![]() ![]() Using the Boulton & Strauss method (c) This sub-figure shows the error between the exact and lower bound for the first five approximation eigenvalues of Using the quadratic method (b) This sub-figure shows the error between the exact and lower bound for the first five approximation eigenvalues of (a) This sub-figure shows the error between the exact and lower bound for the first five approximation eigenvalues of Approximating enclosures for the first five eigenvalues ofīy the: Quadratic, Boulton & Strauss, and Our improvement methods with n = 200, λ low is the lower bound of the enclosing λ and λ upp is the upper bound. The slope of the graphs is close to the value (6) in all cases as n increases ( Figure 3).įigure 3. We calculate the Error between the exact and the lower bound for the first five approximation enclosure eigenvalues of Trial 1: In this trial we use the Quadratic method, Boulton & Strauss method, and our improvement with: n = 200, L = 6, to find the first five approximationĮnclosure eigenvalues (upper and lower eigenvalues) of NOTE: In our improvement method we choose (n = 200, L = 6) to compare our results with Boulton and Strauss method, which approximate the first five eigenvalues of harmonic and anharmonic models with n = 200, L = 6.įigure 1 and Figure 2 show the conjugate pair for each eigenvalue with upper and lower bounds of eigenvalues in All the coefficients of the matrices were found analytically. This equation can be solved explicitly and we can find the approximation eigenvalues using Matlab program by matrices M, N, R where,Īs described before and calculate the eigenvalues enclosure. The Harmonic Oscillator is one of the most important models of quantum theory. We compare between the Quadratic method, Boulton & Strauss and our development on these models to calculate the enclosure eigenvalues to know which one is the best in this field. ![]() Then the equation is related to anharmonic oscillator model. Then the equation is related to harmonic oscillator model. which is called the Dirichlet Boundary Conditions. Where V(x) is called the potential function, and it must be bounded below Then, the general Schrodinger equation is: Our improvement depends on domain expansion around the first five eigenvalues in the spectrum, we will take a value for a less than λ 1 and a value for b greater than λ 5 and calculate the conjugate pairs of eigenvalues ( Boulton and Strauss extended this method to normal operators and optimal convergence rates for eigenvalues and estimated that by an order of magnitude for the harmonic & anharmonic oscillator models, by cut the interval into sub-interval around the eigenvalues in the Spectrum, and the approximation enclosure eigenvalues results of these models are more accurate than the Quadratic method around λ. Properties of second order relative spectra have been studied recently by Bolton & Leviton in and then by Bolton & Strauss (2007, 2011) in. Various implementations, including on models from elasticity, solid state, physics, relativistic quantum mechanics and magneto hydrodynamics confirm that the Quadratic method is a reliable tool for eigenvalue approximation in the spectral pollution regime. It was then suggested by Shargrodsky and subsequently by Levitin and Shargorodsky (2000) in that the second order relative spectra can also be employed for the pollution-free computation of eigenvalues in gaps of the essential spectrum. Second-order relative spectra were first considered by Davies (1998) in the context of resonances for general self-adjoint operators in. The method will be examined by harmonic and anharmonic oscillator models. Then we study our new technique which gives more accurate results, we also follow the results that have been published by Boulton & Hobiny in. The quadratic method, which relies on calculation of the second-order which is providing, certified a priori intervals of spectral enclosure. These methods have used for computing eigenvalue enclosures (upper and lower bounds) of the eigenvalues of self-adjoin operators. At first, we study the second-order relative spectrum (The Quadratic method) in, and Boulton & Strauss method in. This paper shows how to compute enclosures of the eigenvalues of self-adjoint operators by the Quadratic method. ![]() Galerkin method is one of the best methods for determining upper bounds for the eigenvalues of semi-definite operators, unfortunately this method cannot find enclosures eigenvalue. ![]()
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