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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 consider a general class of metric measure spaces equipped with a regular Dirichlet form and then provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of th
We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrodinger operators with random point interactions on $mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the densit
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
For a domain $Omega subset mathbb{R}^n$ and a small number $frak{T} > 0$, let [ mathcal{E}_0(Omega) = lambda_1(Omega) + {frak{T}} {text{tor}}(Omega) = inf_{u, w in H^1_0(Omega)setminus {0}} frac{int | abla u|^2}{int u^2} + {frak{T}} int frac{1}{2
We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. Fr