Wronskian

Now, continuity requires that the wronskian/i/j —fifi is preserved, from which it may be verified that... 

Here W[z] = U(z) - v(< ) is Wronskian> and C] and Cl are arbitrary constants that may be found from the continuity condition of the probability density and the probability current at the origin ... 

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. 

Because our interest is with second-order differential equations, two linearly independent solutions always arise (the Wronskian of solutions is non-zero [490], see Sect. 5) and requires two arbitrary constants to be fixed from the two boundary conditions imposed on p0(r, t) by the physics of the problem being modelled. These boundary conditions determine how much of each of the two linearly independent solutions of the homogeneous equation (317) must be added to the particular integral to ensure that the solution of eqn. (316) is consistent with the boundary conditions. In the next three sections, the method of deriving the particular integral from the two linearly independent solutions of the homogeneous equation are discussed. 

It remains to find the four arbitrary constants A<, B<, A> and B> in eqns. (327) with the four conditions (328) and (329). By comparison with unity, the term ( rc2 /12D) In (src2/AD) is small for small s and so it may be ignored. Now, the Wronskian of the independent solutions is inversely dependent on the radial co-ordinate and... 

Clearly, W(yj, y2) is zero if y L = constant xy2. If y2 andy3 are linearly independent, then the Wronskian is non-zero for some values of sc. In addition, if we multiply eqn. (334a) by y, eqn. (334c) by yl7 subtract and integrate by parts, we have... 

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... 

The standard angular momentum indices ,m, and an implicit spin index, are here summarized by a single collective index L. Relative normalization is such that (Nl W0. // ) = h,, , expressed in terms of Wronskian integrals over surface a defined by... 

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,... 

Introducing a matrix notation for surface Wronskian integrals,... 

The notation Wtl defines a Wronskian integral over ct/2. By the surface matching theorem, xm = X> the interior component of 4> Since 4r is a solution of the Lippmann-Schwinger equation, this implies xout = Xm when evaluated in the interior of Tjj. This is a particular statement of the tail-cancellation condition. To show this in detail, after integration by parts... 

The Schlosser-Marcus variational principle is derived for a single surface a that subdivides coordinate space 9i3 into two subvolumes rm and rout. This generalizes immediately to a model of space-filling atomic cells, enclosed for a molecule by an external cell extending to infinity. The continuity conditions for the orbital Hilbert space require i>out =a i>in This implies a vanishing Wronskian surface integral... 

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... 

This can vanish only if (H — r) b = 0 in both rin and rout. Moreover, this requires that both ( ifrin Wa if/in — fout) and (S j/0Ut Wa i//in — j/out) must vanish when ijrin and fs out are varied independently. By an extension of the surface matching theorem, both these Wronskian integrals must vanish in order to eliminate the value and normal gradient of tf/m — i//""r on o. Practical applications of this formalism use independent truncated orbital basis expansions in adjacent atomic cells, so that the continuity conditions cannot generally be satisfied exactly. 

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 variational equations imply [r — t- =a 0 on each cell boundary ct/2. Given independent expansions ijr = E/. within each atomic cell, and (ct) = J2/1.L Ni(To)Pl on the global matching surface, the coefficients are determined by the implied variational equations. The surface matching theorem implies two independent Wronskian integral conditions for each atomic cell,... 

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