# Stable Fermion Bag Solitons in the Massive Gross-Neveu Model: Inverse Scattering Analysis

###### Abstract

Formation of fermion bag solitons is an important paradigm in the theory of hadron structure. We study this phenomenon non-perturbatively in the dimensional Massive Gross-Neveu model, in the large limit. We find, applying inverse scattering techniques, that the extremal static bag configurations are reflectionless, as in the massless Gross-Neveu model. This adds to existing results of variational calculations, which used reflectionless bag profiles as trial configurations. Only reflectionless trial configurations which support a single pair of charge-conjugate bound states of the associated Dirac equation were used in those calculations, whereas the results in the present paper hold for bag configurations which support an arbitrary number of such pairs. We compute the masses of these multi-bound state solitons, and prove that only bag configurations which bear a single pair of bound states are stable. Each one of these configurations gives rise to an antisymmetric tensor multiplet of soliton states, as in the massless Gross-Neveu model.

###### pacs:

11.10.Lm, 11.15.Pg, 11.10.Kk, 71.27.+a## I Introduction

An important dynamical mechanism, by which fundamental particles acquire masses, is through interactions with vacuum condensates. Thus, a massive particle may carve out around itself a spherical region sphericalbag or a shell shellbag in which the condensate is suppressed, thus reducing the effective mass of the particle at the expense of volume and gradient energy associated with the condensate. This picture has interesting phenomenological consequences sphericalbag ; mackenzie .

This dynamical distortion of the homogeneous vacuum condensate configuration, namely, formation of fermion bag solitons, was demonstrated explicitly by Dashen, Hasslacher and Neveu (DHN) dhn many years ago, in their study of semiclassical bound states in the dimensional Gross-Neveu (GN) model gn .

Fermion bags in the GN model were discussed in the literature several other times since the work of DHN, using alternative methods others ; papa ; josh1 . For a review on these and related matters (with an emphasis on the relativistic Hartree-Fock approximation) see thiesreview . For a more recent review of static fermion bags in the GN model (with an emphasis on reflectionless backgrounds and supersymmetric quantum mechanics) see bagreview . The large- semiclassical DHN spectrum of these fermion bags turns out to be essentially correct also for finite , as analysis of the exact factorizable S-matrix of the GN model reveals Smatrix .

A variational calculation of these effects in the dimensional massive generalization of the Gross-Neveu model, which we will refer to as MGN, was carried in FZMGN a few years ago, and more recently in Thies . In this paper we study static fermion bags in the MGN model using inverse-scattering formalism inverse , thus avoiding the need to choose a trial variational field configuration. The present work is thus a natural extension of the inverse-scattering analysis carried out by DHN to the massive case.

The MGN model is defined by the action

(1) | |||||

describing self interacting massive Dirac fermions carrying a flavor index , which we promptly suppress.

An obvious symmetry of (1) with its Dirac spinors is . Actually, (1) is symmetric under the larger group dhn (see also Section 1 of bagreview ). The fact that the symmetry group of (1) is rather than , indicates that it is invariant against charge-conjugation. It is easy to see this in a concrete representation for matrices, which we choose as the Majorana representation dhn ; bagreview

(2) |

(Henceforth, in this paper we will use this representation for matrices in all explicit calculations.)

In the representation (2), charge-conjugation is realized simply by complex conjugation of the spinor

(3) |

Thus, if is an eigenstate of the Dirac equation

(4) |

with energy (assuming time independent ), then is an energy eigenstate of (4), with energy . Therefore, the MGN model (1), like the GN model, is invariant against charge conjugation, and energy eigenstates of (4) come in pairs.

As usual, the theory (1) can be rewritten with the help of the scalar flavor singlet auxiliary field . Also as usual, we take the large limit holding fixed. Integrating out the fermions, we obtain the bare effective action

(5) |

Noting that , we can rewrite the as . In this paper we focus on static soliton configurations. If is time independent, the latter expression may be further simplified to

(6) |

and where is an infra-red temporal regulator.

