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Register a new userPrograms in Mathematica relevant to Phase Integral Approximation for coupled ODEs of the Schrodinger type
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Authors
Andrzej A. Skorupski
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Three programs in Mathematica are presented, which produce expressions for the lowest order and the higher order corrections of the Phase Integral Approximation. First program is pertinent to one ordinary differential equation of the Schrodinger type. The remaining two refer to a set of two such equations.
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Four generalizations of the Phase Integral Approximation (PIA) to sets of N ordinary differential equations of the Schroedinger type: u_j(x) + Sum{k = 1 to N} R_{jk}(x) u_k(x) = 0, j = 1 to N, are described. The recurrence relations for higher order corrections are given in the form valid in arbitrary order and for the matrix R_{jk} either hermitian or non-hermitian. For hermitian and negative definite R matrices, the Wronskian conserving PIA theory is formulated which generalizes Fullings current conserving theory pertinent to positive definite R matrices. The idea of a modification of the PIA, well known for one equation: u(x) + R(x) u(x) = 0, is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a non-degenerate eigenvalue of the R matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-hermitian R matrices. General theory is illustrated by a few examples generated automatically by using authors program in Mathematica, published in arXiv:0710.5406.
The general construction of quasi-classically concentrated solutions to the Hartree-type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter h (hto0), are constructed with a power accuracy of O(h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator Physics
We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to $x^{alpha}$ for $x>0$; the limit $alphatoinfty$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $alpha =1$ and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action $S$, the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.
We provide a list of explicit eigenfunctions of the trigonometric Calogero-Sutherland Hamiltonian associated to the root system of the exceptional Lie algebra E8. The quantum numbers of these solutions correspond to the first and second order weights of the Lie algebra.
We study the Bloch variety of discrete Schrodinger operators associated with a complex periodic potential and a general finite-range interaction, showing that the Bloch variety is irreducible for a wide class of lattice geometries in arbitrary dimension. Examples include the triangular lattice and the extended Harper lattice.
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