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On singular points in the essential spectrum

Alexander Plakhotnikov

Abstract

The paper investigates the existence of a limit in the operator norm for a family of operators $T_z(H)= F(H-z)^{-1}F^*$ for $z$ tending to the real axis. The conditions for the $H$ operator and the rigging operator $F$ are established, under which the limit exists. Special attention is paid to the separation of cases when the limit point belongs and does not belong to the point spectrum $H$.

On singular points in the essential spectrum

Abstract

The paper investigates the existence of a limit in the operator norm for a family of operators for tending to the real axis. The conditions for the operator and the rigging operator are established, under which the limit exists. Special attention is paid to the separation of cases when the limit point belongs and does not belong to the point spectrum .

Paper Structure

This paper contains 4 theorems, 13 equations.

Key Result

Lemma 3

Let the weight function be $w\in C_0(\mathbb{R})$. Then for any $s > 1/2$, the operator $F(1+|H|)^{-s}:\mathfrak{H}\to\mathfrak{K}$ is compact.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Definition 4
  • Theorem 5: see Yafaev1992
  • proof : Proof scheme
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 2 more