We establish the Lee-Huang-Yang formula for the ground state energy of a dilute Bose gas for a broad class of repulsive pair-interactions in 3D as a lower bound. Our result is valid in an appropriate parameter regime of soft potentials and confirms that the Bogolubov approximation captures the right second order correction to the ground state energy.
For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound e( ρ) ≥ 4πaρ 2 (1 − C ρa 3 ). e( ρ ) ≥ 4π ρ 2 a 1 − C ρa 3 .( 1.4)Our result is the first rigorous lower bound on the hard core potential that gives the correct order for the correction term. (See below for a further discussion of the expected correction term, the so-called Lee-Huang-Yang term [9]).
We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $$\Omega _t:= \Omega \times [0,t]$$ Ω t : = Ω × [ 0 , t ] for $$\Omega \Subset \mathbb {R}^2$$ Ω ⋐ R 2 with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by $$\begin{aligned} \mathcal {E}[M]= \int _{\Omega _t} d^2 |\nabla M|^2+ Q \int _{\Omega _t} (M_1^2+M_2^2) + \int _{\mathbb {R}^3} |\mathcal {H}(M)|^2 - \int _{\mathbb {R}^3} |\mathcal {H}(e_3 \chi _{\Omega _t})|^2, \end{aligned}$$ E [ M ] = ∫ Ω t d 2 | ∇ M | 2 + Q ∫ Ω t ( M 1 2 + M 2 2 ) + ∫ R 3 | H ( M ) | 2 - ∫ R 3 | H ( e 3 χ Ω t ) | 2 , describing the energy difference for a given magnetization $$M: \mathbb {R}^3 \rightarrow \mathbb {R}^3$$ M : R 3 → R 3 with $$|M| = \chi _{\Omega _t}$$ | M | = χ Ω t and the uniform magnetization $$e_3 \chi _{\Omega _t}$$ e 3 χ Ω t . For $$t \ll d$$ t ≪ d , we derive the scaling of the minimal energy and a BV-bound in the critical regime, where the base area of the film has size of order $$|\Omega |^{{\frac{1}{2}}} \sim (Q-1)^{{-\frac{1}{2}}} d e^{\frac{2\pi d}{t} \sqrt{Q-1}}$$ | Ω | 1 2 ∼ ( Q - 1 ) - 1 2 d e 2 π d t Q - 1 . We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.
We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form Ωt := Ω × [0, t] for Ω R 2 with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by
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