As it turns out, the two Schrödinger operators are isospectral (see Appendix A of bagreview and Section 2 of josh1 ) and thus we obtain

(7) | |||||

In contrast to the standard massless GN model, the MGN model studied here is not invariant under the symmetry , , and the physics is correspondingly quite different. As a result of the degeneracy of its vacuum, the GN model contains a soliton (the so called CCGZ kink ccgz ; dhn ; others ; bagreview ; josh1 ) in which the field takes on equal and opposite values at . The stability of this soliton is obviously guaranteed by topological considerations. With any non-zero the vacuum value of is unique and the CCGZ kink becomes infinitely massive and disappears. If any soliton exists at all, its stability has to depend on the energetics of trapping fermions.

Let us briefly recall the computation of the unique vacuum of the MGN model. We shall follow FZMGN . For an earlier analysis of the MGN ground state (as well as its thremodynamics), see klimenko . Setting to a constant we obtain from (7) the renormalized effective potential (per flavor)

(8) |

where is a sliding renormalization scale with and the running couplings. By equating the coefficient of in two versions of , one defined with and the other with , we find immediately that

(9) |

and thus the coupling is asymptotically free, just as in the GN model. Furthermore, by equating the coefficient of in we see that the ratio is a renormalization group invariant. Thus, and have the same scale dependence.

Without loss of generality we assume that and thus the absolute minimum of (8), namely, the vacuum condensate , is the unique (and positive) solution of the gap equation

(10) |

Referring to (1), we see that is the mass of the fermion. Using (9), we can re-write the gap equation as , an observable physical quantity, is a renormalization group invariant. This equation also implies that , which makes sense physically. , which shows manifestly that

Fermion bags correspond to inhomogeneous solutions of the saddle-point equation . In particular, static bags are the extremal configurations of the energy functional (per flavor)

(11) |

subjected to the boundary condition that relaxes to its unique vacuum expectation value at .

The rest of this paper is organized as follows. In the next section we express in terms of the scattering-data. DHN have already done most of the work for us, save for one crucial piece, specific to the MGN model: an expression for in terms of the scattering data. In this paper we derive this relation. The details of this derivation are relegated to the Appendix. We then complete the task of writing down in terms of the scattering data. We prove that the static extrema of are reflectionless, as in the massless GN model dhn , and calculate their masses. Such an extremal configuration, considered as a scalar background in the Dirac equation, typically supports a number of pairs of charge conjugate bound states at energies which bind fermions and antifermions. Each one of these s depends only on the total number of fermions and antifermions it binds, thus giving rise to an rank- antisymmetric tensor multiplet of soliton states. As it turns out, the mass of such a soliton is a convex function of the s. In Section 3, we invoke the convexity of the soliton’s mass formula and prove that only solitons which support a single pair of bound states (i.e., ) are stable against decaying into lighter solitons. These are precisely the configurations studied variationally in FZMGN ; Thies .

## Ii Static Configurations and Inverse-Scattering Analysis

A fermion bag is essentially a trap for fermions. Evidently, an apriori specification of a static fermion bag configuration should indicate the list of bound states and enumerate the fermions it traps. To be specific, we shall evaluate in this section the energy functional (11) of a static configuration , obeying the appropriate boundary conditions at spatial infinity, which supports pairs of bound states of the Dirac equation at energies , (where, of course, ). The bound states at are to be considered together, due to the charge conjugation invariance of the GN model. Due to Pauli’s exclusion principle, we can populate each of the bound states with up to fermions. In such a typical multiparticle state, the negative frequency state is populated by fermions and the positive frequency state contains fermions. In the parlance of Dirac’s hole theory, this is a many fermion state, with particles and holes occupying the pair of charge-conjugate bound states at energies . We shall refer to the total number of particles and antiparticles trapped in the n-th pair of bound states

(12) |

as the valence, or occupation number of that pair.

From (7) and (11) we obtain the bare energy functional as

(13) | |||||

where

(14) |

(Here we used by invoking the boundary conditions .) The expression (13) is divergent. We regulate it, as usual, by subtracting from it the divergent contribution of the vacuum configuration and by imposing a UV cutoff on . Thus, the regulated bare energy functional associated with is

(15) |

where

(16) |

is the hamiltonian corresponding to the vacuum configuration. We are not done yet, as the integrals over in (15) still diverge logarithmically with the UV cutoff . However, this logarithmically divergent term is precisely the one that should be added to the bare in the first term in (15) in order to obtain the renormalized coupling appearing in (9) dhn ; josh1 ; bagreview .

Now that we have written the energy-functional (15) of a static configuration, or a fermion bag, our next step is to identify those fermion bags on which (15) is extremal.

