A transmutation operator method for solving the inverse quantum scattering problem
Vladislav V. Kravchenko, Elina L. Shishkina, Sergii M. Torba
aa r X i v : . [ m a t h - ph ] S e p A transmutation operator method for solving the inverse quantumscattering problem
Vladislav V. Kravchenko , Elina L. Shishkina and Sergii M. Torba Departamento de Matem´aticas, Cinvestav, Unidad Quer´etaro,Libramiento Norponiente Voronezh State University.e-mail: [email protected], [email protected], ∗ September 25, 2020
Abstract
The inverse quantum scattering problem for the perturbed Bessel equation is considered. Adirect and practical method for solving the problem is proposed. It allows one to reduce theinverse problem to a system of linear algebraic equations, and the potential is recovered from thefirst component of the solution vector of the system. The approach is based on a special formFourier-Jacobi series representation for the transmutation operator kernel and the Gelfand-Levitanequation which serves for obtaining the system of linear algebraic equations. The convergence andstability of the method are proved as well as the existence and uniqueness of the solution of thetruncated system. Numerical realization of the method is discussed. Results of numerical tests areprovided revealing a remarkable accuracy and stability of the method.
We present a direct and simple method for practical solution of the inverse quantum scattering problemfor the perturbed Bessel equation Lu := − u ′′ + (cid:18) ℓ ( ℓ + 1) x + q ( x ) (cid:19) u = ρ u, x > ℓ ≥ − / q satisfying Z ∞ ( x µ + x ) | ˜ q ( x ) | dx < + ∞ (1.1)for some µ ∈ [0 , / q ( x ) = ( q ( x ) , ℓ > − / , (cid:0) | log( x ) | (cid:1) q ( x ) , ℓ = − / . (1.2)The problem consists in recovering q from the given scattering data. The bibliography dedicated tothe theory of this problem and applications is vast. We refer to [2], [5], [7], [26], [18] and references ∗ Research was supported by CONACYT, Mexico via the projects 222478 and 284470. Research of VladislavKravchenko was supported by the Regional mathematical center of the Southern Federal University, Russia. ℓ = 0 in [20], [21], [8],[15] and reported in the book [22]. The extension of an approach onto the singular case ℓ = 0 is alwaysa challenge requiring additional ideas and tools. The first important ingredient here is an appropriateFourier-Jacobi series representation for the transmutation operator kernel [24]. It captures singularfeatures of the kernel, such as its behaviour near t = 0 and on the characteristic line t = x , andallows one to recover the potential from the first coefficient of the series. Thus, we do not follow theusual approach of computing the transmutation kernel first and then recovering the potential from it.Instead, we compute the first coefficient of the Fourier-Jacobi series representation, from which thepotential is recovered.The right choice of the orthogonal function system used in the series representation resulted to beof crucial importance in the interplay between the transmutation operator kernel and the Gelfand-Levitan input kernel, which gave us the possibility in the present work to obtain a system of linearalgebraic equations for the coefficients of the series representation with explicit formulas for the entriesof the system matrix.We prove the convergence and stability of the method. This results in the possibility of recoveringthe potential from noisy scattering data. A corresponding numerical example is provided. Moreover,we prove the existence and uniqueness of the solution of the truncated system of equations arising inthe numerical realization of the method.Thus, the method developed in the present work is convergent, stable and possesses an importantadditional advantage. Its numerical implementation is simple and does not require much programmereffort. The numerical examples reveal a remarkable accuracy, stability and fast convergence of themethod.Besides this introduction the paper contains four sections. Section 2 presents some preliminarieson the inverse quantum scattering problem including the example of the square well potential, whichis used later on for one of the numerical tests. In Section 3 the Fourier-Jacobi series representationfor the transmutation operator kernel is presented. It is explained how the potential can be recoveredfrom the first coefficient of the series, and the Gelfand-Levitan equation is recalled. In Section 4we construct the system of linear algebraic equations for the coefficients of the Fourier-Jacobi seriesrepresentation, prove the existence and uniqueness of solutions of corresponding truncated systemsand the convergence of solutions of truncated systems to the exact one. Observing that the obtainedtruncated systems result from applying the Bubnov-Galerkin procedure with a special choice of theorthogonal function system, we prove the stability of the method, which allows one to work efficientlywith noisy scattering data. In Section 5 we discuss the numerical implementation of the method andprovide some numerical examples. They illustrate that indeed the developed approach leads to a directand simple method for accurate recovering of the potential even with few equations in the truncatedsystem and from noisy scattering data. Finally, in Appendix A we present a refined asymptotics ofthe Jost function. 2 Preliminaries
