Riemann sheets

The important point is that v E,k) and A0( ,fe) are multivalued, with a logarithmic branch point at fe = s = 0, while the residual functions m(e, k, zo) and w(e, k) are single valued. The result, as discussed in more detail later, is that the value of the quantum number v depends on the chosen location of the branch cut and on which Riemann sheet is taken, bearing in mind that the branch of arctan( /fe) must be taken according to the appropriate quadrant of the complex k, e) plane. Thus cj) = arctan( /fe) + ti/2 increases smoothly from zero to In around a counterclockwise circle in the k, e) plane, starting at = 0 and < 0. 

The Bohr quantization condition, with quantum number v, corresponds to choosing the lowest Riemann sheet, which requires an additional phase correction of —2ti on crossing the branch cut, which is taken along the positive -axis. Thus the phase term cj) rises to ti at = 0+ and > 0, drops abruptly to —71 on crossing the positive fe = 0 axis, and returns to zero on the negative... 

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

The resolvent can be continued analytically from above through the cut into a second Riemann sheet. The decaying state is now associated with a pole on the second sheet. Let us write the coordinates of the pole as... 

Now, it is clear that since t > 0 in the complex exponential in the integrand of eq. (12-29) we must close the contour in the lower half plane. Thus we analytically continue the functions Rij(E) from the first to the second Riemann sheet through the real T -axis (the branch cut of G(E)), i.e., we set... 

Figure 2.11 3D plot of the Riemann sheet structure of the perturbed Coulomb potential as displayed in Eq (64) with a varying barrier height parameter The imaginary part of Vmax grows toward the viewer. 

Recently,30-31 it has been shown that the lowest Riemann sheet of the twovalued potential-energy surface for a homonuclear triatomic system can be represented by a function which shows the correct analytical structure when expanded in terms of the Dih coordinates (17), (18) near the conical intersection. Such a procedure has been suggested for the analytical continuation of the potential energy from the lower to the upper Riemann sheet.30 By carrying out a similar expansion for the LSTH potential, it is easy to show that it contains improper terms in the sense of ref. 30, thus invalidating its use for the analytical continuation of the energy to the upper Riemann sheet of the H3 surface. 

Due to the considerable success of the functional form used by Truhlar and Horowitz, this has been kept as much as possible to represent the EHF energy term of the H3 potential after correcting for the formally unacceptable terms mentioned above. We note that in the Truhlar-Horowitz parametriza-tion procedure the H2(h3Z, ) potential function is partly obtained from a least-squares fit to the Dxh energies of H3 (Section 1V.B.1). Thus, in order to allow the upper Riemann sheet of the H3 potential to dissociate into the correct triplet state of H2, the following form has been used for the EH F triplet ... 

Fig. 5.1b. Qualitative behavior of —Q2(rf) for m / 0 in the sub-barrier case. The wavy lines are cuts, and Al and Ak are closed contours of integration. The part of Al that lies on the second Riemann sheet is drawn as a broken line. The phases of Q1 2(r ) indicated in the figure refer to the first Riemann sheet.

Therefore, the existence of singularities leads to an infinite number of Riemann sheets on the complex lapse time plane, half of which contribute to the reflection and the other half of which yield transmissive components. The destination of the trajectory—that is, which side is it ending up, transmissive side or the reflective side —changes depending on a choice of integration paths. 

In the case of an arbitrary p, one can introduce in addition branch cuts in the complex / -plane in order to ensure single-valued /( r)> because for 0 R < + , x(p R) can have more than one Riemann sheet. In particular, the following mappings result ... 

Figure 5 (A) Upper halfplane A structured scattering cross section as a function of energy. (B) Complex S-matrix poles in the second Riemann sheet which, together with their residues, are used to describe the cross section.

V (z) describes a decreasing in time quasi-stationary state. Contrary to the Lippmann-Schwinger equation, which requires scattering boundary conditions, V (z) does require outgoing boundary conditions commensurate with the Gammow-Siegert method. It is inherent in the complex technique and defined in a nonambiguous manner as a continued wavefunction in the second Riemann sheet. 

In fhe general case, fhe contour for fhe complex variable z = E irj surrounds fhe specfrum of H counferclock-wise on the first Riemann sheet of E, in which case if is valid for f > 0 and for f < 0. However, in fhe case of fhe decay of an unsfable sfafe of a field-free Hamiltonian, rigor implies that the following physically consfraints musf be imposed on fhe infegrafion of Eq. (7) E > 0 and f > 0 [37,89]. 

When the contour surrounds the spectrum of H on the first Riemann sheet of E, the integral (6c) is valid for f > 0 and for f < 0. Since R(z) is analytic on the first Riemann sheet, any of its possible simple complex poles must appear on the second sheet. Indeed, Eq. (6c) leads naturally to the identification of the isolated unstable state corresponding to (0) with a complex energy of Eq. (2) that is identical to a complex pole of ( (0) R(z) (0))on the second Riemann sheet below the real axis, for t > 0 [1,3,10,11]. 

In potential scattering models the basic "mathematical" reason for exponential decay is a complex pole in the fourth quadrant of the momentum complex plane (second Riemann sheet of the energy plane), which, through its exponentially decaying residue, dominates the dynamics for some time. A simple analytical example of the deviation from exponentiality follows from the integral expression for the survival amplitude. 

Now consider the ground state function E (z). Starting at the origin, trace a path in the z-plane that circles about the point zoi This point is a branch point singularity, which means that a 360° circuit will lead to a new Riemann sheet corresponding to the fimction E > (z). Similarly, following E ) (z) on a path that circles zq2 leads to E (z), and so on. 

FIGURE 13.9 Representations of Riemann surface for f z) = The dashed segments of the loops lie on the second Riemann sheet. 

The Riemann surface for the cube root f(z) = z comprises three Riemann sheets, corresponding to three branches of the function. Analogously, any integer or rational power of z will have a finite number of branches. However, an irrational power such as f(z) = z = will not be periodic in any integer multiple of In and will hence require an infinite number of Riemaim sheets. The same is true of the complex logarithmic function... 

For Bessel functions of noninteger order v, the same integral pertains except that the contour must be deformed as shown in Fig. 13.12, to take account of the multivalued factor z - The contour surrounds the branch cut along the negative real axis, such that it lies entirely within a single Riemann sheet. 

Let us recall that Go is always assumed to be analytically continued in the second Riemann sheet. 

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