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Propagation of singularities for anharmonic Schrödinger equations

Marco Cappiello, Luigi Rodino, Patrik Wahlberg

TL;DR

This work tackles the propagation of microlocal singularities for generalized anharmonic Schrödinger equations, focusing on evolution operators with anisotropic polynomial symbols. It develops a rigorous framework based on anisotropic Shubin–Sobolev modulation spaces $M_{\sigma,s}$ (with $\sigma=\frac{k}{m}$) and a $\sigma$-anisotropic Gabor wave front set $WF_g^{\sigma}$, yielding well-posedness results and precise propagation statements tied to a critical exponent $p_c=\frac{1}{2}\left(\frac{1}{k}+\frac{1}{m}\right)$. For $p\le p_c$, singularities propagate along the Hamilton flow when $p=p_c$, and remain invariant when $p<p_c$, while for $p>p_c$ the usual front set is no longer sufficient, motivating the introduction of a filter of singularities $\mathcal{F}(u)$ that is preserved by the flow. The paper also analyzes the Hamilton flow in 1D, provides a detailed calculus for anisotropic symbols, and proves propagation results in the filter sense, offering a robust microlocal picture for anharmonic Schrödinger dynamics with anisotropic scaling.

Abstract

We consider evolution equations for two classes of generalized anharmonic oscillators and the associated initial value problem in the space of tempered distributions. We prove that the Cauchy problem is well posed in anisotropic Shubin--Sobolev modulation spaces of Hilbert type, and we investigate propagation of suitable notions of singularities.

Propagation of singularities for anharmonic Schrödinger equations

TL;DR

This work tackles the propagation of microlocal singularities for generalized anharmonic Schrödinger equations, focusing on evolution operators with anisotropic polynomial symbols. It develops a rigorous framework based on anisotropic Shubin–Sobolev modulation spaces (with ) and a -anisotropic Gabor wave front set , yielding well-posedness results and precise propagation statements tied to a critical exponent . For , singularities propagate along the Hamilton flow when , and remain invariant when , while for the usual front set is no longer sufficient, motivating the introduction of a filter of singularities that is preserved by the flow. The paper also analyzes the Hamilton flow in 1D, provides a detailed calculus for anisotropic symbols, and proves propagation results in the filter sense, offering a robust microlocal picture for anharmonic Schrödinger dynamics with anisotropic scaling.

Abstract

We consider evolution equations for two classes of generalized anharmonic oscillators and the associated initial value problem in the space of tempered distributions. We prove that the Cauchy problem is well posed in anisotropic Shubin--Sobolev modulation spaces of Hilbert type, and we investigate propagation of suitable notions of singularities.
Paper Structure (7 sections, 30 theorems, 248 equations)

This paper contains 7 sections, 30 theorems, 248 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Preliminaries
  3. Well-posedness
  4. The Hamilton flow for a class of generalized anharmonic oscillators
  5. Proof of Theorem \ref{['thm:propanisogabor']}
  6. The filter of singularities
  7. Proof of Theorem \ref{['thm:propagationannular']}

Key Result

Theorem 1.1

Let $A$ be a positive operator of the form eq:diffoperator such that Then for every $s \in \mathbf R$ and $u_0 \in M_{\sigma,s}(\mathbf R^{d})$ there exists a unique solution $u= u(t,\cdot) \in \mathcal{B} (\mathbf R, M_{\sigma,s}(\mathbf R^{d})) \cap \mathcal{B}^1 (\mathbf R, M_{\sigma,s - k}(\mathbf R^{d}))$ of eq:SchrCP. If $s = 0$ then $\| u(t,\cdot) \|_{L^2} = \

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 61 more