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Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schroedinger operators

Nabile Boussaid, Andrew Comech

Abstract

We develop a general approach to virtual levels in Banach spaces. We show that virtual levels admit several characterizations which are essentially equivalent: (1) there are corresponding virtual states (from a certain larger space); (2) there is no limiting absorption principle in their vicinity (e.g. no weights such that the ``sandwiched'' resolvent is uniformly bounded); (3) an arbitrarily small perturbation can produce an eigenvalue. We provide applications to Schrödinger operators with nonselfadjoint nonlocal potentials and in any dimension, deriving resolvent estimates in the neighborhood of the threshold when the corresponding operator has no virtual level there.

Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schroedinger operators

Abstract

We develop a general approach to virtual levels in Banach spaces. We show that virtual levels admit several characterizations which are essentially equivalent: (1) there are corresponding virtual states (from a certain larger space); (2) there is no limiting absorption principle in their vicinity (e.g. no weights such that the ``sandwiched'' resolvent is uniformly bounded); (3) an arbitrarily small perturbation can produce an eigenvalue. We provide applications to Schrödinger operators with nonselfadjoint nonlocal potentials and in any dimension, deriving resolvent estimates in the neighborhood of the threshold when the corresponding operator has no virtual level there.

Paper Structure

This paper contains 16 sections, 47 theorems, 336 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Virtual levels and virtual states in Banach spaces
  4. General theory and examples
  5. Jensen--Kato lemma: limit of the resolvent as an inverse
  6. Set of relatively compact regularizations
  7. Space of virtual states
  8. Bifurcations from the essential spectrum
  9. Dependence on the choice of regularizing spaces
  10. Derivation of the LAP properties
  11. Virtual levels of the adjoint operator
  12. Schrödinger operators in one dimension
  13. Schrödinger operators in two dimensions
  14. Schrödinger operators in higher dimensions
  15. Appendix: convergence for families of compact and collectively compact operators
  16. ...and 1 more sections

Key Result

Lemma 2.2

Let Assumption ass-virtual be satisfied. Let be the canonical embeddings; here $\mathfrak{D}(A)$ and $\mathfrak{D}(\hat{A})$ are considered as the Banach spaces endowed with the corresponding graph norms (gn) and There is a unique map $\hat{\jmath}\in\mathscr{B}(\mathfrak{D}(A),\mathfrak{D}(\hat{A}))$ such that

Figures (4)

  • Figure 2.1: The map $\hat{\jmath}:\,\mathfrak{D}(A)\to\mathfrak{D}(\hat{A})$ is defined by $\updelta_{\hat{A}}\circ\hat{\jmath}=\jmath\circ\updelta_A$.
  • Figure 2.2: Embeddings $\mathbf{E}_\nu\hookrightarrow\mathbf{X}\hookrightarrow\mathbf{F}_\nu$ and $\mathbf{E}=\mathbf{E}_1\cap\mathbf{E}_2 \hookrightarrow\mathbf{X}\hookrightarrow \mathbf{F}=\mathbf{F}_1+\mathbf{F}_2$.
  • Figure 2.3: Resolvent considered as a mapping $\tilde{\mathcal{R}}:\,\tilde{\mathbf{E}}\to\tilde{\mathbf{F}}$.
  • Figure 4.1: The worst case (the shortest distance from $z$ to $\mathbb{R}_{+}$), when $z=\zeta^2$ is separated from $|z|$ by exactly $|z|/2$. The two isosceles triangles are similar, so the base of the smaller triangle is $|z|/4$, and then one has: $\mathrm{dist}\,(z,\mathbb{R}_{+})=h=\sqrt{(|z|/2)^2-(|z|/8)^2} =|z|\sqrt{15/64} >|z|/3$.

Theorems & Definitions (128)

  • Definition 2.1: Virtual levels
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.2
  • proof
  • Remark 2.7
  • ...and 118 more