A Positive Density Analogue of the Lieb-thirring Inequality
نویسنده
چکیده
The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger operator in terms of an L norm of the potential. These are dual to bounds on the H-norms of a system of orthonormal functions. Here we extend these bounds to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.
منابع مشابه
A Lieb-Thirring inequality for singular values
Let A and B be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of p does the majorisation σ(BpAp) ≺w σ((BA) p) hold, and for which values its reversed inequality σ(BpAp) ≻w σ((BA) p).
متن کاملLieb-thirring Inequality for a Model of Particles with Point Interactions
We consider a model of quantum-mechanical particles interacting via point interactions of infinite scattering length. In the case of fermions we prove a Lieb-Thirring inequality for the energy, i.e., we show that the energy is bounded from below by a constant times the integral of the particle density to the power 5 3 . To Elliott Lieb, in appreciation of many years of fruitful and inspiring co...
متن کاملA Simple Proof of Hardy-lieb-thirring Inequalities
We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schrödinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Sørensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).
متن کاملConnection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane
To determine the sharp constants for the one dimensional Lieb– Thirring inequalities with exponent γ ∈ (1/2, 3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb–Thirring conjecture for γ = 1 and an isporimetric inequality fo...
متن کاملNew Bounds on the Lieb-thirring Constants
are known as Lieb-Thirring bounds and hold true with finite constants Lγ,d if and only if γ ≥ 1/2 for d = 1, γ > 0 for d = 2 and γ ≥ 0 for d ≥ 3. Here and in the following, A± = (|A| ± A)/2 denote the positive and negative parts of a self-adjoint operator A. The case γ > (1 − d/2)+ was shown by Lieb and Thirring in [21]. The critical case γ = 0, d ≥ 3 is known as the Cwikel-Lieb-Rozenblum inequ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012