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Complex translation methods and its application to resonances for quantum walks

Kenta Higuchi, Hisashi Morioka

Abstract

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.

Complex translation methods and its application to resonances for quantum walks

Abstract

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.
Paper Structure (16 sections, 21 theorems, 123 equations, 3 figures)

This paper contains 16 sections, 21 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Complex translation on the pseudo momentum space
  3. Preliminary results of spectra
  4. Complex translation on the pseudo momentum space
  5. Outgoing solution
  6. Trapping and non-trapping trajectory
  7. Elastic scattering of QWs
  8. Eigenvalue of QWs via non-penetrable barriers
  9. Resonance associated with perturbed closed trajectories
  10. Perturbation of closed trajectories
  11. Construction of outgoing solutions
  12. Asymptotic distribution of eigenvalues and resonances
  13. Shape resonance model for QW
  14. Shape resonance model
  15. Resolvent estimate
  16. ...and 1 more sections

Key Result

Lemma 2.1

We have $\sigma (U_0)= \sigma_{ac} (U_0)= \{ e^{-i\lambda} \ ; \ \lambda \in [0,2\pi )\} =S^1$.

Figures (3)

  • Figure 1: The potential $V\in C^{\infty} ({\bf R}^d )$ for the shape resonance model $H(h)= -(h^2 /2)\Delta +V$ on ${\bf R}^d$ with small parameter $h>0$. If $V(x)\to 0$ rapidly as $|x| \to \infty$, $H(h)$ has no positive eigenvalue. However, there exist some resonances of $H(h)$ near the positive semi-axis.
  • Figure 2: The situation of $\Omega^i$, $\Omega^e$, and $K$ for $M_0 = 1$.
  • Figure 3: The situation of $U_{el}$ for the case $m_0 =n_0 =2$.

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Proposition 2.9
  • Definition 3.1
  • ...and 14 more