The Mourre theory for analytically fibered operators
The Mourre method for self-adjoint operators was devised by \textit{E. Mourre} [Commun. Math. Phys. 78, 391-408 (1981; Zbl 0489.47010)]. Let \(H_0\) be a self-adjoint operator acting on a Hilbert space \(H\). The Mourre theory for such an operator is based on the construction of another self-adjoint operator \(A\), called a conjugate operator, such that the following estimate holds: \[ 1_\Delta(H_0)[H_0, iA]1_\Delta(H_0)\geq c_0 1_\Delta(H_0)+ K,\tag{1} \] where \(1_\Delta(H_0)\) denotes the spectral projection on the interval \(\Delta\subset\mathbb{R}\) for the operator \(H_0\), \(c_0\) is a positive constant and \(K\) is a compact operator. The estimate (1) is a Mourre estimate. The main aim of the paper is the construction of the conjugate operator and to prove the strict Mourre estimate \((K=0)\) for an abstract class of self-adjoint operators which are called analytically fibered operators. Three examples of the last class include: 1. matrix valued differential operators with constant coefficients, 2. neutral two-particle systems in a magnetic field, and 3. periodic Schrödinger operators. The paper is interesting and addresses to specialists in mathematical physics.
matrix valued differential operators with constant coefficients
neutral two-particle systems in a magnetic field
scientific article; zbMATH DE number 1121259
scientific article