The energy functional (15) is, in principle, a complicated and generally unknown functional of and of its derivatives. Thus, the extremum condition , as a functional equation for , seems intractable. The considerable complexity of the functional equations that determine the extremal configurations is the source of all difficulties that arise in any attempt to solve the model under consideration.

DHN found a way around this difficulty in the case of the GN model dhn . They have used inverse scattering techniques inverse to express the energy functional (15) in terms of the so-called “scattering data” associated with, e.g., the hamiltonian in (6) (and thus with ), and then solved the extremum condition on (15) with respect to those data.

The scattering data associated with are inverse the reflection amplitude of at momentum , the number of bound states in and their corresponding energies , and also additional parameters , where has to do with the normalization of the th bound state wave function of . More precisely, the th bound state wave function, with energy , must decay as as , where

(17) |

If we impose that be normalized, this will determine the constant coefficient as . (With no loss of generality, we may take .) Recall that , since the Schrödinger potential is real. Thus, only the values of for enter the scattering data. The scattering data are independent variables, which determine uniquely, assuming belongs to a certain class of potentials which fall-off fast enough towards infinity.

Since the MGN does not bear topological solitons, neither nor can have a normalizable zero energy eigenstate jackiwrebbi . (See also Section A.1.1 in Appendix A of bagreview .) Thus, all the are strictly positive.

We can apply directly the results of DHN in order to write down that part of (15) which is common to the MGN and GN models, i.e., (15) with its term proportional to removed, in terms of the scattering data. In what follows we briefly summarize their results. (See Sections 2, 3, and Appendix B of dhn for details.)

Using the trace identities of the spectral theory of , they were able to show that

(18) |

This takes care of the first term in (15). Onto the spectral determinants: These encode the contribution of the Fermi fields to the energy of the bag relative to the vacuum. To account for it correctly, we should put our field theory in a big spatial box of length so as to make the spectrum discrete, which will enable us matching each mode of fermion fluctuations around the vacuum, with its counterpart, obtained as is turned on adiabatically. We shall take the limit only in the end. Thus, we obtain the Fermi field part of the energy (per flavor) as

(19) |

The first sum runs over all positive-energy scattering states, where is the energy of the scattering state to which the Fermi mode energy flows to as the vacuum configuration is deformed adiabatically to . The second sum in (II) accounts for the first scattering states above the threshold which migrate into the gap to become the (positive energy) bound states as is switched on. In the limit their energies are indistinguishable from . Note the minus sign in front of these two sums, as appropriate for fermion zero-point energy. The last sum in (II) is evidently the contribution of valence fermions and antifermions trapped inside the bag at the bound states . By carefully counting scattering modes in the box, DHN arrived at the fairly standard result

(20) |

where is the scattering phase-shift. Then, changing to momentum space (), and using a dispersion integral representation for , DHN derived that

(21) | |||||

The second integral in (21) can be calculated explicitly. In the limit we obtain

(22) |

DHN’s results, Eqs. (II) - (21), correspond to all terms of (15), save for the term proportional to , . This integral cannot be expressed in terms of the scattering data based on the trace identities of the Schrödinger operator discussed in Appendix B of dhn . Evidently, new analysis is required to obtain its representation in terms of the scattering data. Happily enough, we were able to obtain such a representation. In the Appendix we prove that

(23) |

Combining (II)- (21) and (II) we obtain the desired expression for the energy functional (15) in terms of the scattering data as

(24) |

where

(25) |

We shall now extremize (24) with respect to the scattering data, to obtain the self-consistent static fermion bags in the MGN model. Let us vary with respect to (with ) first. As is evident from (II), (21) and (II), , where is a calculable function, which does not vanish identically. (For example, it can be shown that .) Thus, is the unique solution of the variational equation . Static extremal bags in the MGN model are reflectionless, as their counterparts in the GN model.