We consider the perturbed Bessel equation Lu := − u ′′ + (cid:18) ℓ ( ℓ + 1) x + q ( x ) (cid:19) u = ρ u, x > q , often called the potential, being a real valued function satisfying the condition( x µ + x )˜ q ( x ) ∈ L (0 , ∞ ) for some 0 ≤ µ < / , (2.2)Here ˜ q is given by (1.2). Sometimes, potentials satisfying (2.2) at infinity are said to belong to theMarchenko class. The spectral parameter ρ ∈ C is chosen so that Im ρ ≥ ℓ ≥ − / q ( x ) in the perturbed Bessel equation from so-called scattering data whichinclude the eigenvalues, the corresponding norming constants and the Jost function F ℓ ( ρ ), ρ ∈ [0 , ∞ ).Notice that we suppose the Jost function to be given, although in a usual study of the inverse problemit is obtained first from the S -function (the scattering function) which is supposed in its turn to beknown as a part of the scattering data.The unique solvability of such inverse quantum scattering problem follows from [18, Theorem 5.1],where a more general class of potentials is considered for arbitrary ℓ ≥ − /
2. Additional restrictionson the potential imposed in this paper are needed to guarantee that the problem possesses at most afinite number of eigenvalues, to use the Gelfand-Levitan equation and to be sure that the solution ofthe Gelfand-Levitan equation is square-integrable. For the case of integer ℓ one can consult a lot ofadditional details, e.g., in [36] and [5].We remind that the set of eigenvalues, if it is not empty, consists of a finite set of numbers ρ j ≤ j = 1 , . . . , N , which are such that equation (2.1) admits a square integrable solution on (0 , ∞ ), see [5,(II.1.10a)], [32, Theorem 5.1] and [37, Section 9.7]. Thus, ρ j = iτ j , τ j ≥
0. For recalling the definitionof the norming constants and of the Jost function we proceed with some necessary notations.A solution ϕ ℓ ( ρ, x ) of (2.1) satisfying the asymptotic relation at the originlim x → ℓ +1 √ π Γ (cid:18) ℓ + 32 (cid:19) x − ( ℓ +1) ϕ ℓ ( ρ, x ) = 1 , is called the regular solution. Note that for integer values of ℓ one has ℓ +1 √ π Γ (cid:0) ℓ + (cid:1) = (2 ℓ + 1)!!.The last formula is known as the extension of the double factorial symbol to complex arguments. Tosimplify notations, later in this paper we will use (2 ℓ + 1)!!.In the case when ρ = ρ j is an eigenvalue, the regular solution ϕ ℓ ( ρ j , x ) is an eigenfunction, andthe norming constants are defined as c j := 1 R ∞ ϕ ℓ ( ρ j , x ) dx . A solution f ℓ ( ρ, x ) of (2.1) satisfying the asymptotic relation at infinitylim x →∞ (cid:16) e − iπℓ e − iρx f ℓ ( ρ, x ) (cid:17) = 1is called the Jost solution. The uniqueness and the existence of both regular and Jost solutions is awell known fact (see, e.g., [5], and for non-integer values of ℓ , [19], [13] and references therein).The function F ℓ ( ρ ) which can be represented as a Wronskian of the solutions F ℓ ( ρ ) = ( − ρ ) ℓ W [ f ℓ ( ρ, x ) , ϕ ℓ ( ρ, x )]3s known as the Jost function. In fact, the Jost function contains information on the behaviour ofthe Jost solution at the origin. The following asymptotic relation is valid for ℓ > − / ℓ it can be established using the results from [19]) F ℓ ( ρ ) = lim x → ( − ρx ) ℓ (2 ℓ − f ℓ ( ρ, x ) , (2.3)while for ℓ = − / F − / ( ρ ) = lim x → − √ π ( − ρ ) − / √ x log x f − / ( ρ, x ) . (2.4)Note that F ℓ is analytic in the upper half-plane, F ℓ ( ρ ) = 1 + o (1) when ρ → ∞ , Im ρ ≥
0, and F ℓ ( − ρ ) = F ℓ ( ρ ) for ρ ∈ R [18, Lemma B.5]. Moreover, for ℓ > − / q such that q ∈ L (0 , ∞ ) the asymptotic formula is valid F ℓ ( ρ ) = 1 + i ρ Z ∞ q ( x ) dx + o ( ρ − ) , | ρ | → ∞ , see [19, Remark 2.14]. In Appendix A we prove a refinement of this formula, namely, that F ℓ ( ρ ) = 1 + i ρ Z ∞ q ( x ) dx + O ( ρ − ) , | ρ | → ∞ , (2.5)for any ℓ ≥ − / q ∈ L (0 , ∞ ) ∩ BV [0 , ∞ ). Here BV denotes functions of boundedvariation vanishing at infinity.Denote by b ℓ ( ρx ) a solution of the Bessel equation − u ′′ + ℓ ( ℓ + 1) x u = ρ u, x > b ℓ ( ρx ) ∼ ( ρx ) ℓ +1 (2 ℓ + 1)!! , x → . (2.6)It has the form b ℓ ( ρx ) = ρxj ℓ ( ρx )where j ℓ stands for the spherical Bessel function of the first kind (see [1, Section 10.1]), j ℓ ( z ) := p π z J ℓ + ( z ). Example . Consider the square well potential q of the form q ( x ) = ( − Q , x ≤ R, , x > R (2.7)where Q is a positive constant. Denote ω := p ρ + Q . Then the Jost solution has the form f ℓ ( ρ, x ) = ( a ( ρ ) b ℓ ( ωx ) + b ( ρ ) ωxh (1) ℓ ( ωx ) , x ≤ R, ( − ℓ iρxh (1) ℓ ( ρx ) , x > R a ( ρ ) and b ( ρ ) are found from the condition of continuity of the solution f ℓ ( ρ, x )and of its derivative at x = R , which leads to the following system of equations a ( ρ ) b ℓ ( ωR ) + b ( ρ ) ωRh (1) ℓ ( ωR ) = ( − ℓ iρRh (1) ℓ ( ρR ) ,a ( ρ ) (cid:0) ω ( ℓ + 1) j ℓ ( ωR ) − ω Rj ℓ +1 ( ωR ) (cid:1) + b ( ρ ) (cid:16) ω ( ℓ + 1) h (1) ℓ ( ωR ) − ω Rh (1) ℓ +1 ( ωR ) (cid:17) = ( − ℓ iρ (cid:16) ( ℓ + 1) h (1) ℓ ( ρR ) − ρRh (1) ℓ +1 ( ρR ) (cid:17) . From (2.3) we find that F ℓ ( ρ ) = ( − ℓ +1 ib ( ρ ) (cid:16) ρω (cid:17) ℓ . A solution u ℓ ( ρ, x ) of (2.1), satisfying the asymptotic relation (2.6) admits the following representation u ℓ ( ρ, x ) = T [ b ℓ ( ρx )] := b ℓ ( ρx ) + Z x K ℓ ( x, t ) b ℓ ( ρt ) dt where the integral kernel K ℓ ( x, t ) is a square integrable function of the variable t , independent of ρ .This Volterra integral operator of the second kind is known as a transmutation (or transformation)operator. The existence of such K ℓ ( x, t ) for the potentials satisfying condition (1.1) at zero was provedin [36]. Properties of K ℓ ( x, t ) were studied in several publications (see, e.g., [38], [7], [6], [12], [16],[23], [33], [34]). For the purpose of the present work the following statement is crucial. Theorem 3.1 ([24]) . Let q satisfy the condition R b x µ | q ( x ) | dx < ∞ for some ≤ µ < / . Then thekernel K ℓ ( x, t ) admits the following series representation K ℓ ( x, t ) = ∞ X n =0 β n ( x ) x ℓ +2 t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) , (3.1) where P ( α,β ) n stands for the Jacobi polynomial and the coefficients β n ( x ) can be calculated by a recurrentintegration procedure, starting with β ( x ) = (2 ℓ + 3) (cid:18) u ℓ, ( x ) x ℓ +1 − (cid:19) (3.2) where u ℓ, ( x ) is a regular solution of the equation Lu = 0 (3.3) normalized by the asymptotic condition u ℓ, ( x ) ∼ x ℓ +1 , x → .For any x > , the series in (3.1) converges in L (0 , x ) . Suppose additionally that q is absolutelycontinuous on [0 , b ] . Then the series in (3.1) converges absolutely and uniformly with respect to t ∈ [0 , x − ε ] for an arbitrarily small ε > . If additionally q ∈ W [0 , b ] , then the series convergesabsolutely and uniformly with respect to t on the whole [0 , x ] .Remark . The condition on the potential q in the theorem is equivalent to condition (1.1) at theorigin. The recurrent integration procedure mentioned in the theorem is superfluous for the presentwork and can be consulted in [24]. 5 emark . Equality (3.2) gives us the possibility to recover the potential q if β is known. Indeed,we have that u ℓ, ( x ) = (cid:18) β ( x )(2 ℓ + 3) + 1 (cid:19) x ℓ +1 , (3.4)and since u ℓ, is a solution of (3.3), we obtain q = xβ ′′ ( x ) + 2( ℓ + 1) β ′ ( x ) x ( β ( x ) + 2 ℓ + 3) . (3.5) Remark . The following orthogonality property of the Jacobi polynomials is valid [24] Z x t ℓ +2 P ( ℓ +1 / , n (cid:18) − t x (cid:19) P ( ℓ +1 / , m (cid:18) − t x (cid:19) dt = x ℓ +3 m + 2 ℓ + 3 δ nm (3.6)with δ nm standing for the Kronecker delta. Consequently, for any x > p n ( x ; t ) := √ n + 2 ℓ + 3 x ℓ +3 / t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) (3.7)is a complete orthonormal system in L (0 , x ). Hence the series (3.1) is an expansion of the kernel K ℓ ( x, t ) with respect to the basis of L (0 , x ) represented by the system of functions { p n ( x ; t ) } ∞ n =0 , K ℓ ( x, t ) = ∞ X n =0 α n ( x ) p n ( x ; t ) (3.8)with α n ( x ) = β n ( x ) √ n + 2 ℓ + 3 √ x . In the following we assume that zero is not an eigenvalue of the problem. Then the transmutationkernel K ℓ ( x, t ) is related to the scattering data via the Gelfand-Levitan integral equation K ℓ ( x, y ) + Ω ℓ ( x, y ) + Z x K ℓ ( x, t )Ω ℓ ( t, y ) dt = 0 , x > y (3.9)where the input kernel Ω ℓ ( x, y ) has the formΩ ℓ ( x, y ) = N X j =1 C j b ℓ ( iτ j x ) b ℓ ( iτ j y ) + 2 π Z ∞ b ℓ ( ρx ) b ℓ ( ρy ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ,C j := c j ( iτ j ) ℓ +2 . Note that under condition (1.1) the integral kernel K ℓ satisfies [36] for any finite a > ≤ x ≤ a k K ℓ ( x, · ) k L (0 ,x ) < ∞ . The function Ω ℓ is symmetric, and it can be easily obtained from (3.9) that Ω ℓ ( x, · ) ∈ L (0 , x ) andΩ ℓ ∈ L ((0 , x ) × (0 , x )). 6 A system of linear algebraic equations for the coefficients β n ( x ) Denote A m,n ( x ) := N X j =1 C j j ℓ +2 n +1 ( iτ j x ) j ℓ +2 m +1 ( iτ j x ) + 2 π Z ∞ j ℓ +2 n +1 ( ρx ) j ℓ +2 m +1 ( ρx ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ, (4.1)and B m ( x ) := − N X j =1 C j b ℓ ( iτ j x ) j ℓ +2 m +1 ( iτ j x ) − π Z ∞ b ℓ ( ρx ) j ℓ +2 m +1 ( ρx ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ. (4.2) Theorem 4.1.
The coefficients β n from (3.1) satisfy the following infinite system of linear algebraicequations β m ( x )(4 m + 2 ℓ + 3) x + ∞ X n =0 β n ( x ) A m,n ( x ) = B m ( x ) , for all m = 0 , , . . . . (4.3) Proof.
