Hokkaido Mathematical Journal

The zero modes and zero resonances of massless Dirac operators

Yoshimi SAITŌ and Tomio UMEDA

Full-text: Open access

Abstract

The zero modes and zero resonances of the Dirac operator $H=\alpha\cdot D + Q(x)$ are discussed, where $\alpha= (\alpha_1, \, \alpha_2, \, \alpha_3)$ is the triple of $4 \times 4$ Dirac matrices, $D=\frac{1}{i} ∇_x$, and $Q(x)=( q_{jk} (x))$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C \langle x \rangle^{-\rho}$, $\rho >1$. We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$ and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $ρ > 3/2$.

Article information

Source
Hokkaido Math. J., Volume 37, Number 2 (2008), 363-388.

Dates
First available in Project Euclid: 21 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1253539560

Digital Object Identifier
doi:10.14492/hokmj/1253539560

Mathematical Reviews number (MathSciNet)
MR2415906

Zentralblatt MATH identifier
1144.35458

Subjects
Primary: 35Q40: PDEs in connection with quantum mechanics
Secondary: 35P99: None of the above, but in this section 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords
Dirac operators Weyl-Dirac operators zero modes zero resonances the limiting absorption principle

Citation

SAITŌ, Yoshimi; UMEDA, Tomio. The zero modes and zero resonances of massless Dirac operators. Hokkaido Math. J. 37 (2008), no. 2, 363--388. doi:10.14492/hokmj/1253539560. https://projecteuclid.org/euclid.hokmj/1253539560


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