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Register a new userFaber-Krahn inequalities for Schrodinger operators with point and with Coulomb interactions
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Added by
Vladimir Lotoreichik
Publication date
2020
fields
Physics
and research's language is
English
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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.
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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 the classical Liebs inequality on metric measure spaces and (ii) uniqueness of nonnegative super-solutions on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [LS18].
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 density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrodinger operators in the continuum. The special structure of resolvent of Schrodinger operators with point interactions facilitates the proof of the Minami estimate for these models.
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.
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} | abla w|^2 - w ] be a modification of the first Dirichlet eigenvalue of $Omega$. It is well-known that over all $Omega$ with a given volume, the only sets attaining the infimum of $mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $Omega$ with the same volume and barycenter as $B_R$ and whose boundaries are parametrized as small $C^2$ normal graphs over $partial B_R$ with bounded $C^2$ norm, [ int |u_{Omega} - u_{B_R}|^2 + |Omega triangle B_R|^2 leq C [mathcal{E}_0(Omega) - mathcal{E}_0(B_R)] ] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $Omega$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_{Omega}$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $Omega$. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.
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. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.
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