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Semiclassical Birkhoff-Gustavson normal forms and spectral asymptotics for nearly resonant Schrödinger operators

Abdelkader Bourebai, Kaoutar Ghomari, San Vu Ngoc

Abstract

The concept of near resonances for harmonic approximations of semiclassical Schrödinger operators is introduced and explored. Combined with a natural extension of the Birkhoff-Gustavson normal form, we obtain formulas for approaching the discrete spectrum of such operators which are both accurate and easy to implement. We apply the theory to the physically important case of the near Fermi 1:2 resonance, for which we propose explicit expressions and numerical computations.

Semiclassical Birkhoff-Gustavson normal forms and spectral asymptotics for nearly resonant Schrödinger operators

Abstract

The concept of near resonances for harmonic approximations of semiclassical Schrödinger operators is introduced and explored. Combined with a natural extension of the Birkhoff-Gustavson normal form, we obtain formulas for approaching the discrete spectrum of such operators which are both accurate and easy to implement. We apply the theory to the physically important case of the near Fermi 1:2 resonance, for which we propose explicit expressions and numerical computations.

Paper Structure

This paper contains 16 sections, 10 theorems, 124 equations, 11 figures.

Table of Contents

  1. Introduction and motivation
  2. Preparation: smooth diagonalization
  3. Birkhoff-Gustavson normal form in near resonance
  4. $\varepsilon$-Birkhoff-Gustavson normal form theorem
  5. Normalizing the near Fermi resonance
  6. Spectral analysis in near Fermi resonance
  7. Creation and annihilation operators
  8. Bargmann representation
  9. Spectrum in near Fermi resonance
  10. Numerical simulations for the near Fermi resonance
  11. Numerical computation of the spectrum of $\hat{P}_\varepsilon$
  12. $V_\varepsilon$ has a global minimum
  13. Joint spectrum of $\hat{H}_{2,0}$ and $\hat{K}_{3,\varepsilon}$
  14. The case $\gamma=0$
  15. The case $\gamma=1$
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $A(\varepsilon)$ be a family of $n\times n$ real symmetric, respectively hermitian, matrices depending in a smooth (ie.$C^{\infty}$) way on a small parameter $\varepsilon$. Then there exists a smooth family of orthogonal, respectively unitary, matrices $U(\varepsilon)$ and smooth functions $\var

Figures (11)

  • Figure 1: Quality of the numerical spectrum in terms of $M$. With $\hbar=0.01$, $\varepsilon=0$, and $E=10\hbar$, the value given by \ref{['equ:M']} is $M=25$. We plot here $\left\|\textup{sp}_M - \textup{sp}_{M_{\textup{max}}}\right\|_{\infty}$ with $M_{\textup{max}} = 35$, and $\textup{sp}_M$ is the spectrum below $E$ obtained by diagonalizing the truncated matrix of $\hat{P}_\varepsilon$ in the basis $\mathcal{B}_M$. The same data with a $\log_{10}$ scale is plotted on the right picture. We see that, at $M\geq 25$, the error is indeed negligible. Interestingly, we also see that simply choosing $M=\frac{3E}{2\hbar}=15$ would not be sufficient.
  • Figure 2: Spectrum of $\hat{P}_\varepsilon$ (green boxes, solid line) on top of the spectrum of the $1:2$ harmonic oscillator $H_{2,0}$ (red discs, dotted line).
  • Figure 3: Comparison of the spectrum of $\hat{P}_\varepsilon$ with the eigenvalues obtained from the Birkhoff normal form $H_{2,0}+K_{3,\varepsilon}$. Here $\hbar=0.005$ (left) or $\hbar=0.0005$ (right), and $\gamma=0,\varepsilon=0$, hence $K_{3,\varepsilon}=0$.
  • Figure 4: Error(log scale) between the spectrum of $\hat{P}_\varepsilon$ and that of $H_{2,0}+K_{3,\varepsilon}$. Here $\gamma=0,\varepsilon=0$. When $\hbar$ is small enough, we observe the theoretical slope of 2 (green line), corresponding to an error of order $\mathcal{O}(\hbar^2)$.
  • Figure 5: Comparison of the spectrum of $\hat{P}_\varepsilon$ with the eigenvalues obtained from the Birkhoff normal form $H_{2,0}+K_{3,\varepsilon}$. Here $\hbar=0.005$ (left) or $\hbar=0.0005$ (right), and $\gamma=0,\varepsilon=\sqrt{\hbar}$, hence $K_{3,\varepsilon}=0$.
  • ...and 6 more figures

Theorems & Definitions (11)

  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 4.1: bargmann
  • Proposition 4.1: bargmann
  • Theorem 4.2
  • ...and 1 more