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Spectral inequalities for Schrödinger equations and quantitative propagation of smallness in the plane

Eugenia Malinnikova, Jiuyi Zhu

TL;DR

This paper analyzes spectral inequalities for one-dimensional Schrödinger operators $H=-\partial_x^2+V(x)$ with potentials growing to infinity and bounded between increasing weight functions. It develops a novel quantitative propagation of smallness in the plane, leveraging a sharp Cauchy-uniqueness estimate for elliptic equations and a lifting (ghost-dimension) construction to connect 1D eigenfunction combinations with 2D holomorphic-type controls. The authors establish explicit, parameter-dependent exponent bounds for spectral inequalities over thick and generalized thick sensor sets, including cases of polynomial-growth potentials and $(s,\tau)$-thick sets, and derive logarithmic refinements for sets of positive measure. These results have implications for null controllability of the associated heat equation and enrich the theory of uncertainty principles for Schrödinger operators with nontrivial growth conditions.

Abstract

This paper deals with spectral inequalities for one-dimensional Schrödinger operators with potentials bounded between two increasing functions (weights). The spectral inequality allows one to estimate the norm of a function with bounded spectrum by its values on a certain sensor set. We say that a measurable subset of the real line is thick if the measure of the intersection of this set with any interval of fixed length is bounded from below. First, we consider thick sensor sets a large class of pairs of weights. For potentials constrained between two polynomials, spectral inequalities for a broad class of so-called generalized thick sets are analyzed. A quantitative dependence of the constants in the spectral inequalities on the density of the sensor sets, the growth rate of the potentials, and the spectral interval is established. The proofs rely on a new quantitative propagation of smallness (or quantitative Cauchy uniqueness) for elliptic equations in the plane.

Spectral inequalities for Schrödinger equations and quantitative propagation of smallness in the plane

TL;DR

This paper analyzes spectral inequalities for one-dimensional Schrödinger operators with potentials growing to infinity and bounded between increasing weight functions. It develops a novel quantitative propagation of smallness in the plane, leveraging a sharp Cauchy-uniqueness estimate for elliptic equations and a lifting (ghost-dimension) construction to connect 1D eigenfunction combinations with 2D holomorphic-type controls. The authors establish explicit, parameter-dependent exponent bounds for spectral inequalities over thick and generalized thick sensor sets, including cases of polynomial-growth potentials and -thick sets, and derive logarithmic refinements for sets of positive measure. These results have implications for null controllability of the associated heat equation and enrich the theory of uncertainty principles for Schrödinger operators with nontrivial growth conditions.

Abstract

This paper deals with spectral inequalities for one-dimensional Schrödinger operators with potentials bounded between two increasing functions (weights). The spectral inequality allows one to estimate the norm of a function with bounded spectrum by its values on a certain sensor set. We say that a measurable subset of the real line is thick if the measure of the intersection of this set with any interval of fixed length is bounded from below. First, we consider thick sensor sets a large class of pairs of weights. For potentials constrained between two polynomials, spectral inequalities for a broad class of so-called generalized thick sets are analyzed. A quantitative dependence of the constants in the spectral inequalities on the density of the sensor sets, the growth rate of the potentials, and the spectral interval is established. The proofs rely on a new quantitative propagation of smallness (or quantitative Cauchy uniqueness) for elliptic equations in the plane.
Paper Structure (30 sections, 21 theorems, 292 equations)

This paper contains 30 sections, 21 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Propagation of smallness lemma
  3. Auxiliary estimate
  4. Main Lemma
  5. Spectral inequalities: case of thick sets
  6. Localization of linear combinations of eigenfunctions and the lift construction
  7. Proof of Theorem \ref{['th-mn1']}
  8. Examples of admissible potentials
  9. Generalized thick sets
  10. Definition
  11. Case 1: $s<1$
  12. Case 2: $s=1$
  13. Spectral inequalities for potentials of polynomial growth
  14. Proof of Theorem \ref{['th1']}
  15. Spectral inequalities on sets of positive measure
  16. ...and 15 more sections

Key Result

Theorem 1

Suppose that $H=-\partial^2_x+V(x)$, where $V$ satisfies eq-phi-psi, $\Phi$ has sub-exponential growth, i.e., for any $\beta>0$, there is $C_\beta$ such that $\Phi(|x|)\le C_\beta e^{\beta |x|}$, and Then the spectral inequality spec-in holds for any thick set $\omega$. Moreover, one can choose $C=C_0 a|\log \delta|$, where $a$ and $\delta$ are as in (thick-2).

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Corollary 1
  • Example 1
  • Example 2
  • ...and 31 more