Prolate spheroidal wave function

This property is well-known for prolate spheroidal wave functions (the basic SVD functions in Fresnel or far-field approximation [2,7]), but, as it was shown [8], the double-orthogonality property is quite common for different physical Green fimctions. This property can be used for simple estimation of the noise (or stray light) impact on resolution enhancement [9]. 

The eigenfunctions of the free electron confined in the same prolate spheroids are expressed as products of regular radial and angular spheroidal wave functions [16] Chapter 21, in the respective coordinates u and v, and the eigenfunctions of Equations (34) and (35). The radial functions are expressed as infinite series of spherical Bessel functions of order m + s and argument kfu. Its eigenvalues are determined by the boundary condition on the radial factor,... 

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

In Figs. 3.82 and 3.83, we show calculations of Re ifg and Im Jfs as functions of the size parameter x = kg max a, b for oblate and prolate spheroids. The spheroids are assumed to be oriented with their axis of symmetry along the Z-axis or to be randomly oriented. Since the incident wave is also assumed to propagate along the Z-axis, the medium is not anisotropic and is characterized by a single wave number Kg. The axial ratios are a/b = 0.66 for oblate spheroids and a/b =1.5 for prolate spheroids. As before, the fractional... 

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