论文标题
在Lieb-loss模型中，Pauli-Fierz Hamiltonian的紫外线极限
On the Ultraviolet Limit of the Pauli-Fierz Hamiltonian in the Lieb-Loss Model
论文作者
论文摘要
二十年前，Lieb和Lieb和损失提议近似于自由，非偏移性电子的基态能量，并由所有期望值的$ e_ {α，λ} $耦合到量化的辐射场的所有期望值$ \ langle ϕ_ {el} _ {el} \ otimesψ_{pH} h_ {α，λ}（ϕ_ {El} \ otimesψ_{ph}）\ rangle $，其中$ h_ {α，λ} $是具有良好结构常数$α> 0 $ $α> 0 $的相应的hamiltonian和ulteraviolet cutoff $λ<\λ<\λ<\\\\\\\\\\\\\\\\\\\\\\ $ nrormation uson and ph}光子波分别功能。 Lieb和损失表明，$Cα^{1/2}λ^{3/2} \ leq e_ {α，λ} \ leq c^{ - 1}α^{2/7}λ^{12/7} $对于某些常数$ c> 0 $。在本文中，我们证明存在常数$ c <\ infty $，以便 \ begin {align*} \ bigg | \ frac {e_ {α，λ}} {f [1] \，α^{2/7} \，λ^{12/7}} -1 \ bigg | \ \ leq \ c \，α^{4/105} \，λ^{ - 4/105} \ end {align*}保持true，其中$ f [1]> 0 $是明确的通用号码。该结果表明，Lieb and Loss的上限实际上是锋利的，并且在极限$α\至0 $中均匀地给出了$ e_ {α，λ} $，并且在紫外线限制$λ\ to \ infty $中。
Two decades ago, Lieb and Loss proposed to approximate the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum $E_{α, Λ}$ of all expectation values $\langle ϕ_{el} \otimes ψ_{ph} | H_{α, Λ} (ϕ_{el} \otimes ψ_{ph}) \rangle$, where $H_{α, Λ}$ is the corresponding Hamiltonian with fine structure constant $α>0$ and ultraviolet cutoff $Λ< \infty$, and $ϕ_{el}$ and $ψ_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c α^{1/2} Λ^{3/2} \leq E_{α, Λ} \leq c^{-1} α^{2/7} Λ^{12/7}$ for some constant $c >0$. In the present paper we prove the existence of a constant $C < \infty$, such that \begin{align*} \bigg| \frac{E_{α, Λ}}{F[1] \, α^{2/7} \, Λ^{12/7}} - 1 \bigg| \ \leq \ C \, α^{4/105} \, Λ^{-4/105} \end{align*} holds true, where $F[1] >0$ is an explicit universal number. This result shows that Lieb and Loss' upper bound is actually sharp and gives the asymptotics of $E_{α, Λ}$ uniformly in the limit $α\to 0$ and in the ultraviolet limit $Λ\to \infty$.
