Wronskian function

The integral in Eq. (3.34) is bir >mm due to the orthonormal property of the spherical harmonics. The second line is the Wronskian of the spherical modified Bessel functions. 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Here H is an Hermitian linear integral operator over a that can be constructed variationally from basis solutions (J), of the Schrodinger equation that are regular in the enclosed volume. The functions , do not have to be defined outside the enclosing surface and in fact must not be constrained by a fixed boundary condition on this surface [270], This is equivalent to the Wronskian integral condition (4>i WG f) = 0, for all such ,. When applied to particular solutions of the form = J — Nt on a,... 

Schlosser and Marcus [359] showed that for variations about such a continuous trial function, the induced first-order variations of Zr and Za exactly cancel, even if the orbital variations are discontinuous at a or have discontinuous normal gradients. After integration by parts, the variation of Zr about an exact solution is a surface Wronskian integral... 

Matrix elements ZJ L are Wronskian surface integrals on interface cr/a, between adjacent cells r/( and xv, evaluated for basis functions ,

basis functions in cell pu are all evaluated by integrating the local Schrodinger equation at the same energy, the site-diagonal matrix elements Z/[Pg.109]

The derivation given above of the stationary Kohn functional [ K] depends on logic that is not changed if the functions Fo and l< of Eq. (8.5) are replaced in each channel by any functions for which the Wronskian condition mm — m 0 = l is satisfied [245, 191]. The complex Kohn method [244, 237, 440] exploits this fact by defining continuum basis functions consistent with the canonical form cv() = I.a = T, where T is the complex-symmetric multichannel transition matrix. These continuum basis functions have the asymptotic forms... 

Other asymptotic forms consistent with unit Wronskian define different but equally valid Green functions, with different values of the asymptotic coefficient of u>i. In particular, if w k 2 exp i(kr — ln), this determines the outgoing-wave Green function, and the asymptotic coefficient of w is the single-channel F-matrix, F sin ij. This is the basis of the T-matrix method [342, 344], which has been used for electron-molecule scattering calculations [126], It is assumed that Avf is regular at the origin and that Ad vanishes more rapidly than r 2 for r — oo. 

The multichannel model functions wosq and wlsq are real-valued continuum solutions of the model problem, in a A -matrix formalism. They are normalized to the matrix Wronskian condition... 

In this / -matrix theory, open and closed channels are not distinguished, but the eventual transformation to a A -matrix requires setting the coefficients of exponentially increasing closed-channel functions to zero. Since the channel functions satisfy the unit matrix Wronskian condition, a generalized Kohn variational principle is established [195], as in the complex Kohn theory. In this case the canonical form of the multichannel coefficient matrices is... 

Here p = /E( 1 + E/c2) is the momentum, the functions h (pr) are the relativistic Hankel functions of the first and second kind (Rose 1961) and [ ]r denotes the relativistic form of the Wronskian evaluated at r outside the potential well. Finally, the single-site r-matrix t(E) is obtained from the expression (Ebert and Gyorffy 1988) ... 

The Wronskians of these functions have been calculated in [1] and the appropriate choice of independent solutions can be made for each boundary condition imposed. 

The first of these, which connects the logarithmic derivatives of the spherical Bessel and Neumann functions, is a direct consequence of the Wronskian relation nj - jn = S [5.4], while the latter may be derived quite generally in analogy to (4.17), to which it reduces when the two boundary conditions are - i - 1 and . In addition, we renormalise the structure constants... 

The Wronskian of the two independent solutions is 2i and the Green s function, which should vanish when r or r equals zero or infinity, is... 

A remark on flux conservation. In the traditional inelastic scattering theory, one shows that the logarithmic derivative matrix Z is a symmetric matrix, because of the symmetry of coupling matrices. Moreover, the Wronskian of asymptotic functions is unity and it is easy to show that the K-matrix is symmetric and the S-matrix symmetric and unitary. In reactive collisions, the symmetry of the logarithmic derivative matrix is destroyed by the transformation (3.24) because U. Also, the matrix Wronskian of the asymptotic... 

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