Explicit formulas for reflectionless configurations with an arbitrary number of pairs of bound states are displayed in Appendix B of bagreview . In particular, the one which supports a single pair of bound states at energies (), the one originally discovered by DHN, is

(26) | |||||

We see that the formidable problem of finding the extremal configurations of the energy functional (15), is reduced to the simpler problem of extremizing an ordinary function with respect to the 2 parameters that determine the reflectionless background . If we solve this ordinary extremum problem, we will be able to calculate the mass of the fermion bag. Let us write down this function explicitly:

(27) |

The logarithmically divergent term in (27) should remind us that this equation is expressed in terms of the bare couplings and . As it turns out, the renormalization procedure of the effective potential for the constant condensate, which we reviewed in the Introduction, applies also in the case of inhomogeneous background : The bare coupling in (27) is related to the renormalized one at the energy scale via an equation identical to (9), namely,

(28) |

(Due to the anisotropic cutoff implied in (21), the cutoff scale in (28) is rather than just .) In addition, since the ratio is a renormalization group invariant all the way up to the cutoff scale, we can replace the coefficient of the last term in (27) by the common value of that invariant, namely, . With the help of these two relations, we obtain the renormalized form of (27) as

(29) |

Next, note that the energy functional evaluated at a reflectionless , is independent of the normalization coefficients , that do affect the shape of . The ’s are thus flat directions of in the space of all reflectionless configurations. In fact, the ’s (or more precisely, their logarithms) are collective translational coordinates of the fermion bag (see e.g., Appendix B in bagreview ). One of these coordinates, corresponds, of course, to global translations of the bag as a whole.

Finally, we are left with the task of extremizing (29) with respect to the bound state energies of the reflectionless background . To this end, we follow DHN and parametrize these energies as , with . (Note that cannot attain the end-point values of its range: plunges into the continuum, and is possible only if is topologically nontrivial, which is not the case in the MGN model.) From (17) we see that . In terms of the angular variables, we may write (29) as

(30) |

where we have defined the renormalization group invariant

(31) |

Finally, extremizing (30) with respect to yields

(32) |

thus fixing as a function of the filling fraction

(33) |

according to

(34) |

The fact that the extremal value of is determined by the total number of particles and holes trapped in the bound states of the Dirac equation at , and not by the numbers of trapped particles and holes separately, is a manifestation of the underlying symmetry, which treats particles and holes symmetrically. As explained in witten and in Appendix D of bagreview , this fact indicates that this pair of bound states gives rise to an antisymmetric tensor multiplet of rank of soliton states. The soliton as a whole is therefore the tensor product of all these antisymmetric tensor multiplets. The fermion number operator is one of the generators of . Its expectation value in the background of the extremal fermion bag in one of its states is (see Section 3 of bagreview )

(35) |

which is simply the sum over the individual valence fermion numbers

(36) |

of each of the states in the individual antisymmetric factors. Evidently, for each of these antisymmetric representations

(37) |

in accordance with charge conjugation invariance.

The left-hand side of (34) is a monotonically increasing function. Therefore, (34) has a unique solution in the interval . This solution is evidently smaller than , the corresponding value of in the GN model for the same occupation number. Thus, the corresponding bound state energy in the MGN model is higher than its GN counterpart, and thus less bound.

Substituting the extremal ’s from (34) in (30) we find that the mass of our soliton (namely, evaluated at the extremal point) is

(38) |

In the case of a single pair of bound states, , (32)- (34) and (38) coincide with the corresponding results of variational calculations presented in FZMGN ; Thies , which were based on (26) as a trial configuration. In fact, it was realized in Thies that the trial configuration (26) is an exact solution of the extremum condition , provided (34) is used to fix . This choice of trial configuration was very successful indeed!

We should mention that renormalization of the energy functional (15) in the background of a generic reflectionless and its extremization with respect to the s can be carried out using an alternative method based on the diagonal resolvent of the Dirac operator bagreview , which is basically a generalization of the calculations in FZMGN ; josh1 to the case of an arbitrary number of pairs of bound states. As it turns out, there are simple explicit formulas for the diagonal resolvents of the Dirac operator and of in reflectionless backgrounds, which make these computations possible, and lead to (32)- (38) FHunpublished .

## Iii Investigating Stability of Extremal Static Fermion Bags

The extremal static soliton multiplets which we encountered in the previous section, correspond, in the limit , to exact eigenstates of the hamiltonian of the MGN model. However, at large but finite , we expect some of these states to become unstable and thus to acquire small widths, similarly to the behavior of baryons in QCD with a large number of colors wittenlargeN . The latter are also solitonic objects and are analogous to the “multi-quark” bound states of the MGN and GN models. In particular, Section 9 of wittenlargeN offers a sketch of the expansion of two-dimensional QCD in the Coulomb gauge, both in the baryon and meson sectors, which is similar to the corresponding expansion of the GN and MGN models, in the presence of fermion bags. (This expansion is based on the so-called bilocal condensate formalism, which was later developed in papa ; rdmc ; bilocal .)

