arXiv subjects

Danko Aldunate

Publications and source records attributed to Danko Aldunate.

3 recordsLinked to original sources

arXiv2025

On gap properties for the linearized 1D Dirac--Soler model

We study spectral properties of the Dirac operator $L_0$ arising as the upper-right off-diagonal block in the linearization around standing wave solutions of the one-dimensional Soler model with power nonlinearity $f(s)=s|s|^{p-1}$, $p>0$. Our main results concern the so-called gap property: we show that if $p \geq 1$, then the only eigenvalues of $L_0$ are its ground state energies, $-2\omega$ and $0$. In contrast, for $p<1$, additional eigenvalues appear from the thresholds of the essential spectrum. Furthermore, we prove that the thresholds never admit eigenvalues and that they have at most one resonance.

math-ph↗

arXiv2025

Asymptotics for eigenvalues of one-dimensional Dirac operators in the weak coupling limit

In this paper, we derive new results on the asymptotic behavior of eigenvalues of perturbed one-dimensional massive Dirac operators in the weak coupling limit. Two classes of potentials are considered. For bounded Hermitian potentials $V$ satisfying $|V(x)| \lesssim |x|^{-1}$ for large $|x|$, we recover the leading term, which may include a logarithmic correction if $V(x) \sim |x|^{-1}$ at infinity. For possibly non-Hermitian $L^1$ potentials satisfying a suitable moment condition, we obtain the second term in the asymptotic expansion. The first result is based on a min-max principle adapted to the non-relativistic limit, while the second result is obtained via the Birman-Schwinger principle and resolvent expansions.

math-ph↗

arXiv2022

Results on the spectral stability of standing wave solutions of the Soler model in 1-D

We study the spectral stability of the nonlinear Dirac operator in dimension $1+1$, restricting our attention to nonlinearities of the form $f(\langleψ,βψ\rangle_{\mathbb{C}^2}) β$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-iωt} ϕ_0$. For the case of power nonlinearities $f(s)= s |s|^{p-1}$, $p>0$, we obtain a range of frequencies $ω$ such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition $\langleϕ_0,βϕ_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schrödinger case.

math-ph↗