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Register a new userBound State Solution Schr{o}dinger Equation for Extended Cornell Potential at Finite Temperature
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In this paper, we study the finite temperature-dependent Schr{o}dinger equation by using the Nikiforov-Uvarov method. We consider the sum of the Cornell, inverse quadratic, and harmonic-type potential as the potential part of the radial Schr{o}dinger equation. Analytical expressions for the energy eigenvalues and the radial wave function are presented. Application of the results for the heavy quarkonia and $B_c$ meson masses are good agreement with the current experimental data except for zero angular momentum quantum numbers. Numerical results for the temperature dependence indicates a different behaviour for different quantum numbers. Temperature-dependent results are in agreement with some QCD sum rule results from the ground states.
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We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.
Principal component analysis (PCA) has achieved great success in unsupervised learning by identifying covariance correlations among features. If the data collection fails to capture the covariance information, PCA will not be able to discover meaningful modes. In particular, PCA will fail the spatial Gaussian Process (GP) model in the undersampling regime, i.e. the averaged distance of neighboring anchor points (spatial features) is greater than the correlation length of GP. Counterintuitively, by drawing the connection between PCA and Schrodinger equation, we can not only attack the undersampling challenge but also compute in an efficient and decoupled way with the proposed algorithm called Schrodinger PCA. Our algorithm only requires variances of features and estimated correlation length as input, constructs the corresponding Schrodinger equation, and solves it to obtain the energy eigenstates, which coincide with principal components. We will also establish the connection of our algorithm to the model reduction techniques in the partial differential equation (PDE) community, where the steady-state Schrodinger operator is identified as a second-order approximation to the covariance function. Numerical experiments are implemented to testify the validity and efficiency of the proposed algorithm, showing its potential for unsupervised learning tasks on general graphs and manifolds.
An analytical solution of the Dirac equation with a Cornell potential, with identical scalar and vectorial parts, is presented. The solution is obtained by using the linear potential solution, related to Airy functions, multiplied by another function to be determined. The energy levels are obtained and we notice that they obey a band structure.
We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.
A number of papers over the past eight years have claimed to solve the fractional Schr{o}dinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schr{o}dinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schr{o}dinger equation for general fractional parameters $alpha$. On a more positive note, we present a solution to the fractional Schr{o}dinger equation for the one-dimensional harmonic oscillator with $alpha=1$.
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