Rayleigh theorem for eigenvalues

<p>In mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an <a href="/facts/Eigenvalue_equation/qAwPXmpR">eigenvalue equation</a> as the number of <a href="/facts/Basis_function/Bt7jqjY2">basis functions</a> employed in its resolution increases. Rayleigh, <a href="/facts/Lord_Rayleigh/HcJyYdk5">Lord Rayleigh</a>, and <a href="/facts/3rd_Baron_Rayleigh/HcJyYdk5">3rd Baron Rayleigh</a> are the titles of <a href="/facts/John_William_Strutt%2c_3rd_Baron_Rayleigh/HcJyYdk5">John William Strutt</a>, after the death of his father, the 2nd Baron Rayleigh. Lord Rayleigh made contributions not just to both theoretical and experimental physics, but also to applied mathematics. The Rayleigh theorem for eigenvalues, as discussed below, enables the energy minimization that is required in many self-consistent calculations of electronic and related properties of materials, from <a href="/facts/Atom/69syB8dH">atoms</a>, <a href="/facts/Molecule/N9ljSfFq">molecules</a>, and <a href="/facts/Nanostructure/rkeQGqt7">nanostructures</a> to <a href="/facts/Semiconductor/zFf3mGTE">semiconductors</a>, <a href="/facts/Insulator_(electricity)/SfSXkrh5">insulators</a>, and <a href="/facts/Metal/pqQVs8QU">metals</a>.  Except for metals, most of these other materials have an energy or a <a href="/facts/Band_gap/IjIeh95l">band gap</a>, i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For <a href="/facts/Crystal/e2R2mk3E">crystals</a>, the energy spectrum is in bands and there is a band gap, if any, as opposed to <a href="/facts/Energy_gap/VMvKsyna">energy gap</a>. Given the diverse contributions of Lord Rayleigh, his name is associated with other theorems, including <a href="/facts/Parseval%2527s_theorem/413aRLTs">Parseval's theorem</a>. For this reason, keeping the full name of  "Rayleigh Theorem for Eigenvalues" avoids confusions.
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Rayleigh theorem for eigenvalues open-in-new

Rayleigh theorem for eigenvalues