We review two results in which trial states for bosonic Hamiltonians were discussed. The problem of finding a trial state for a system with a hard-core potential in the Gross-Pitaevskii regime was recently solved by proving a link with the problem of finding a trial state for a system with a more regular potential in a less singular scaling, one of the type $N^{-1+3β}V(N^β\cdot)$ with $β\in(0,1)$ . For both of these models we present the main result for the upper bound to the ground state energy, and discuss the key steps in the proof.We review two results in which trial states for bosonic Hamiltonians were discussed. The problem of finding a trial state for a system with a hard-core potential in the Gross-Pitaevskii regime was recently solved by proving a link with the problem of finding a trial state for a system with a more regular potential in a less singular scaling, one of the type $N^{-1+3β}V(N^β\cdot)$ with $β\in(0,1)$. For both of these models we present the main result for the upper bound to the ground state energy, and discuss the key steps in the proof.

arXiv:2212.00144Nov 30, 20220 citations

Dynamics of two-dimensional Bose gases in the focusing case

Lukas Reichmann

We prove the propagation of Bose--Einstein condensation and a norm approximation for the dynamics of a Bose gas with localized initial data in two dimensions with singular interactions that may have a small attractive part. We allow the scaling of the interaction to be any polynomial scaling $N^β$ for $0<β<\infty$ and even exponential scaling $e^{N^κ}$ for $0<κ<1/4$ . Our approach covers external potentials that are time-dependent, at most quadratic, and bounded below, including the situation without an external potential.We prove the propagation of Bose--Einstein condensation and a norm approximation for the dynamics of a Bose gas with localized initial data in two dimensions with singular interactions that may have a small attractive part. We allow the scaling of the interaction to be any polynomial scaling $N^β$ for $0<β<\infty$ and even exponential scaling $e^{N^κ}$ for $0<κ<1/4$. Our approach covers external potentials that are time-dependent, at most quadratic, and bounded below, including the situation without an external potential.

arXiv:2610.33212Oct 23, 20260 citations

Scaling in the ideal Bose gas

Carol K. Hall

Journal of Statistical PhysicsAug 1, 197517 citations

N^{-1+3β}V(N^β\cdot)

with

β\in(0,1)

. For both of these models we present the main result for the upper bound to the ground state energy, and discuss the key steps in the proof.

N^β

for

0<β<\infty

and even exponential scaling

e^{N^κ}

for

0<κ<1/4

. Our approach covers external potentials that are time-dependent, at most quadratic, and bounded below, including the situation without an external potential.