Displaying similar documents to “Spectral bisection algorithm for solving Schrödinger equation using upper and lower solutions.”

Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting

Marta M. Betcke, Heinrich Voss (2007)

Applications of Mathematics

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In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient...

Universal monotonicity of eigenvalue moments and sharp Lieb–Thirring inequalities

Joachim Stubbe (2010)

Journal of the European Mathematical Society

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We show that phase space bounds on the eigenvalues of Schr¨odinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb– Thirring inequalities.

New bounds for the minimum eigenvalue of 𝓜-tensors

Jianxing Zhao, Caili Sang (2017)

Open Mathematics

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A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.

Unified Spectral Bounds on the Chromatic Number

Clive Elphick, Pawel Wocjan (2015)

Discussiones Mathematicae Graph Theory

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One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. The various known bounds...

Recent results on Lieb-Thirring inequalities

Ari Laptev, Timo Weidl (2000)

Journées équations aux dérivées partielles

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We give a survey of results on the Lieb-Thirring inequalities for the eigenvalue moments of Schrödinger operators. In particular, we discuss the optimal values of the constants therein for higher dimensions. We elaborate on certain generalisations and some open problems as well.