Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

Lucrezia Cossetti, Luca Fanelli, David Krejcirik

TL;DR

The paper addresses spectral stability of the biharmonic operator $H_V=\Delta^2+V$ in $L^2(\mathbb{R}^d)$ for $d\ge 5$ under complex perturbations. It extends the multiplier method to fourth-order operators, using a decomposition $\Delta^2-z=(\Delta-\sqrt{z})(\Delta+\sqrt{z})$ and a gauge change to handle near the essential spectrum, to obtain absence of eigenvalues and uniform resolvent estimates in weighted spaces. The authors provide explicit smallness/repulsivity conditions (including Rellich-type potentials) for both non-self-adjoint and self-adjoint cases, along with cone-type and total-absence results and detailed a priori inequalities. These results yield spectral stability and robust resolvent control for high-order PDE models under perturbations, extending prior Schrödinger- and Dirac-type analyses to the biharmonic context.

Abstract

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

TL;DR

The paper addresses spectral stability of the biharmonic operator in for under complex perturbations. It extends the multiplier method to fourth-order operators, using a decomposition and a gauge change to handle near the essential spectrum, to obtain absence of eigenvalues and uniform resolvent estimates in weighted spaces. The authors provide explicit smallness/repulsivity conditions (including Rellich-type potentials) for both non-self-adjoint and self-adjoint cases, along with cone-type and total-absence results and detailed a priori inequalities. These results yield spectral stability and robust resolvent control for high-order PDE models under perturbations, extending prior Schrödinger- and Dirac-type analyses to the biharmonic context.

Abstract

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.
Paper Structure (11 sections, 16 theorems, 119 equations, 2 figures)

This paper contains 11 sections, 16 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main results
  3. The main ideas
  4. More on self-adjoint perturbations
  5. Organisation of the paper
  6. Multipliers identities
  7. Self-adjoint setting
  8. Non-self-adjoint setting
  9. Absence of eigevalues outside a cone: Proof of Theorem \ref{['thm:absence-out']}
  10. Total absence of eigenvalues: Proof of Theorem \ref{['thm:main-nsa']}
  11. A priori and resolvent estimates: Proof of Theorems \ref{['thm:resolvent-est']} and \ref{['thm:apriori']}

Key Result

Theorem 1.1

Let $d\geq 5.$ Suppose that $V\colon \mathbb{R}^d \to \mathbb{C}$ is such that $V\in L^1_\textup{loc}(\mathbb{R}^d)$ and $r^2 V\in L^2_\textup{loc}(\mathbb{R}^d)$. Moreover, assume that where $a$ is such that with $C_\textup{H}$ being the Hardy constant (cfr. eq:Hardy). Then $\sigma_\mathrm{p}(H_V)=\varnothing.$

Figures (2)

  • Figure 1: The region $\mathcal{S}_\textup{neg}$ (left) and $\mathcal{S}_\textup{pos}$ (right) in the $\mathop{\mathrm{Re}}\nolimits z/\mathop{\mathrm{Im}}\nolimits z$ plane
  • Figure 2: The region $\mathcal{S}_{\textup{neg}}:=\{0\leq \mathop{\mathrm{Re}}\nolimits \sqrt{z} < |\mathop{\mathrm{Im}}\nolimits \sqrt{z}|\}$ (left) and $\mathcal{S}_\textup{pos}:=\{\mathop{\mathrm{Re}}\nolimits \sqrt{z}\geq |\mathop{\mathrm{Im}}\nolimits\sqrt{z}|\}$ (right) in the $\mathop{\mathrm{Re}}\nolimits \sqrt{z}/\mathop{\mathrm{Im}}\nolimits \sqrt{z}$ plane

Theorems & Definitions (36)

  • Theorem 1.1: Total absence of eigenvalues
  • Remark 1.1
  • Theorem 1.2: Absence of non-positive eigenvalues
  • Remark 1.2
  • Theorem 1.3: Uniform resolvent estimates
  • Theorem 1.4: A priori estimates
  • Remark 1.3
  • Theorem 1.5: Absence of eigenvalues: self-adjoint
  • Theorem 1.6: Uniform resolvent estimates: self-adjoint
  • Theorem 1.7: A priori estimates: self-adjoint
  • ...and 26 more