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arXiv 1402.1065math-phmath.MP
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半相对论Pauli-Fierz模型I的谱

Spectrum of the semi-relativistic Pauli-Fierz model I

Takeru Hidaka, Fumio Hiroshima

备注：
We revised Assumption 2.5
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摘要：

研究了量子电动力学中半相对论性Pauli-Fierz哈密顿量$$\HHH=\sqrt{(p\otimes \one -A)^2+M^2}+V\otimes \one +\one\otimes \hf,\quad M\geq 0,$$的HVZ型定理。这里，$H$是希尔伯特空间中的自伴算子$\LR\otimes \fff\cong \int^\oplus_{\RR^d}\fff {\rm d}x$，并且$A=\int^\oplus_{\RR^d} A(x) {\rm d}x$是量子化辐射场，并且$\hf$是由色散关系$ω:\RR^d\to \RR$的第二量子化定义的自由场哈密顿量。强调了无质量情况$M=0$也包括在内。令$E=\inf σ(\HHH)$为$\HHH$的谱的底部。假设$ω$的下确界是$m>0$。然后证明了$σ_{\rm ess}(\HHH)=[E+m, \infty)$ .特别地，可以证明$\HHH$的基态的存在。

英文摘要：

HVZ type theorem for semi-relativistic Pauli-Fierz Hamiltonian, $$\HHH=\sqrt{(p\otimes \one -A)^2+M^2}+V\otimes \one +\one\otimes \hf,\quad M\geq 0,$$ in quantum electrodynamics is studied. Here $H$ is a self-adjoint operator in Hilbert space $\LR\otimes \fff\cong \int^\oplus_{\RR^d}\fff {\rm d}x$, and $A=\int^\oplus_{\RR^d} A(x) {\rm d}x$ a quantized radiation field and $\hf$ the free field Hamiltonian defined by the second quantization of a dispersion relation $ω:\RR^d\to \RR$. It is emphasized that massless case, $M=0$, is included. Let $E=\inf σ(\HHH)$ be the bottom of the spectrum of $\HHH$. Suppose that the infimum of $ω$ is $m>0$. Then it is shown that $σ_{\rm ess}(\HHH)=[E+m, \infty)$. In particular the existence of the ground state of $\HHH$ can be proven.

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