Zusammenfassung
The Golden-Thompson trace inequality which states that $Tr\, e^H+K łeq
Tr\, e^H e^K$ 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, $H=\Delta$
or $H= --\Delta +m$ and $K=$ potential, $Tr\, e^H+(1-u)Ke^uK$ is a
monotone increasing function of the parameter $u$ for $0u 1$. Our
proof utilizes an inequality of Ando, Hiai and Okubo (AHO): $Tr\,
X^sY^tX^1-sY^1-t Tr\, XY$ for positive operators X,Y and for
$12 s,\,t 1 $ and $s+t 32$. The obvious
conjecture that this inequality should hold up to $s+t1$, was proved false
by Plevnik. We give a different proof of AHO and also give more counterexamples
in the $32, 1$ 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.
Beschreibung
A trace inequality of Ando, Hiai and Okubo and a monotonicity property of the Golden-Thompson inequality
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