Let us substitute the representation (3.1) into (3.9). Consider Z x K ℓ ( x, t )Ω ℓ ( t, y ) dt = 1 x ℓ +2 ∞ X n =0 β n ( x ) Z x t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) Ω ℓ ( t, y ) dt. (4.4)The possibility of changing the order of summation and integration follows from the observation thatthis equality is nothing but a concrete realization of the general Parseval identity [4, p. 16]. Indeed,with the aid of Remark 3.4 we have Z x K ℓ ( x, t )Ω ℓ ( t, y ) dt = h K ℓ ( x, · ) , Ω ℓ ( · , y ) i L (0 ,x ) = ∞ X n =0 h K ℓ ( x, · ) , p n ( x ; · ) i L (0 ,x ) h p n ( x ; · ) , Ω ℓ ( · , y ) i L (0 ,x ) = ∞ X n =0 α n ( x ) h p n ( x ; · ) , Ω ℓ ( · , y ) i L (0 ,x ) = ∞ X n =0 β n ( x ) √ n + 2 ℓ + 3 √ x Z x √ n + 2 ℓ + 3 x ℓ +3 / t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) Ω ℓ ( t, y ) dt = 1 x ℓ +2 ∞ X n =0 β n ( x ) Z x t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) Ω ℓ ( t, y ) dt. In order to proceed with the integral in (4.4), we need the following result [24] Z x t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) b ℓ ( ρt ) dt = x ℓ +2 j ℓ +2 n +1 ( ρx ) . (4.5)Hence Z x t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) Ω ℓ ( t, y ) dt = x ℓ +2 (cid:18) N X j =1 C j j ℓ +2 n +1 ( iτ j x ) b ℓ ( iτ j y ) + 2 π Z ∞ j ℓ +2 n +1 ( ρx ) b ℓ ( ρy ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ (cid:19) , (4.6)7nd Z x K ℓ ( x, t )Ω ℓ ( t, y ) dt = ∞ X n =0 β n ( x ) (cid:18) N X j =1 C j j ℓ +2 n +1 ( iτ j x ) b l ( iτ j y ) + 2 π Z ∞ j ℓ +2 n +1 ( ρx ) b ℓ ( ρy ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ (cid:19) . Thus, equation (3.9) can be written in the form y ℓ +1 x ℓ +2 ∞ X n =0 β n ( x ) P ( ℓ +1 / , n (cid:18) − y x (cid:19) + ∞ X n =0 β n ( x ) (cid:18) N X j =1 C j j ℓ +2 n +1 ( iτ j x ) b ℓ ( iτ j y ) + 2 π Z ∞ j ℓ +2 n +1 ( ρx ) b ℓ ( ρy ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ (cid:19) = − N X j =1 C j b ℓ ( iτ j x ) b ℓ ( iτ j y ) − π Z ∞ b ℓ ( ρx ) b ℓ ( ρy ) (cid:16) | F ℓ ( ρ ) | − − (cid:17) dρ (4.7)Multiplying (4.7) by y ℓ +1 P ( ℓ +1 / , m (cid:16) − y x (cid:17) , integrating with respect to y from 0 to x , and using(4.5) and (3.6) we obtain (4.3). The series in (4.3) converges again due to the general Parsevalidentity because it is a scalar product of the functions K ℓ ( x, t ) and R x Ω ℓ ( t, y ) p m ( x ; t ) dt in the space L (0 , x ).The functions β m ( x ) √ m +2 ℓ +3 √ x are the Fourier coefficients of the function K ℓ ( x, · ) with respect to thesystem (3.7), see (3.8). It follows from (4.6) that the functions √ m + 2 ℓ + 3 √ x · B m ( x ) are the Fouriercoefficients of the function − Ω ℓ ( x, · ) with respect to the system (3.7). Finally, multiplying (4.6) by y ℓ +1 P ( ℓ +1 / , m (cid:16) − y x (cid:17) , integrating with respect to y from 0 to x and using (4.5) we obtain that x ℓ +4 A n,m ( x ) = Z x Z x t ℓ +1 P ( ℓ +1 / , n (cid:18) − t x (cid:19) y ℓ +1 P ( ℓ +1 / , m (cid:18) − y x (cid:19) Ω ℓ ( t, y ) dt dy, or that √ n + 2 ℓ + 3 √ m + 2 ℓ + 3 · xA m,n ( x ) = Z x Z x p n ( t ) p m ( y )Ω ℓ ( t, y ) dt dy. The last equality means that the functions √ n + 2 ℓ + 3 √ m + 2 ℓ + 3 · xA m,n ( x ) are the Fouriercoefficients of the function Ω ℓ with respect to the system p n × p m .Hence for each fixed x > ξ j − λ ∞ X k =0 a jk ξ k = b j , j = 0 , , . . . , (4.8)where λ = − ξ j = β j ( x ) √ j + 2 ℓ + 3 √ x , b j = p j + 2 ℓ + 3 √ x · B j ( x ) , a jk = p j + 2 ℓ + 3 √ k + 2 ℓ + 3 · xA j,k ( x ) . The coefficient vectors satisfy { b j } ∞ j =0 ∈ ℓ , { a j,k } ∞ j,k =0 ∈ ℓ ⊗ ℓ and the unknown vector { ξ j } ∞ j =0 is sought to belong to ℓ . The systems of such type, with coefficients from ℓ , were studied in [14,Chapter 14, § roposition 4.2. Let x > be fixed. Consider the system (4.3) truncated to M + 1 equations, i.e., weconsider m, n ≤ M . Then for sufficiently large M the truncated system has a unique solution whichwe denote by { β ( M ) m ( x ) } Mm =0 and M X m =0 | β m ( x ) − β ( M ) m ( x ) | (4 m + 2 ℓ + 3) x + ∞ X m = M +1 | β m ( x ) | (4 m + 2 ℓ + 3) x → , M → ∞ , from which it also follows that β ( M )0 ( x ) → β ( x ) , M → ∞ . The same truncated system results from the application of the Bubnov-Galerkin procedure to theintegral equation (3.9) with respect to the system (3.7), see [29, § β is necessary to recover the potential.Also we point out that the special form of the function system (3.7) allowed us to transform the scalarproducts arising in the Bubnov-Galerkin procedure into the form (4.1) and (4.2). As a consequenceof the general theory presented in [29, §
14] we obtain a stability result for the proposed method.Let I M be the ( M + 1) × ( M + 1) identity matrix, L M = ( a jk ) Mj,k =0 be the coefficient matrix ofthe truncated system and R M = ( b j ) Mj =0 be the truncated right-hand side. Following [29, §
9] considera system (called non-exact system)( I M + L M + Γ M ) v = R M + δ M , where Γ M is an ( M + 1) × ( M + 1) matrix representing errors in the coefficients a jk , and δ M is acolumn-vector representing errors in the coefficients b j . Let U M denote the solution of the exacttruncated system (with Γ M = 0 and δ M = 0) and V M the solution of the non-exact system. Notethat U M = n β Mm ( x ) √ m +2 ℓ +3 √ x o Mm =0 , see Proposition 4.2. The solution of the Bubnov-Galerkin procedureis called stable if there exist constants c , c and r independent of M such that for k Γ M k ≤ r andarbitrary δ M the non-exact system is solvable and the following inequality holds k U M − V M k ≤ c k Γ M k + c k δ M k . From [29, Theorems 14.1 and 14.2] the following result follows.
Proposition 4.3.
The approximate solution n β Mm ( x ) √ m +2 ℓ +3 √ x o Mm =0 of system (4.8) is stable. Moreover,the condition numbers of the coefficient matrices I M + L M are bounded. This result allows one to recover the potential from inexact to a certain point or noisy scatteringdata.
Theorem 4.1 and Proposition 4.2 lead to a direct and simple method for solving the inverse quantumscattering problem. 9. Given the Jost function, the eigenvalues and the norming constants. Choose a number of equa-tions M + 1, so that the truncated system β m ( x )(4 m + 2 ℓ + 3) x + M X n =0 β n ( x ) A m,n ( x ) = B m ( x ) , for all m = 0 , . . . , M (5.1)is to be solved.2. Compute B m ( x ) and A m,n ( x ) according to the formulas (4.2) and (4.1).3. Solve the system (5.1) to find β ( x ).4. Compute q with the aid of (3.5) or by computing first the particular solution u ℓ, using (3.4). Remark . Since the condition numbers of truncated systems (4.8) are bounded, see Proposition4.3, it may be worth converting the system (5.1) into the truncated system of the form (4.8) for largevalues of M . Calculation of the integrals in (4.1) and (4.2) is one of the key steps in the proposed method. Sincefor ρ ∈ R | j ν ( ρx ) | = cos( ρx − πν − π ) | ρx | + O (cid:18) | ρx | (cid:19) , | ρ | → ∞ , see [1, (9.2.1)] and | F ℓ ( ρ ) | − − ρ (cid:18)Z ∞ q ( x ) dx (cid:19) + O (cid:18) ρ (cid:19) = O (cid:18) ρ (cid:19) , | ρ | → ∞ , (5.2)see (2.5), we have (cid:12)(cid:12)(cid:12) j ℓ +2 n +1 ( ρx ) j ℓ +2 m +1 ( ρx ) (cid:16) | F ℓ ( ρ ) | − − (cid:17)(cid:12)(cid:12)(cid:12) ≤ c x ρ and (cid:12)(cid:12)(cid:12) b ℓ ( ρx ) j ℓ +2 m +1 ( ρx ) (cid:16) | F ℓ ( ρ ) | − − (cid:17)(cid:12)(cid:12)(cid:12) ≤ c xρ , ρ → + ∞ . As one can see, the integral in (4.2) can converge slowly. The convergence can be improved to someextent subtracting leading term and integrating it explicitly. Note that due to (5.2), ρ (cid:16) | F ℓ ( ρ ) | − − (cid:17) = O (1) , | ρ | → ∞ , (5.3)that is, a bounded term. Numerical experiments suggest that this bounded term is a sum of a constant,an oscillating function and an o (1) function. The value of the constant, which we will denote by ˜ F ℓ ,can be easily estimated numerically. For example, one can compute the expression (5.3) for some setof points { ρ k } Kk =0 and take for ˜ F ℓ an average of the obtained values. See Figure 1 for an illustration.Note also that (see [31, 2.12.31.2]) Z ∞ j ℓ +2 n +1 ( ρx ) j ℓ +2 m +1 ( ρx ) ρ dρ = πx ℓ +2 n +1 / , if n = m, πx ℓ + n + m +1 / , if n = m ± , , if | n − m | ≥ , (5.4)10nd Z ∞ b ℓ ( ρx ) j ℓ +2 m +1 ( ρx ) ρ dρ = ( πx ℓ +1 / , if m = 0 , , if m > , (5.5)where ( x ) n stands for the Pochhammer symbol. Hence instead of computing integrals (4.1) and (4.2)one can compute the integrals Z ∞ j ℓ +2 n +1 ( ρx ) j ℓ +2 m +1 ( ρx ) | F ℓ ( ρ ) | − − − ˜ F ℓ ρ ! dρ (5.6)and Z ∞ b ℓ ( ρx ) j ℓ +2 m +1 ( ρx ) | F ℓ ( ρ ) | − − − ˜ F ℓ ρ ! dρ (5.7)and afterwards add expressions (5.4) and (5.5) multiplied by ˜ F ℓ . If the integrals are truncated andcomputed on a segment [0 , K ], the proposed modification leads to a more accurate result due to theintegral tail taken into account (the oscillating and o (1) parts in (5.3) are expected to result in smallervalues in comparison with the part given by the constant ˜ F ℓ ). We would like to mention that theproposed modification is nothing more than an adaptation of the method presented in [30, (9.98)]with first two terms taken into account. Note also that in the case ℓ = − / n = m = 0, and the expression (5.5) for m = 0 (due to the divergence at the origin).One should use directly expressions (4.1) and (4.2). For ℓ < F ℓ oscillates a lot even for simplest potentials, see Figure 1. For that reason weare not expecting a simple approximation of the term | F ℓ ( ρ ) | − − Figure 1: Plot of the function ρ (cid:16) | F ℓ ( ρ ) | − − − ˜ F ℓ ρ (cid:17) for the square well potential from Example 2.1with ℓ = 1, Q = 1 and R = π/
2. The parameter ˜ F ℓ is estimated numerically to be 1 . The numerical illustrations presented below were obtained in Matlab2017. For the numerical integra-tion on step 2 a sufficiently large interval was chosen and the Matlab routine trapz was used. Onthe last step, for recovering q we used (3.5). Here the differentiation was performed by representing11he computed function β ( x ) in the form of a spline with the aid of the Matlab routine spapi with aposterior differentiation with the Matlab command fnder . Example . Consider the potential (2.7) with ℓ = 2, Q = 1 and R = π/
2. On Figure 2 the recoveredpotential (on the left) and its absolute error (on the right) are shown in the cases M = 0, M = 1, M = 4and M = 9 that corresponds to 1, 2, 5 and 10 equations in the truncated system (5.1), respectively.Thus, a very reduced number of equations from the system (5.1) is sufficient even in the case of adiscontinuous potential. For the numerical integration we have used the interval ρ ∈ [0 , , /
10 for the trapz function still allowed us to recover the potentialwith 4–5 decimal figures. q r e c o v e r ed -10 -5 ab s . e rr o r Figure 2: On the left: the square well potential from Example 5.2 with ℓ = 2 recovered on the interval(0 , π ] with M = 0, M = 1 and M = 4 that corresponds to 1, 2 and 5 equations in the truncatedsystem (5.1), respectively. On the right: absolute error of the recovered potential for M ∈ { , , , } corresponding to 1, 2, 5 and 10 equations in the truncated system (5.1), respectively.Note that the error increase in the recovered potential closer to the discontinuity point x = π/ β . As one can appreciate, the error remainssmall almost up to the discontinuity point x = π/
2. So one can expect that applying numericaldifferentiation without using values of β from both sides of the discontinuity point, e.g., the finitedifference or constructing a spline using the data from [0 , π/
2] only, can reduce the error for values of x close to π/
2. Indeed, on Figure 3, right plot, we show the error of the recovered potential when thecoefficient β was approximated by a spline separately on [0 , π/
2] and on [ π/ , π ]. One can appreciatea higher accuracy close to x = π/ Example . The method gives excellent results for negative values of ℓ and for larger values of ℓ aswell. Let us consider the same potential as in Example 5.2 but for ℓ = − / ℓ = e .Note that for ℓ = − / ρ ≈ − . c ≈ . Example . Let us consider the equation with the Hulth´en effective potential L u := − u ′′ + ℓ ( ℓ + 1) (cid:18) δ − e − δx (cid:19) e − δx − δe − δx − e − δx ! u = ρ u, x > . (5.8)12 -10 -5 A b s . e rr o r o f r e c o v e r ed -10 -5 A b s . e rr o r o f r e c o v e r ed q whole intervalfirst halfsecond half Figure 3: On the left: absolute error of the recovered coefficient β for the square well potential fromExample 5.2 with ℓ = 2, M = 4 on the interval (0 , π ]. On the right: absolute error of the recoveredpotential depending on the choise of the interval used for spline interpolation of the coefficient β anddifferentiation.Here 0 < δ < q H ( r ) = − δe − δx − e − δx is known as a potential providing a better approximationto the screened Coulomb (Yukawa) potential q sc ( r ) = − e − δr r than the ordinary Coulomb potential − r , see, e.g., [27], [11]. However, it can be exactly solved only for zero angular momentum, i.e.,for ℓ = 0. Greene and Aldrich [11] considered the Hulth´en effective potential as an exactly solvableapproximation for all values of ℓ . Equation (5.8) can be transformed into the form (2.1) if one considers q ( x ) = ℓ ( ℓ + 1) (cid:18) δ − e − δx (cid:19) e − δx − ℓ ( ℓ + 1) x − δe − δx − e − δx . Note that q ( x ) ∼ − x + δℓ ( ℓ +1) x as x → q ( x ) ∼ − ℓ ( ℓ +1) x as x → ∞ , so the potential q does notsatisfy the condition (2.2). Nevertheless, the spectral problem for the original equation (5.8) possessesat most a finite number of negative eigenvalues, see, e.g., [35], so the corresponding quantum scatteringproblem can be solved by the same method, see [5].To simplify the consideration below in what follows we assume that 2 ℓ Z . Then the generalsolution of (5.8) has the form u ( x ) = Ay − ℓ e iρx F ( − ℓ − a ρ − s ρ , − ℓ − a ρ + s ρ ; − ℓ ; y )+ By ℓ +1 e iρx F ( ℓ + 1 − a ρ − s ρ , ℓ + 1 − a ρ + s ρ ; 2 ℓ + 2; y ) , (5.9)where y = 1 − e − δx , a ρ = i ρδ and s ρ = √ δ − ρ δ . This expression was obtained solving transformedequation (7) from [11] using Wolfram Mathematica 10.Note that y → x →
0, hence the regular solution of (5.8) has the form ϕ ℓ ( ρ, x ) = (1 − e − δx ) ℓ +1 e iρx δ ℓ +1 (2 ℓ + 1)!! F (cid:16) ℓ + 1 − a ρ − s ρ , ℓ + 1 − a ρ + s ρ ; 2 ℓ + 2; 1 − e − δx (cid:17) . (5.10)On the other hand, y → x → + ∞ , so the values of the hypergeometric functions in (5.9) are notdefined by their series expansions and to find the Jost solution we need to apply the following analytic13 = − / ℓ = e -6 -4 -2 ab s o l u t e e rr o r -10 -5 ab s o l u t e e rr o r Figure 4: Absolute errors of the recovered square well potential from Example 5.3 having ℓ = − / ℓ = e (on the right). The potential was recovered on the interval (0 , π ] and 3, 5 or10 equations were used in the truncated system (5.1).continuation [9, (2.10.1)] F ( a, b ; c ; z ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) F ( a, b ; a + b − c + 1; 1 − z )+ Γ( c )Γ( a + b − c )Γ( a )Γ( b ) (1 − z ) c − a − b F ( c − a, c − b ; c − a − b + 1; 1 − z ) . Then u ( x ) = A y − ℓ e iρx Γ( − ℓ )Γ(2 a ρ )Γ( − ℓ + a ρ + s ρ )Γ( − ℓ + a ρ − s ρ ) F ( − ℓ − a ρ − s ρ , − ℓ − a ρ + s ρ ; 1 − a ρ ; e − δx )+ A y − ℓ e − iρx Γ( − ℓ )Γ( − a ρ )Γ( − ℓ − a ρ − s ρ )Γ( − ℓ − a ρ + s ρ ) F ( − ℓ + a ρ + s ρ , − ℓ + a ρ − s ρ ; 1 + 2 a ρ ; e − δx )+ B y ℓ +1 e iρx Γ(2 ℓ + 2)Γ(2 a ρ )Γ( ℓ + 1 + a ρ + s ρ )Γ( ℓ + 1 + a ρ − s ρ ) F ( ℓ + 1 − a ρ − s ρ , ℓ + 1 − a ρ + s ρ ; 1 − a ρ ; e − δx )+ B y ℓ +1 e − iρx Γ(2 ℓ + 2)Γ( − a ρ )Γ( ℓ + 1 − a ρ − s ρ )Γ( ℓ + 1 − a ρ + s ρ ) F ( ℓ + 1 + a ρ + s ρ , ℓ + 1 + a ρ − s ρ ; 1 + 2 a ρ ; e − δx ) . The first and the third terms behave like constant by e iρx when x → ∞ , while the second and the forthterms behave like constant by e − iρx when x → ∞ . Hence for the solution u to be the Jost solution,the coefficients A and B have to satisfy the following system A · Γ( − ℓ )Γ(2 a ρ )Γ( − ℓ + a ρ + s ρ )Γ( − ℓ + a ρ − s ρ ) + B · Γ(2 ℓ + 2)Γ(2 a ρ )Γ( ℓ + 1 + a ρ + s ρ )Γ( ℓ + 1 + a ρ − s ρ ) = e iπℓ/ ,A · Γ( − ℓ )Γ( − a ρ )Γ( − ℓ − a ρ − s ρ )Γ( − ℓ − a ρ + s ρ ) + B · Γ(2 ℓ + 2)Γ( − a ρ )Γ( ℓ + 1 − a ρ − s ρ )Γ( ℓ + 1 − a ρ + s ρ ) = 0 . − z )Γ( z ) = π sin πz we obtain that A = e iπℓ/ Γ(1 − a ρ )Γ(2 ℓ + 1)Γ( ℓ + 1 − a ρ + s ρ )Γ( ℓ + 1 − a ρ − s ρ ) ,B = − e iπℓ/ Γ( − ℓ )Γ(1 − a ρ )(2 ℓ + 1)Γ( − ℓ − a ρ + s ρ )Γ( − ℓ − a ρ − s ρ ) . The Jost function is given by F ℓ ( ρ ) = lim x → ( − ρx ) ℓ (2 ℓ − Ae iρx (1 − e − δx ) ℓ = A (2 ℓ − (cid:16) − ρδ (cid:17) ℓ , hence F ℓ ( ρ ) = e iπℓ/ (2 ℓ − (cid:0) − i ρδ (cid:1) Γ(2 ℓ + 1)Γ (cid:16) ℓ + 1 − i ρδ + √ δ − ρ δ (cid:17) Γ (cid:16) ℓ + 1 − i ρδ − √ δ − ρ δ (cid:17) (cid:16) − ρδ (cid:17) ℓ . (5.11)Note that this expression is also well defined for values ℓ satisfying 2 ℓ ∈ N .The eigenvalues ρ j = iτ j , τ j ≥ F ℓ . One can easily see thatall such zeros coincide with the values of ρ for which Γ (cid:16) ℓ + 1 − i ρδ − √ δ − ρ δ (cid:17) = ∞ , i.e., when theargument of the gamma function is a non-positive