Lieb-Thirring inequalities with improved constants
نویسنده
چکیده
where V+ = (|V |+ V )/2 is the positive part of V . Eden and Foias have obtained in [3] a version of a one-dimensional generalised Sobolev inequality which gives best known estimates for the constants in the inequality (2) for 1 ≤ γ < 3/2. The aim of this short article is to extend the method from [3] to a class of matrix-valued potentials. By using ideas from [6] this automatically improves on the known estimates of best constants in (2) for multidimensional Schrödinger operators. Lieb-Thirring inequalities for matrix-valued potentials for the value γ = 3/2 were obtained in [6] and also in [2]. Here we state a result corresponding to γ = 1. Theorem 1. Let V ≥ 0 be a Hermitian m × m matrix-function defined on R and let λn be all negative eigenvalues of the operator (1). Then
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