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Lieb–Thirring Inequality
In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. They are named after Elliott H. Lieb, E. H. Lieb and Walter Thirring, W. E. Thirring. The inequalities are useful in studies of quantum mechanics and differential equations and imply, as a corollary, a lower bound on the kinetic energy of N quantum mechanical particles that plays an important role in the proof of stability of matter. Statement of the inequalities For the Schrödinger operator -\Delta+V(x)=-\nabla^2+V(x) on \Reals^n with real-valued potential V(x) : \Reals^n \to \Reals, the numbers \lambda_1\le\lambda_2\le\dots\le0 denote the (not necessarily finite) sequence of negative eigenvalues. Then, for \gamma and n satisfying one of the conditions :\begin \gamma\ge\frac12&,\,n=1,\\ \gamma>0&,\,n=2,\\ \gamma\ge0&,\,n\ge3, \end there exists a constant L_, which only depends on \g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Araki–Lieb–Thirring Inequality
In mathematics, there are many kinds of inequality (mathematics), inequalities involving matrix (mathematics), matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with Trace (linear algebra), traces of matrices.E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005). Basic definitions Let H''n'' denote the space of Hermitian matrix, Hermitian × matrices, H''n''+ denote the set consisting of Positive-definite matrix#Negative-definite, semidefinite and indefinite matrices, positive semi-definite × Hermitian matrices and H''n''++ denote the set of Positive-definite matrix#Negative-definite, semidefinite and indefinite matrices, positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |