On Green's functions for Hamiltonians with potentials possessing singularity at the origin: application to the zero-range potential formalism
aa r X i v : . [ m a t h - ph ] M a y On Green’s functions for Hamiltonians withpotentials possessing singularity at the origin:application to the zero-range potential formalism
S L Yakovlev, V A Gradusov
Department of Computational Physics, St Petersburg State University, 198504, StPetersburg, RussiaE-mail: [email protected]
E-mail: [email protected]
Abstract.
We evaluate the short-range asymptotic behavior of Green’s function fora Hamiltonian when its potential energy part has an inverse power singularity atthe origin. The analytically solvable case of sharply screened Coulomb potential isconsidered firstly. For this potential the additional logarithmic singular term has beenfound in the short-range asymptote of the Green function as in the case of the pureCoulomb potential. The case of a short-range potential of an arbitrary form withinverse power singularity is treated on the basis of the integral Lippmann-Schwingerequation. It is shown that, if the singularity is weaker than the Coulomb one, theGreen function has only standard singularity. For the case of r − ρ singularity of thepotential with 1 ≤ ρ < ρ = 1 the additional logarithmic singularity hasthe same form as in the case of the pure Coulomb potential. In the case of 1 < ρ < r − ρ +1 . These results are applied for extending the zero-range potential formalism onHamiltonians with singular potentials.PACS numbers: 03.65.Nk, 34.80.-Bm Submitted to:
J. Phys. A: Math. Gen. hort-range plus zero-range potentials
1. Introduction and formulation of the problem
In this paper we concentrate on those attributes of Green’s function G ( z ) of aHamiltonian H , which are needed for evaluating the short-range asymptotic behaviorof the Green function in the configuration space. As an operator G ( z ) is defined as G ( z ) = ( H − z ) − for z ∈ C . The Hamiltonian is assumed to have the form H = − ∆ + V ( r ) , (1)where ∆ stands for the Laplacian over r ∈ R . While the three-dimensionalconfiguration space is considered, the case of arbitrary dimension d > V ( r ) is supposed to represent a short-range potential,i.e. it is a real-valued smooth function for all r ≡ | r | > V ( r ) ∝ r − − δ , δ > ‡ when r → ∞ . More precisely, we assume that there exists aconstant C > | V ( r ) | ≤ C (1 + r ) − − δ , δ > r except of a small neighborhood of the origin r = 0. The specific concernof the paper is the singular short-range behavior of the potential V ( r ) ∝ r − ρ , when r →
0. More precisely, we assume that the potential V ( r ) in a neighborhood of theorigin r = 0 can be represented as V ( r ) = r − ρ W ( r ) , (3)where W ( r ) is a smooth bounded function with the finite limitlim r → W ( r ) = V . (4)In what follows this class of potentials will be referred to as the V ( ρ, δ ) class. ForHermiticity of H it is sufficient to require ρ < V C ( r ) = V r − .This modification is represented in the short-range behavior of the solution φ of theSchr¨odinger equation φ ∼ α π [1 /r + V log( r )] + β, r → r − one. Themodification appears in fact as the result of the interplay between singularities of theCoulomb potential and the zero-range potential. Before treating the general case wewill deal with the sharply screened Coulomb potential V C R ( r ) = V C ( r ) θ ( R − r ), where ‡ This condition for δ can be weakened up to δ >
0. However we will use the stronger condition δ > hort-range plus zero-range potentials R > θ -function is defined as θ ( t ) = 1 (0) when t ≥ < V C R , which is alsoanalytically solvable [3] as V C , in order to emphasize that only the short-range behaviorof the Coulomb potential is responsible for the effect of that interplay and therefore thelong-range behavior of the tail of the Coulomb potential does not affect the zero-rangepotential structure.One of the approaches for constructing the zero-range potential in the general caseof V ∈ V ( ρ, δ ) consists in inserting a delta-functional term into the Schr¨odinger equation[2] (cid:2) − ∆ + V ( r ) − k (cid:3) φ ( r , k ) + λδ ( r ) β = 0 , (6)where λ is a coupling constant and β is actually a linear functional of φ [1]. Then thesolution of (6) can be given by the Lippmann-Schwinger integral equation φ ( r , k ) = φ ( r , k ) − λ Z d r ′ G + ( r , r ′ , k ) δ ( r ′ ) β. (7)Here by G + ( k ) we denote lim ǫ → G ( k +i ǫ ) and this notation will be used systematicallythroughout the paper. In (7) the function φ is the solution to the equation (cid:2) − ∆ + V ( r ) − k (cid:3) φ ( r , k ) = 0 , (8)obeying the asymptotic boundary condition φ ( r , k ) ∼ exp (i k · r ) + Ar − exp (i kr ) (9)as r → ∞ . The dot-product here and further means the scalar product of vectors in R .The integration in (7) is performed easily due to the delta-function which yields φ ( r , k ) = φ ( r , k ) − λG + ( r , , k ) β. (10)As it will be shown below the limit of φ as r → V ∈ V ( ρ, δ ) and thereforethe non trivial short-range asymptotic of φ is completely determined by Green’s functionterm in (10). Hence, from (10) it is seen that the principal features of the zero-rangepotential formalism follow from the short-range behavior in r of the Green function G + ( r , , k ) as it takes place in the case of regular potentials [1]. In consecutive sectionswe will evaluate the respective asymptote of the Green function and as the result thezero-range potential will be constructed.The paper is organized as follows. In section 2 we derive the closed formrepresentation for the Green function of the Hamiltonian with sharply screened Coulombpotential V C R and infer the asymptote from it. As to the best of our knowledge, thisrepresentation for r, r ′ < R has been obtained here for the first time. The section 3 isdevoted to studying the general case of potentials from V ( ρ, δ ). In the section 4 theasymptotes of the Green function are used for evaluating the short-range behavior ofthe solution of (6) and establishing the zero-range potentials for different values of ρ .Also in this section the respective pseudo-potentials are constructed. The section 5 givesconcluding remarks. hort-range plus zero-range potentials
2. Green’s function for Screened Coulomb potential
The Green function is defined by the solution to the inhomogeneous equation (cid:2) − ∆ + V C R ( r ) − k (cid:3) G + R ( r , r ′ , k ) = δ ( r − r ′ ) . (11)One of the convenient ways for constructing the Green function for radial potentials isthe use of the partial wave decomposition [4]. The Green function is represented thenas the series in terms of Legendre polynomials P ℓ G + R ( r , r ′ , k ) = 14 π ∞ X ℓ =0 (2 ℓ + 1) G Rℓ ( r, r ′ , k ) rr ′ P ℓ ( ˆ r · ˆ r ′ ) , (12)where ˆ r = r r − . The partial Green function G Rℓ obviously obeys the one-dimensionalequation (cid:20) − d d r + ℓ ( ℓ + 1) r + V C R ( r ) − k (cid:21) G Rℓ ( r, r ′ , k ) = δ ( r − r ′ ) (13)with natural boundary condition G Rℓ = 0 as r = 0. The radiation boundary conditionas r → ∞ requires the outgoing wave asymptote G Rℓ ∝ exp(i kr − i πℓ/ G Rℓ can be constructed following the standard procedure [5] G Rℓ ( r, r ′ , k ) = − u ℓ ( r < ) v ℓ ( r > ) W ( u ℓ , v ℓ ) , (14)where r > = max { r, r ′ } , r < = min { r, r ′ } and W ( u ℓ , v ℓ ) means the Wronskian of solutionsto the equation (cid:20) − d d r + ℓ ( ℓ + 1) r + V R ( r ) − k (cid:21) u ( r ) = 0 . (15)The particular solutions u ℓ and v ℓ should be defined by boundary conditions u ℓ (0) = 0,and v ℓ ( r ) → exp { i kr − i πℓ/ } as r → ∞ . The exact representations for both u ℓ and v ℓ depend on whether the coordinate r is in or out the interval 0 < r ≤ R . Theserepresentations can be obtained by the matching technique as in [3]. For u ℓ one gets u ℓ ( r ) = F l ( η, kr ) , r ≤ R,u ℓ ( r ) = a ˆ j l ( kr ) + b ˆ n l ( kr ) , r > R. (16)For v ℓ the solution takes the form v ℓ ( r ) = a F ℓ ( η, kr ) + b G ℓ ( η, kr ) , r ≤ Rv ℓ ( r ) = ˆ h + ℓ ( kr ) , r > R. (17)In these representations the Zommerfeld parameter η is defined by the standardexpression η = V / (2 k ). By ˆ j l , ˆ n l and ˆ h + ℓ we denote the Riccati-Bessel, Riccati-Neumann and Riccati-Hankel functions which are related to the respective sphericalBessel functions as for example ˆ j l ( z ) = zj l ( z ). For spherical Bessel functions and forthe regular F ℓ and irregular G ℓ Coulomb functions we use the normalization of [6]. hort-range plus zero-range potentials a , b , a and b should be determined in such a way that both functions u ℓ , v ℓ and their first derivatives are continuous at r = R . This yields a = − W R ( F ℓ , ˆ n ℓ ) /k, b = W R ( F ℓ , ˆ j ℓ ) /k,a = − W R (ˆ h + ℓ , G ℓ ) /k, b = W R (ˆ h + ℓ , F ℓ ) /k, (18)where W R stands for Wronskian that is calculated at r = R . Now the Wronskian W ( u ℓ , v ℓ ) from (14) can easily be computed and takes the form W ( u ℓ , v ℓ ) = W R ( F l , ˆ h + ℓ ) . (19)Equations (14) – (19) completely determine the partial Green function G Rℓ .For our needs of evaluating the short-range asymptotic behavior of the Greenfunction G R , if R is well separated from zero, the region should be considered where r < R and r ′ < R . In this case, by inserting into (14) the quantities calculated abovewe finally represent the partial Green function by the sum of two terms G ℓ ( r, r ′ , k ) = 1 k F ℓ ( η, kr < ) H + ℓ ( η, kr > ) + χ Rℓ ( k ) k F ℓ ( η, kr ) F ℓ ( η, kr ′ ) , (20)where χ Rℓ ( k ) is given by χ Rℓ ( k ) = − W R (ˆ h + ℓ , H + ℓ ) /W R (ˆ h + ℓ , F ℓ ) . (21)Here the Coulomb outgoing wave H + ℓ is introduced according to the definition H + ℓ = G ℓ + i F ℓ .Now we have all components which are needed for calculating the Green function G + R by the formula (12). In view of (20) the representation for G + R in the region where r, r ′ < R is given by the sum of two terms G + R ( r , r ′ , k ) = G C ( r , r ′ , k + i0) + Q R ( r , r ′ , k ) . (22)The first term is the conventional Coulomb Green function which is calculated by thepartial wave series [4] as G C ( r , r ′ , k + i0) = 14 πk ∞ X ℓ =0 (2 ℓ + 1) F ℓ ( η, kr < ) H + ℓ ( η, kr > ) rr ′ P ℓ ( ˆ r · ˆ r ′ ) . (23)The second term Q R reads Q R ( r , r ′ , k ) = 14 πk ∞ X ℓ =0 (2 ℓ + 1) χ Rℓ ( k ) F ℓ ( η, kr ) F ℓ ( η, kr ′ ) rr ′ P ℓ ( ˆ r · ˆ r ′ ) . (24)The last formula can be rewritten in terms of the Coulomb Green functions taken on theupper and lower rims of the cut along the positive real semi axis of the energy complexplane. This can be made by using the formula F ℓ = ( H + ℓ − H − ℓ ) / (2i) and the methodof the paper [3]. The result reads Q R ( r , r ′ ) = 12i Z − d ζ Z R ( ξ, ζ )[ G C ( r, r ′ , ζ , k +i0) − G C ( r, r ′ , ζ , k − i0)] . (25) hort-range plus zero-range potentials ξ is defined as ξ = ˆ r · ˆ r ′ . The kernel Z R is given by the decompositionin Legendre polynomials Z R ( ξ, ζ ) = ∞ X ℓ =0 ( ℓ + 1 / χ Rℓ ( k ) P ℓ ( ξ ) P ℓ ( ζ ) . (26)As it may be seen from the Hostler representation [7], the Coulomb Green function G R ( r , r ′ , z ) actually depends on r , r ′ and the angle between vectors r and r ′ throughthe expression ˆ r · ˆ r ′ . We have reflected this fact in notations in the integrand of (25)where ζ stands for ˆ r · ˆ r ′ . The formulae (20-26) are valid only in that part of configurationspace where r, r ′ < R . Nevertheless, this representation of the Green function in thatregion completely defines the transition operator T R ( z ) = V R − V R G R ( z ) V R . Indeed,with (22) the operator T R takes the form T R ( z ) = V R − V R G C ( z ) V R − V R Q R ( z ) V R . (27)It is interesting to see what is the consequence of (27) when R → ∞ . Evaluating theright hand side of (21) asymptotically when kR ≫ ℓ ( ℓ + 1) and kR ≫ ℓ ( ℓ + 1) + η [8]we come to the expression for χ Rℓ ( k ) χ Rℓ ( k ) = i η exp(2i θ ℓ ) / ( kR ) + O (1 /R ) , (28)where θ ℓ = kR − η log(2 kR ) − πℓ/ σ ℓ and σ ℓ = arg Γ( ℓ + 1 + i η ) is the Coulomb phaseshift. From (26) we have for the L ( − ,
1) norm of the kernel Z R k Z R k = max ℓ | χ Rℓ ( k ) | = η/ ( kR ) + O ( R − ) . (29)Hence, the last term in (27) is negligible when R → ∞ and therefore T R ( z ) = V R − V R G C ( z ) V R + O ( R − ) . (30)This can be used for calculating the limit of T R when R → ∞ and it will be done inanother publication.In the last part of this section we calculate the asymptote of G + ( r , , k ) when r →
0. We start from the second term in (22). Since [8] F ℓ ( η, x ) = C ℓ ( η ) x ℓ +1 (cid:0) ηx/ ( ℓ + 1) + O ( x ) (cid:1) , (31)as x →
0, the leading order behavior of Q R reads Q R ( r , , k ) = C ( η ) χ R ( k ) F ( η, kr ) / (4 πr ) + O ( kr ) (32)as kr →
0. From this it is seen that Q R ( r , , k ) has the finite limitlim r → Q R ( r , , k ) = kC ( η ) χ R ( k ) / (4 π ) , (33)where C ( η ) = 2 πη ( e πη − − . This analysis shows us that the singular behaviorof G + R ( r , , k ) at small r comes exclusively from the first term in (22), i.e. from theCoulomb Green function, which as r → G C ( r , , k + i0) = 14 π [1 /r + V log r ] + C ( k ) + O ( r ln r ) . (34) hort-range plus zero-range potentials C ( k ) is given by C ( k ) = i k π + V π [log( − k ) + ψ (1 + i η ) + 2 γ − , (35)where γ is the Euler-Mascheroni constant and ψ ( z ) is the digamma function [8].Collecting together the expressions obtained above we arrive at the following resulton the short-range behavior of the Green function G + R G + R ( r , , k ) = 14 π [1 /r + V log r ] + C ( k ) + kC ( η ) χ R ( k )4 π + O ( r ln r ) . (36)The latter finalizes our study of Green’s function for the sharply screened Coulombpotential. It is apparent that in this case the zero-range potential will be identical tothat of the case of the pure Coulomb potential [2].
3. Green’s function behavior in the case of V ( ρ, δ ) class potentials In this section we study the Green function short-range asymptote for potentials of the V ( ρ, δ ) class. The Lippmann-Schwinger integral equation G + ( r , r ′ , k ) = G +0 ( r , r ′ , k ) − Z d q G +0 ( r , q , k ) V ( q ) G + ( q , r ′ , k ) (37)with G +0 ( r , r ′ , k ) = 14 π exp(i k | r − r ′ | ) | r − r ′ | (38)in this case has the unique solution [4, 9] which completely defines the Green function G + ( k ) . For our purpose of evaluating the short-range behavior we set r ′ = 0 anditerate (37) one time which results G + ( r , , k ) = G +0 ( r , , k ) − Z d q G +0 ( r , q , k ) V ( q ) G +0 ( q , , k )+ Z d q G +0 ( r , q , k ) V ( q ) Z d q ′ G +0 ( q , q ′ , k ) V ( q ′ ) G + ( q ′ , , k ) . (39)Now we consecutively consider the short-range behavior of right hand side terms. Thefirst term asymptote as r → G +0 ( r , , k ) = 14 πr + i k π + O ( r ) . (40)For evaluating the second term on the right hand side of (39) it is useful to split theintegral into two parts in order to separate the short-range and long-range contributionsof the integrand. Let us consider the integrals I j ( r ) = Z Ω j d q G +0 ( r , q , k ) V ( q ) G +0 ( q , , k ) (41)over domains Ω j ∈ R defined as Ω = { q : q < ( > ) r } . The radius r can be chosenas any positive bounded number well separated from zero and will be specified below. hort-range plus zero-range potentials I ( r ) we can expand the Green function factors of the integrandby using the Taylor decomposition up to quadratic terms as G +0 ( r , q , k ) = 1 / (4 π | r − q | ) + i k/ (4 π ) + O ( | r − q | ) (42)and similar expression holds for G +0 ( q , , k ) with r set to 0. For the potential V ( q ) inΩ we assume that r is chosen such that the formula (3) can be used and the W ( q )factor can be given by its Taylor decomposition W ( q ) = V + q · ∇ W (0) + O ( q ) . (43)Then the most singular term of I ( r ) as r → j = 1 the leading terms of integrand constituents which are defined by (42) and(43). This will lead us to the integral I s ( r ) = V / (4 π ) Z Ω d q | r − q | − q − ρ − . (44)Since we need to find the short-range behavior of this integral when r → r such that r < r . In this case the evaluation of the integral in (44) is easy toperform with the help of the formula1 | q − q ′ | = 1 q > ∞ X ℓ =0 q ℓ< q ℓ> P ℓ ( ˆ q · ˆ q ′ ) , (45)where as usual q > = max { q, q ′ } and q < = min { q, q ′ } . The results are naturally separatedin two ones, i.e. if ρ = 1 it reads I s ( r ) = V π (2 − ρ )( ρ − r − ρ +1 + V π (1 − ρ ) r − ρ +10 (46)and when ρ = 1 the integral I s takes the form I s ( r ) = − V π log( r ) + V π [1 + log( r )] . (47)From (46) it is seen that if 1 < ρ < I s has the polar singularity r − ρ +1 whereasif ρ < r → I s is regular and has a finitelimit. From this analysis of the integral (44) it becomes clear that taking into accountthe less singular terms in the expressions for the integrand of the integral I ( r ) one willobtain non singular contributions as r → I ( r ). For the modulus of I ( r ) we can easilyarrive at the inequality | I ( r ) | ≤ π ) Z Ω d q | V ( q ) | q | r − q | . (48)Let us now suppose that r is chosen such that the inequality (2) can be used for q > r then with the help of (45) the right hand side of (48) can be estimated as Z Ω d q | V ( r ) | q | r − q | ≤ C Z ∞ r d q (1 + q ) − − δ . (49)Since the last integral converges, the integral I ( r ) is uniformly bounded for all r suchthat r < r . hort-range plus zero-range potentials q ′ byits structure is quite similar to the integrals considered above if G +0 ( q ′ , , k ) standsinstead of G + ( q ′ , , k ). In this case the inner integral as the function of q may havea singularity that is not stronger than q − ρ +1 one. As it has been already shown sucha singularity in the integrand of the outer integral over q will lead to the nonsingularbehavior of the result as the function of r in the vicinity of the point r = 0. On usingthe iterative arguments this result can easily be extended on the case of the genuineintegrand in the third term of (39) [9]. Thus, the last term in (39) should have a finitelimit as r → G + ( r , , k ) for the case of 1 < ρ < G + ( r , , k ) = 14 π (cid:2) /r + A /r ρ − (cid:3) + B + o (1) , (50)for the case of ρ = 1 as G + ( r , , k ) = 14 π [1 /r + V log( r )] + B + o (1) , (51)and for the case of ρ < G + ( r , , k ) = 14 πr + B + o (1) . (52)Here the constant A is given by A = V (2 − ρ )(1 − ρ )and all finite contributions from respective integrals are denoted by B j , j = 1 , ,
4. Application to zero-range potential formalism
The zero-range potential is introduced by implementation of a special singular boundarycondition on the solution of the Schr¨odinger equation at small inter-particle distances[1]. This boundary condition can be enforced [2] if the delta functional term is insertedinto equation (6). In this case the solution is represented by φ ( r , k ) = φ ( r , k ) − λG + ( r , , k ) β. (53)Here φ is defined according to (8, 9). Since the asymptotic behavior of Green’s functionhas been studied in details in preceding section it remains to estimate the short rangebehavior of φ . It can be done on the basis of the subsequent Lippmann-Schwingerintegral equation φ ( r , k ) = exp(i k · r ) − Z d q G +0 ( r , q , k ) V ( q ) φ ( q , k ) . (54)It is easy to see that for a potential V ∈ V ( ρ, δ ) with ρ < δ > r = 0 and so is the solution. For completeness hort-range plus zero-range potentials J and J defined by J j ( r ) = Z Ω j d q G +0 ( r , q , k ) V ( q ) exp(i q · k ) , j = 1 , . (55)As in the previous section, we can obtain the following estimate for the modulus of J ( r )for all r such that r < r | J ( r ) | ≤ C π Z Ω d q q − (1 + q ) − − δ . (56)The integral on the right hand side of (56) converges. Thus, J ( r ) is uniformly boundedin the neighborhood of the origin. The integral J ( r ) can be treated by similar methodwhich was used in the previous section for the integral I ( r ). Particularly, if the mostsingular terms in the integrand are kept then the leading order of the integral J ( r )takes the form J ( r ) ∼ V π Z Ω d q | r − q | − q − ρ . (57)The integral on the right hand side of this formula coincides with the integral from (44)if ρ + 1 in (44) is replaced by ρ . Then from the analysis made in the previous section itfollows that the integral J ( r ) is regular as r →
0. As the result we can conclude thatfor the potentials of the class V ( ρ, δ ) the short-range asymptote of the wave function φ ( r , k ) is regular as r → φ ( r , k ) has a finite limit at the origin.The singular behavior of the full solution φ is therefore completely determined bythe singularities of the Green function. Accordingly, we have three following cases ofwave function asymptotes depending on the value of ρ . If 2 > ρ > φ ( r , k ) = α π (cid:2) /r + A /r ρ − (cid:3) + β + o (1) . (58)For the case of ρ = 1 it is given by φ ( r , k ) = α π [1 /r + V log( r )] + β + o (1) , (59)and for the case of ρ < φ ( r , k ) = α πr + β + o (1) . (60)Here the constants are given by following expressions A = V (2 − ρ )(1 − ρ ) ,α j = − λβ j and β j = φ (0 , k ) − λB j β j for j = 1 , ,
3. The case of ρ = 1 completelycoincides with the case of the zero-range potential for the Coulomb potential [2].Alternatively, the zero range potential is defined by the pseudo-potential. Followingthe procedure described in the paper [2] the form of the pseudo-potential is calculated hort-range plus zero-range potentials λW j ( r ) = λδ ( r ) dd ω j ω j (61)with variables ω j determined by formulae1 /ω = r − + A r − ρ +1 , (62)1 /ω = r − + V log( r ) , (63)1 /ω = r − . (64)
5. Conclusion
We have shown that the Green function of a Hamiltonian with the potential possessingsingular behavior O ( r − ρ ) at small inter-particle distances receives the additionalsingularity O ( r − ρ +1 ) except for the case of ρ = 1 when the logarithmic singularityappears. We have considered the class V ( ρ, δ ) of potentials with ρ < δ >
1. Thischoice of parameters is not exhausting. With little effort the weaker condition δ > I and J which should be treated as non absolutely convergent integrals. Wehave left out such an analysis in order not to make the paper too long. Acknowledgments
This work was partially supported by St Petersburg State University under project No.11.0.78.2010. The support by the Ministry of Education and Science of the RussianFederation under project No. 2012-1.5-12-000-1003-016 is also acknowledged.
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