Complex Eigenvalue Splitting for the Dirac Operator

Abstract

We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

“…Klaus and Shaw proved in [21] that all the eigenvalues are simple and purely imaginary under the bell-shaped conditions (A1), (A2). Recently Hirota and Wittsten refined Theorem 2.2 to show that the eigenvalues are still pure imaginary even if we only impose an "energy-local" bell-shaped condition for small enough ǫ (see [11]). Now let us focus on the asymptotic behavior of the functions R(λ, ε) and m(µ, ε) when λ > 0 or µ > 0 tends to 0 together with ε.…”

“…Suppose that ε b(µ) is small enough and let r(µ, ǫ) := log − 1 m(µ,ε) where the logarithm is defined near 1 with log 1 = 0. Then the Bohr-Sommerfeld quantization condition (9) is equivalent to (11) S(µ) = (2n + 1)πǫ + iǫr(µ, ǫ),…”

Semiclassical WKB problem for the non-self-adjoint Dirac operator with a decaying potential

“…We conjecture that the forementioned hypothesis is always true for all potentials A satisfying Assumption 4.1. It has been proved in the case of two lobes [9] and it looks probable that the argument can be extended to the multi-hump case. Hence, from now on we always assume that 0 < < 0 .…”

“…In general, a non-self-adjoint operator like D has complex EVs. For such an operator (with a potential A satisfying Assumption 4.1), we know the following about its spectrum (see article [13] by Klaus and Shaw and [9] by Hirota and Wittsten).…”

“…Working on the same lines, we now discard the evenness assumption and additionally let the potential have multiple "humps" (instead of just a single assumed in [8]). We should point out that our methods are necessarily different from the ones found in [7] and [9]. Those works use the exact WKB method which requires analyticity.…”