Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological Levinson's theorem in presence of embedded thresholds and discontinuities of the scattering matrix

V. Austen, D. Parra, A. Rennie, S. Richard

Abstract

A family of discrete Schroedinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibit changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C*-algebra, whose K-theory is carefully analysed. An index theorem is deduced from these investigations, which corresponds to a topological version of Levinson's theorem in presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the second half of the paper, very detailed computations for the simplest realisation of this family of operators are provided. In particular, a surface of resonances is exhibited, probably for the first time. For Levinson's theorem, it is shown that contributions due to resonances at the lowest value and at the highest value of the continuous spectrum play an essential role.

Topological Levinson's theorem in presence of embedded thresholds and discontinuities of the scattering matrix

Abstract

A family of discrete Schroedinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibit changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C*-algebra, whose K-theory is carefully analysed. An index theorem is deduced from these investigations, which corresponds to a topological version of Levinson's theorem in presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the second half of the paper, very detailed computations for the simplest realisation of this family of operators are provided. In particular, a surface of resonances is exhibited, probably for the first time. For Levinson's theorem, it is shown that contributions due to resonances at the lowest value and at the highest value of the continuous spectrum play an essential role.
Paper Structure (25 sections, 27 theorems, 200 equations, 5 figures)

This paper contains 25 sections, 27 theorems, 200 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The model and the existing results
  3. The C*-algebraic framework
  4. K-theory for the quotient algebra
  5. Topological Levinson's theorem
  6. Explicit computations for N=2
  7. Values of S(lambda) at thresholds
  8. The scattering matrix S(lambda)
  9. Case: $\lambda \in (\lambda_1^\theta-2, \lambda_2^\theta-2)$.
  10. Case: $\lambda \in (\lambda_2^\theta-2,\lambda_1^\theta+2)$.
  11. Case: $\lambda \in (\lambda_1^\theta+2,\lambda_2^\theta+2)$.
  12. Computations of some total variations
  13. The special case b=0.
  14. Variation on $(\lambda_1^\theta-2,\lambda_2^\theta-2)$.
  15. Variation on $(\lambda_2^\theta-2,\lambda_1^\theta+2)$ .
  16. ...and 10 more sections

Key Result

Theorem 2.3

For any $\theta\in(0,\pi)$, one has the equality with $\mathfrak K^\theta \in\mathscr K(\mathfrak h)$ and where $\mathfrak X$ and $\mathfrak D$ are representations of the canonical position and momentum operators in the Hilbert space $\mathfrak h$.

Figures (5)

  • Figure 1: A representation of the quotient algebra $\mathcal{Q}^\theta$.
  • Figure 2: Representation of the algebra $Q_4$, with the biggest square representing $C_4$, while the part of the left of this square corresponds to $A(3)\oplus A_4$, and the part on the right corresponds to $B_4\oplus B(3)$. For comparison with the algebra $\mathcal{Q}^\theta$, the dashed lines represent the support of the functions $\phi_{\downarrow_j}$ while the dotted lines represent the support of the functions $\phi_{\uparrow^j}$. The remaining lines and squares correspond to the support of the scattering matrix.
  • Figure 3: The horizontal axis corresponds to $v(1)$ while the vertical axis corresponds to $v(2)$. The two white regions correspond to points $(v(1),v(2))$ for which there exists a unique $\theta_0\in (0,\pi)$ leading to a resonant case.
  • Figure 4: The values $\theta_0$ that satisfy the resonant case (in the left white region of Figure \ref{['fig:SN2']}) is represented, with the value $v(1)$ and $v(2)$ on the $x$ and $y$ axes. The 2 scales are different, and the red line represent the diagonal on this quadrant. The $z$--axis corresponds to $\theta$ in $(0,\pi)$, and the surface represents the values of $\theta_0$, as a function of $v(1)$ and $v(2)$.
  • Figure 5: Visual representation of the number of eigenvalues below the essential spectrum of $H^\theta$: For each fixed $\theta$, each interface between a blue and a red region corresponds to an eigenvalue. The magnified picture represents the region close to $\theta=\pi$, where a second interface appears.

Theorems & Definitions (50)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 40 more