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Register a new userSymmetry breaking for Schrodinger-Poisson-Slater energy
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We study the asymptotic behavior of ground state energy for Schrodinger-Poisson-Slater energy functional. We show that ground state energy restricted to radially symmetric functions is above the ground state energy when the number of particles is sufficiently large.
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We prove the existence of ground states for the semi-relativistic Schrodinger-Poisson-Slater energy $$I^{alpha,beta}(rho)=inf_{substack{uin H^frac 12(R^3) int_{R^3}|u|^2 dx=rho}} frac{1}{2}|u|^2_{H^frac 12(R^3)} +alphaintint_{R^{3}timesR^{3}} frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-betaint_{R^{3}}|u|^{frac{8}{3}}dx$$ $alpha,beta>0$ and $rho>0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $alpha, beta>0$ such that $I^{alpha, beta}(rho)>-infty$ for every $rho>0$, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant $S>0$ such that $$frac{1}{S}frac{|varphi|_{L^frac 83(R^3)}}{|varphi|_{dot H^frac 12(R^3)}^frac 12}leq left (intint_{R^3times R^3} frac{|varphi(x)|^2|varphi(y)|^2}{|x-y|}dxdyright)^frac 18 $$ for all $varphiin C^infty_0(R^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $|u|^2_{H^frac 12(R^3)}$ by the homogeneous one $|u|_{dot H^frac 12(R^3)}$.
We consider the cubic Gross-Pitaevskii (GP) hierarchy in one spatial dimension. We establish the existence of an infinite sequence of observables such that the corresponding trace functionals, which we call ``energies, commute with respect to the weak Lie-Poisson structure defined by the authors in arXiv:1908.03847. The Hamiltonian equation associated to the third energy functional is precisely the GP hierarchy. The equations of motion corresponding to the remaining energies generalize the well-known nonlinear Schrodinger hierarchy, the third element of which is the one-dimensional cubic nonlinear Schrodinger equation. This work provides substantial evidence for the GP hierarchy as a new integrable system.
The compressible Navier-Stokes-Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.
We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrodinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schrodinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrodinger operators in $mathbb{R}^d,, dgeq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.
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