Problems in spectral analysis of canonical Hamiltonian systems

Abstract

This note focuses on recent results in spectral analysis of canonical systems of differential equations obtained via the approach developed in our previous papers \cite{MIF1, MP3, etudes, etudes2, PZ, Direct}. Many of our results are motivated by the pioneering research of Barry Simon and his co-authors; see, for instance, the papers cited in the main text. We discuss solutions to the inverse spectral problem (ISP) for canonical Hamiltonian systems and mixed spectral problems for Schrödinger operators. One of our goals is to show connections of ISP with classical tools of analysis, such as the Hilbert transform, orthogonal polynomials, the gap problem and solutions to the Riemann-Hilbert problem. We illustrate our results with examples and discuss further questions.

Figures (5)

• Figure 1: From left to right: $h^{\mu}_{\pi}$, $h^{\mu}_{2\pi}$, $h^{\mu}_{4\pi}$, $h^{\mu}_{8\pi}$.
• Figure 2: Periodization for $d\mu(x) = (1 + \frac{\sin x}{x}) dx$. From left to right: $h^{\mu}_{\pi}$, $h^{\mu}_{2\pi}$, $h^{\mu}_{4\pi}$, $h^{\mu}_{8\pi}$.
• Figure 3: Periodization for $d\mu(x) = \left( 1 + \sin(x^2) \right) dx$. From left to right: $h^{\mu}_{\pi}$, $h^{\mu}_{2\pi}$, $h^{\mu}_{4\pi}$, $h^{\mu}_{8\pi}$.
• Figure 4: Periodization for $d\mu(x) = (1 + |x|^\frac{1}{2})dx$. From left to right: $h^{\mu}_{\pi}$, $h^{\mu}_{2\pi}$, $h^{\mu}_{4\pi}$, $h^{\mu}_{8\pi}$.
• Figure 5: Periodization for $\mu(x) = (1 + |x|)^{\frac{1}{4}}m + \delta_0$. From left to right: $h^{\mu}_{\pi}$, $h^{\mu}_{2\pi}$, $h^{\mu}_{4\pi}$, $h^{\mu}_{8\pi}$.

Theorems & Definitions (80)

• Theorem 2.1: etudes
• Corollary 2.2: etudes
• Proposition 2.3: etudes
• Theorem 2.4: etudes
• Remark 2.5
• Theorem 3.1: etudes
• Theorem 3.2: etudes
• Theorem 3.3: etudes
• Proposition 4.1: etudes
• Proposition 4.2: etudes