Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians
Let \({\mathcal H}\) and \({\mathcal K}\) be Hilbert spaces and \({\mathcal F}\) the tensor product \({\mathcal H}\otimes {\mathcal F}_b({\mathcal K})\), where \({\mathcal F}_b({\mathcal K})\) be the Bosonic Fock space over \({\mathcal K}\). Let \(A\) be a self-adjoint opertor on \({\mathcal H}\) which is bounded below, \(S\) a nonnegative injective self-adjoint operator on \({\mathcal K}\), \(d\Gamma(S)\) its second quantization, \(H_0= A\otimes I+ I\otimes d\Gamma(S)\) and \(H_I\) a symmetric operator on \({\mathcal F}\), with \({\mathcal D}(H_0)\cap{\mathcal D}(H_I)\) dense in \({\mathcal F}\). The paper is concerned with the symmetric operator \(H= H_0+ H_I\). In the case when \(H\) is self-adjoint, theorems are established on the location of the essential spectrum \(\sigma_{\text{ess}}(H)= [E_0(H),\infty)\), where \(E_0(H)= \inf\sigma(H)\). Applications to quantum field models are given.
scientific article; zbMATH DE number 1496351