satisfies the limit-absorption principle describes the flux of quasiparticles from a source emitting \(a|w|^2/2\) electrons per unit time. In particular, this means that their density and also their current vector components in the semiclassical limit \(h\to+0\) are nonzero functions in the caustic domain (Fig. 1).

In other words, given the factor \(h^{3/2}\) in Eq. (1), the leading term of the semiclassical asymptotic solution is of degree zero in the small parameter \(h\). This also holds for the Helmholtz equation with variable coefficients under certain regularity conditions, which follows from the results in [9].

We verify the above facts in a small neighborhood of the source. In the usual way, we reduce Eq. (1) with frozen coefficients to the Helmholtz equation, whose fundamental solution is well known and is expressed in terms of the Hankel function.

Our explicit test is as follows. We examine Eq. (1) in a small neighborhood of the source. Freezing the coefficients and moving the source to the origin, we obtain the equation

$$(-a\sigma_0+\sigma_x \hat{p} _x+\sigma_y \hat{p} _y)\psi(x,y)=h^{3/2}\delta(x,y) \mathrm{w} ,$$
which is symmetric under rotations. Therefore, without loss of generality, we consider the case where \(w_1=1\) and \(w_2=0\):
$$\begin{aligned} \, &-a\psi_1-ih\, \partial _x\psi_2-h\, \partial _y\psi_2=h^{3/2}\delta, \\ &-ih\, \partial _x\psi_1+h\, \partial _y\psi_1-a\psi_2=0. \end{aligned}$$
This system reduces to the Helmholtz equation
$$\biggl( \partial _x^2+ \partial _y^2+\frac{a^2}{h^2}\biggr)\psi_1= -\frac{a}{\sqrt{h}}\delta,\qquad \psi_2=\frac{h}{a}(-i\, \partial _x+ \partial _y)\psi_1, $$
(15)
whose solution satisfying the limit-absorption principle is well known, can be expressed in terms of the Hankel function, and has the asymptotic form
$$\psi_1(x,y)=\frac{ia}{4\sqrt{h}}H_0^{(1)}\biggl(\frac{ar}{h}\biggr)\sim \frac{1+i}{4}\sqrt{\frac{a}{\pi r}}e^{iar/h},\quad h\to+0,\qquad r=\sqrt{x^2+y^2}.$$
Substituting the resulting asymptotic formula for the function \(\psi_1\) in formula (15) and expressing the second component \(\psi\) in terms of the first, we obtain
$$\psi_2(x,y)\sim\frac{1+i}{4}\sqrt{\frac{a}{\pi r}} \frac{x+iy}{r}e^{iar/h},\quad h\to+0.$$
Replacing \(a\) with \(a+i \varepsilon \) as in Sec. 3.3, we verify that the limit-absorption principle is indeed satisfied for the given solution. Hence, degrees in \(h\) of the obtained asymptotic formulas are indeed equal to zero.

Using the obtained asymptotic formulas, we calculate the limits as \(h\to+0\) of the density and current of Dirac particles:

$$\begin{aligned} \, & \varrho =|\psi|^2=\psi^*\sigma_0\psi= \bar\psi_1\psi_1+\bar\psi_2\psi_2\to\frac{a}{4\pi r}, \\ &j=(\psi^*\sigma_x\psi,\psi^*\sigma_y\psi)= (\bar\psi_1\psi_2+\bar\psi_2\psi_1,-i\bar\psi_1\psi_2+ i\bar\psi_2\psi_1)\to\frac{a}{4\pi r^2}(x,y), \\ &\mathrm{div}\,j\to\frac{a\delta}{2}. \end{aligned}$$
The first two formulas clearly show that the density limits of \( \varrho \) and both components of \(j\) are indeed nonzero functions. The last formula means that in the particular case where \(w_1=1\) and \(w_2=0\), the source intensity is \(a/2\). In the general case, it is also multiplied by \(| \mathrm{w} |^2=|w_1|^2+ |w_2|^2\) and is equal to \(a| \mathrm{w} |^2/2\), as claimed.

Remark 7.

If we take the complex conjugate instead of the indicated fundamental solution \(\psi_1\) of the Helmholtz equation in formulas (15), then we obtain a solution of Dirac equation (1) that does not satisfy the limit-absorption principle and describes the flow of electrons arriving from infinity and disappearing at the origin.

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Bogaevsky, I.A. Fundamental solution of the stationary Dirac equation with a linear potential. Theor Math Phys 205, 1547–1563 (2020). https://doi.org/10.1134/S0040577920120016

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Keywords

  • massless Dirac equation
  • semiclassical asymptotic behavior
  • fundamental solution
  • Green’s matrix
  • parabolic cylinder function
  • graphene
  • quasiparticle