Furthermore, we can imagine perturbing the MGN action (1) by a small singlet perturbation (e.g., by adding to (1) a term ), and ask which of the extremal fermion bags of the previous section are stable against such perturbations. (The perturbations and correspond merely to a redefinition of the bare quantities and , and are thus not interesting. The higher perturbations, with , are non-renormalizable. However, we could think of the resulting perturbed lagrangian as an effective one.) Under these circumstances, all possible decay channels of a given unstable soliton multiplet must conserve energy, momentum and quantum numbers.

It turns out that non-trivial results concerning stability may be established without getting into all the details of decomposing representations, by imposing a simple necessary condition on the spectrum of the fermion number operator in the multiplets involved in a given decay channel. This way of arguing (as described in detail below) has led, in the case of the GN model (see Section 4 of bagreview ), to specification of all topologically-trivial stable fermion bags, consistent with the exact results of Smatrix . Thus, it is reasonable to expect (and conjecture) that applying these stability considerations to the MGN model should lead to the correct list of fermion bags in this model which remain stable at finite as well. Unfortunately, exact results analogous to Smatrix and valid for finite , are not available for the MGN model, so this conjecture has to be verified by explicit calculation of corrections. We shall now make these stability considerations explicit.

As we have learned so far, a given static soliton multiplet in our model is a direct product of antisymmetric tensors. The decay products of this soliton also correspond to a direct product of antisymmetric tensors. According to (37), the spectrum of in an antisymmetric tensor representation is symmetric, namely, . When we compose two such representations , the spectrum of in the composite representation , will obviously have the range . In particular, each of the possible eigenvalues in this range, will appear in at least one irreducible representation in the decomposition of . More generally, the spectrum of will have the range .

Consider now a decay process, in which a parent static soliton, which belongs to a (possibly reducible) representation , decays into a bunch of other solitons, such that the collection of all irreducible representations associated with the decay products is (in which a given irreducible representation may occur more than once). By symmetry, the representation must occur in the decomposition of . Thus, according to the discussion in the previous paragraph, if this decay process is allowed, we must have

(39) |

which is the necessary condition for symmetry we sought for. (Obviously, similar necessary conditions arise for the other components of the highest weight vectors of the representations involved.) The decay process under consideration must respect energy conservation, i.e., it must be exothermic. Thus, we supplement (39) by the requirement

(40) |

on the masses of the particles involved.

For each of the static soliton multiplets discussed in the previous section, we will scan through all decay channels (into static solitons) and check which of these decay channels are necessarily closed, simply by requiring that the two conditions (39) and (40) be mutually contradictory.

In order to complete our argument, we must make the plausible assumption that for any time dependent stable soliton with given quantum numbers (should such a soliton exist), the static soliton with the same quantum numbers is lighter. In the GN this is in fact true for the known time dependent DHN breathers dhn .

For all the multiplets with a single pair of bound state, , we will find in this way that all decay channels are necessarily closed, thus establishing their stability. That these are stable multiplets is almost obvious to begin with - they are the lightest solitons, given their quantum numbers. We cannot establish in this way that all decay channels are necessarily closed for the higher solitons , and they are presumably unstable.

Consider the function

(41) |

where is a solution of (34). From (34) and (41) we obtain that

(42) |

Thus, is a monotonically increasing convex function in the range of interest, satisfying . In terms of (41), we may write the soliton mas (38) simply as

(43) |

Now, we are ready to start the stability analysis. Consider the decaying parent soliton to be a static soliton with pairs of bound states, corresponding to the direct product of antisymmetric tensor representations of ranks . The mass of this soliton is , and according to (37), the maximal fermion number eigenvalue occurring in this representation is .

Following the strategy which we laid above, we shall now scan through all imaginable decay channels of this parent soliton (into final states of purely static solitons), and identify those channels which are necessarily closed. Thus, assume that the parent soliton under consideration decays into a configuration of lighter solitons, with quantum numbers of the direct product of antisymmetric tensor representations . The way in which these multiplets are arranged into extremal fermion bags is of no consequence to our discussion. Thus, we discuss all decay channels consistent with these quantum numbers in one sweep.

