A certified exact-arithmetic counterexample shows the stationary quantum mechanical bootstrap at level three is not tight in two dimensions, while eigenstate constraints seem to close the gap.
A Strict Positivstellensatz for the Weyl Algebra
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Let $c$ be an element of the Weyl algebra $W(d)$ which is given by a strictly positive operator in the Schr"odinger representation. It is shown that, under some conditions, there exist elements $b_1,...,b_d$ in $W(d)$ such that $b_1 c b_1^* + ... + b_d c b_d^*$ is a finite sum of squares.
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Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone
A certified exact-arithmetic counterexample shows the stationary quantum mechanical bootstrap at level three is not tight in two dimensions, while eigenstate constraints seem to close the gap.