Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Harmonic Approximation and Resolvent Estimates for Non-Self-Adjoint Operators

Stepan Malkov

TL;DR

The paper develops semiclassical resolvent estimates for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime by averaging the real part of the symbol along the Hamilton flow of the imaginary part. A non-unitary conjugation with an exponential weight combined with a complex deformation and the FBI–Bargmann transform transfer the problem to a holomorphic setting where the quadratic model governs the local spectrum. A local resolvent bound is proven for the quadratic model on Bargmann space, while exterior estimates are obtained away from the characteristic set, and the two are glued to produce a full resolvent bound of order 1/h in an O(h) neighborhood of the origin. The results yield spectral localization near the origin described by the lattice of eigenvalues of the quadratic approximation and have applications to complex Schrödinger operators with V and W, enabling a precise low-energy spectral description and paving the way for full asymptotic expansions via Grushin problems in future work.

Abstract

We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the operator, which we obtain by imposing a dynamical condition on the average of the real part of the symbol along the Hamiltonian flow generated by its imaginary part. An application of the resolvent estimate to a family of semiclassical Schrödinger operators with complex potentials allows us to localize the spectral problem to an $O(h)$-sized neighborhood of the origin.

Harmonic Approximation and Resolvent Estimates for Non-Self-Adjoint Operators

TL;DR

The paper develops semiclassical resolvent estimates for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime by averaging the real part of the symbol along the Hamilton flow of the imaginary part. A non-unitary conjugation with an exponential weight combined with a complex deformation and the FBI–Bargmann transform transfer the problem to a holomorphic setting where the quadratic model governs the local spectrum. A local resolvent bound is proven for the quadratic model on Bargmann space, while exterior estimates are obtained away from the characteristic set, and the two are glued to produce a full resolvent bound of order 1/h in an O(h) neighborhood of the origin. The results yield spectral localization near the origin described by the lattice of eigenvalues of the quadratic approximation and have applications to complex Schrödinger operators with V and W, enabling a precise low-energy spectral description and paving the way for full asymptotic expansions via Grushin problems in future work.

Abstract

We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the operator, which we obtain by imposing a dynamical condition on the average of the real part of the symbol along the Hamiltonian flow generated by its imaginary part. An application of the resolvent estimate to a family of semiclassical Schrödinger operators with complex potentials allows us to localize the spectral problem to an -sized neighborhood of the origin.
Paper Structure (9 sections, 11 theorems, 198 equations)

This paper contains 9 sections, 11 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Statement of Main Result
  3. Order Function Estimates
  4. Method of Averaging and the Exponential Weight Function
  5. The FBI Transform and Bargmann Space Norms
  6. Spectra of Quadratic Operators and Local Resolvent Estimates
  7. Resolvent Estimates Away from the Characteristic Set
  8. Full Resolvent Estimate for the Low-Lying Spectrum
  9. The Quantization-Multiplication Formula

Key Result

Theorem 1.1

Let $\Omega \subseteq \mathbb{C}$ be an open neighborhood of $\text{Spec}(Q).$ Then, for each $C>1,$ there exists some $h_0>0$ such that for all $0 < h \leq h_0$ and $\lambda \in D(0,Ch) \setminus h\Omega,$ the resolvent $(P-\lambda)^{-1}$ exists and satisfies

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 9 more