The necessary conditions (39) and (40) for possible decay imply

(44) |

where all . The two pairs of boundary hypersurfaces in (44) are

(45) |

where are hypersurfaces in -space and are the corresponding hypersurfaces in -space.

Consider the behavior of (43) over the hyperplane

(46) |

where is an integer and . This hyperplane corresponds, of course, to solitons with a fixed value of the maximal fermion number . We will now prove that attains its absolute minimum on in the positive orthant, at the vertices of the intersection of and the hypercube , namely, the points

(47) |

with all possible choices of coordinates out of .

We prove this as follows (see Appendix F of bagreview ): consider a sequence , subjected to (46). Assume that for some , . We will show that there exists another sequence of ’s, with the same sum, but with a lower sum of the s. Thus, let be chosen such that and , i.e., . Modify the sequence under consideration by replacing by and by , keeping the other terms unaltered. The new sequence thus obtained has the same sum as the original sequence, and thus defines another point on . We must show that . Indeed, . Clearly, , and also in the relevant range of . Thus, increases monotonically with , and reaches its maximum at , where, depending on the initial condition at , either or . Thus, the sequence of ’s constrained to , which minimizes (43), cannot have more than one element in the interior of . Thus, due to (46), the absolute minimum is the sequence in which the largest ’s are , one is and the rest are zero, namely, the vertices (47). This is just the statement that the mass function (43), being the sum of the convex functions , is convex inside the cube .

Therefore, a parent soliton corresponding to a point in the interior of the intersection of and the hypercube , can decay into a final state with quantum numbers corresponding to the points (47), i.e., and in (44). In fact, by continuity, such a parent soliton can decay also at least into the set of final states contained in small pockets above the vertices (47), which correspond to and in (44), where is some small calculable quantity, or into final states corresponding to in (44), with for and for , and where is another small calculable quantity.

On the other hand, the parent soliton which corresponds to the vertices (47) has no open channel to decay through. Hence it is potentially stable. Indeed, if it could decay through a channel corresponding to , then, from the requirement that these parameters satisfy (44), we would have

(48) |

Define the hyperplane

(49) |

From the analysis above we know that the absolute minimum of over the intersection of and the hypercube is . Thus, the points which satisfy the second inequality in (III) are bounded by , in contradiction with the first inequality in (III). This completes the proof that parent solitons which correspond to the vertices (47) are stable.

Each such vertex represents a soliton which cannot decay through the channel under consideration, and is thus potentially stable. More precisely, all these vertices correspond to the same soliton, since the coordinates of these vertices are just permutations of each other, and thus all of them correspond to the same set of parameters, in which

(50) |

Does such a soliton exist? To answer this question let us recall a few basic facts: The parent soliton under discussion is topologically trivial. As such, it must bind fermions to be stabilized, and none of its bound state energies may vanish. Thus, all the ranks occurring in it must satisfy . Finally, note, that due to the elementary fact, that the spectrum of the one dimensional Schrödinger operator cannot be degenerate, all the ’s must be different from each other, and so must be the s. Thus, the only physically realizable parent solitons, which are necessarily stable against the decay channel in question, correspond to and . These are, of course, the static solitons studied in FZMGN ; Thies .

A corollary of our analysis so far is that these solitons are stable against decaying into free fermions or antifermions (i.e., fundamental representations) plus solitons corresponding to higher antisymmetric tensor representations. In particular, it is stable against complete evaporation into free fermions. (Strictly speaking, this argument is valid only for values of which are a finite fraction of , since our mass formula (43) is the leading order in the expansion, while removing a finite number of particles from the parent soliton is a perturbation of the order relative to (43).) Stability of the solitons was studied in FZMGN in detail. In particular, the regimes and were investigated. It was noted in FZMGN that the mass of the most stable soliton, obtained as the filling fraction , is nonanalytic around (with leading nonanalytic behavior ). It was shown in FZMGN that this nonanalyticity was related to the enhanced symmetry of the GN model at . The opposite regime, may be attained by making the four-fermi interactions weak. In this regime the theory should be described in terms of quasi-free massive fermions of mass . We thus expect that the binding energy of bags will tend to zero as , which was indeed verified in FZMGN .

## Iv Appendix: Derivation of (Ii)

Consider Schrödinger equation

(51) |

with given by (14). Due to the boundary conditions on at spatial infinity, . Let