论文标题
Ando，Hiai和Okubo的痕迹不平等以及Golden-Thompson不等式的单调性属性
A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality
论文作者
论文摘要
黄金thompson痕迹不等式指出$ tr \，e^{h+k} \ leq tr \，e^h e^k $已被证明在量子统计机制中非常有用。 Golden用它表明经典的自由能小于量子。在这里，我们通过证明某些操作员（尤其是量子力学感兴趣的运营商），$ h =δ$或$ h = - \ sqrt {-Δ+m} $和$ k = $势，$ h =δ$或$ h = - \ leq 1 $。我们的证明利用了Ando，Hiai和Okubo（AHO）的不平等：$ tr \，X^sy^sy^tx^{1-s} y^{1-t} \ leq tr \，xy $，xy $，用于正运算符x，y和$ \ tfrac {1） \ tfrac {3} {2} $。 Plevnik证明了这种不平等最多应持续到$ S+T \ LEQ 1 $的明显猜想。我们给出了不同的AHO证明，还提供了$ \ tfrac {3} {2}，1 $范围的更多反例。更重要的是，我们表明，如果$ x，y $具有一定的阳性属性，AHO中的不平等确实确实存在于此范围内 - 它确实适用于量子机械操作员，从而使我们能够证明我们的G-T单调定理。
The Golden-Thompson trace inequality which states that $Tr\, e^{H+K} \leq Tr\, e^H e^K$ has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here we make this G-T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, $H=Δ$ or $H= -\sqrt{-Δ+m}$ and $K=$ potential, $Tr\, e^{H+(1-u)K}e^{uK}$ is a monotone increasing function of the parameter $u$ for $0\leq u \leq 1$. Our proof utilizes an inequality of Ando, Hiai and Okubo (AHO): $Tr\, X^sY^tX^{1-s}Y^{1-t} \leq Tr\, XY$ for positive operators X,Y and for $\tfrac{1}{2} \leq s,\,t \leq 1 $ and $s+t \leq \tfrac{3}{2}$. The obvious conjecture that this inequality should hold up to $s+t\leq 1$, was proved false by Plevnik. We give a different proof of AHO and also give more counterexamples in the $\tfrac{3}{2}, 1$ range. More importantly we show that the inequality conjectured in AHO does indeed hold in this range if $X,Y$ have a certain positivity property -- one which does hold for quantum mechanical operators, thus enabling us to prove our G-T monotonicity theorem.
