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Eigenfunctions with double exponential rate of localization

S. Krymskii, A. Logunov, F. Pagano

Abstract

We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\nabla u)=λu$, $λ>0,$ in the cylinder $\mathbb{T}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $|u(x,y,t)|\leq C e^{-c e^{c|t|}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=Δu + B \nabla u$ in a half-cylinder with continuous and uniformly bounded $B$, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed.

Eigenfunctions with double exponential rate of localization

Abstract

We construct a real-valued solution to the eigenvalue problem , in the cylinder with a real, uniformly elliptic, and uniformly matrix such that for some . We also construct a complex-valued solution to the heat equation in a half-cylinder with continuous and uniformly bounded , which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed.
Paper Structure (33 sections, 35 theorems, 413 equations, 6 figures)

This paper contains 33 sections, 35 theorems, 413 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Landis conjecture
  3. Main results
  4. Interpretation of our results
  5. Few subtleties related to Landis' conjecture
  6. Notation
  7. Structure of the paper
  8. Acknowledgments
  9. Transforming solutions
  10. The equation near $\{t=T\}$
  11. Constructions of rapidly decaying solutions
  12. Definitions, notations
  13. Informal explanation of the proof of Theorem \ref{['EigenTheorem']}
  14. Plan of the proof
  15. Proof of Theorem \ref{['EigenTheorem']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

In the 3-dimensional cylinder $\mathbb{T}^2\times\mathbb{R}^+$, there exists a $C^1$-uniformly smooth and uniformly elliptic real-valued matrix $A$ and a non-zero uniformly $C^2$ real function $u$ such that and $u$ has double exponential decay: for any $T\gg 1,$ for some numericals $c, C>0$.

Figures (6)

  • Figure 1: Transforming $f$ into $g$
  • Figure 2: A graph of the function $\theta$
  • Figure 3: Main step
  • Figure 4: Changing a coefficient
  • Figure 5: Transformation into a slower decaying but faster oscillating solution
  • ...and 1 more figures

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Corollary 2.2
  • Remark 2.3
  • Claim 2.4
  • proof : Proof of the claim
  • Claim 2.5
  • ...and 59 more