Spectrum of the semi-relativistic Pauli-Fierz model I

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• 作者：Takeru Hidaka¹ ; ^{t-hidaka@math.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Fumio Hiroshima ; ² ; ^{hiroshima@math.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ground state ; Embedded eigenvalues ; Pauli&ndash ; Fierz model ; HVZ theorem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：437
• 期：1
• 页码：330-349
• 全文大小：447 K

文摘

A HVZ type theorem for the semi-relativistic Pauli–Fierz Hamiltonian, in quantum electrodynamics is studied. Here H   is a self-adjoint operator in Hilbert space $L^2({\mathbb{R}}^d)\otimes \mathcal{F}\cong {\int }_{{\mathbb{R}}^d}^{\oplus }\mathcal{F}\mathrm{d}x$, $A={\int }_{{\mathbb{R}}^d}^{\oplus }A(x)\mathrm{d}x$ is a quantized radiation field and H_f${\mathrm{H}}_{\mathrm{f}}$ is the free field Hamiltonian defined by the second quantization of a dispersion relation ω:R^d→R$\omega :{\mathbb{R}}^d\to \mathbb{R}$. It is emphasized that massless case, M=0$M=0$, is included. Let E=inf⁡σ(H)$E=\inf \sigma (H)$ be the bottom of the spectrum of H. Suppose that the infimum of ω   is m>0$m>0$. Then it is shown that σ_ess(H)=[E+m,∞)${\sigma }_{\mathrm{ess}}(H)=[E+m,\infty )$. In particular the existence of the ground state of H can be proven.