integer, which is equivalent to the equation ℓ + 1 + τδ − √ δ + τ δ = − m, m ∈ N . Squaring the equation we find that( ℓ + 1 + m ) + τ δ + 2 τδ ( ℓ + 1 + m ) = 2 δ + τ δ , or τ = 1 ℓ + 1 + m − δ ℓ + 1 + m ) , m ∈ N . Recalling that τ must be non-negative, we find that the set of eigenvalues is given by τ j = 1 ℓ + j − δ ℓ + j ) , j = 1 , . . . , "r δ − ℓ , where [ · ] is the integer part function. The corresponding eigenfunctions are given by ϕ ℓ ( iτ j , x ) = (1 − e − δx ) ℓ +1 e iρx δ ℓ +1 (2 ℓ + 1)!! F (cid:18) − j + 1 , ℓ + 1 + 2 δ ( ℓ + 1) ; 2 ℓ + 2; 1 − e − δx (cid:19) . Note that the first argument of the hypergeometric function is a non-positive integer, so the hypergeo-metric function reduces to a polynomial. The corresponding norming constants can be easily obtainednumerically.On Figure 5 we show the recovered potential and the absolute error. In this example the function | F ℓ ( ρ ) | − − /ρ , and not as 1 /ρ as was considered in Subsection 5.2. However, a similarprocedure was implemented to improve the computation of the integrals. The interval ρ ∈ [0 , M . As one can see, the condition number,which is equal to the quotient of these eigenvalues, remains bounded independently of the numberof equations used. On the right we show the potential recovered from the noisy data { τ j , c j } j =1 and F ℓ ( ρ ), ρ ∈ (cid:8) k (cid:9) k =0 , 10% noise was added to all the values.15 q r e c o v e r ed -6 -4 -2 ab s o l u t e e rr o r Figure 5: Recovered Hulth´en effective potential from Example 5.4 with ℓ = 1 / δ = 1 /
10 (on theleft) and the absolute error (on the right). The potential was recovered on the interval [ ,
3] usingup to 20 equations in the truncated system (5.1).
A direct and simple method for solving the inverse quantum scattering problem for an arbitraryangular momentum ℓ ≥ − / A On the asymptotic behaviour of the function F ℓ According to [18, Lemma B.5] the function F ℓ admits the following integral representation F ℓ ( ρ ) = 1 + Z ∞ ψ ℓ ( ρ, x ) ϕ ℓ ( ρ, x ) q ( x ) dx, (A.1)where ϕ ℓ is the regular solution of (2.1) considered in Section 2 and ψ ℓ denotes the Jost solution of(2.1) with q ≡
0. We also denote the regular solution for q ≡ ϕ ℓ .First, we recall some estimates from [18] and [19]. The solutions ϕ ℓ and ψ ℓ of the unperturbedequation are given by ϕ ℓ ( ρ, x ) = ρ − ℓ − / r πx J ℓ +1 / ( ρx ) , (A.2) ψ ℓ ( ρ, x ) = iρ ℓ +1 / r πx H (1) ℓ +1 / ( ρx ) . (A.3)Here H ℓ +1 / is the Hankel function of the first kind. The following estimates hold for ρ ∈ R , ℓ > − / | ϕ ℓ ( ρ, x ) | ≤ C (cid:18) x | ρ | x (cid:19) ℓ +1 , | ψ ℓ ( ρ, x ) | ≤ C (cid:18) x | ρ | x (cid:19) − ℓ . (A.4)For ℓ = − / | ψ − / ( ρ, x ) | ≤ C (cid:18) x | ρ | x (cid:19) / (cid:18) − log | ρ | x | ρ | x (cid:19) . (A.5)16
20 40 60 80 100
M+1 || min. eigenvaluemax. eigenvalue q exact potentialrecovered potential Figure 6: The Hulth´en effective potential from Example 5.4 having ℓ = 1 / δ = 1 /
10 is considered.On the left: the smallest and the largest eigenvalues of the coefficient matrix of the truncated systemfor x = 3 as the function of M . On the right: potential recovered from the noisy data.Considering the difference between the regular solutions ϕ ℓ and ϕ ℓ , the following estimate followsfrom [17, (2.21) and (2.23)] and [18, Lemma B.2] | ϕ ℓ ( ρ, x ) − ϕ ℓ ( ρ, x ) | ≤ ∞ X n =1 C n +1 n ! (cid:18) x | ρ | x (cid:19) ℓ +1 e | Im ρ | x (cid:18)Z x y | ¯ q ( y ) | | ρ | y dy (cid:19) n = C (cid:18) x | ρ | x (cid:19) ℓ +1 e | Im ρ | x (cid:18) exp (cid:18) C Z x y | ¯ q ( y ) | | ρ | y dy (cid:19) − (cid:19) , (A.6)here ¯ q ( x ) = q ( x ) if ℓ > − / q ( x ) = (cid:0) − log( x x ) (cid:1) q ( x ) if ℓ = − / ϕ ℓ and ψ ℓ can be approximated using the asymptoticformulas for the functions J ν and H (1) ν . We have (see [1, (9.2.5)–(9.2.10)] for z ∈ R J ν ( z ) = r πz (cid:0) P ( ν, z ) cos χ − Q ( ν, z ) sin χ (cid:1) , H (1) ν ( z ) = r πz (cid:0) P ( ν, z ) + iQ ( ν, z ) (cid:1) e iχ , (A.7)where χ = z − ( ν + ) π and P ( ν, z ) = [ ν/ / X k =0 ( − k ( ν, k )(2 z ) k + θ ( z ) ( ν, ν/ / z ) ν/ / , (A.8) Q ( ν, z ) = [ ν/ / X k =0 ( − k ( ν, k + 1)(2 z ) k +1 + θ ( z ) ( ν, ν/ /
4] + 1)(2 z ) ν/ / , (A.9)with | θ , | ≤
1. From (A.7)–(A.9) it follows that for all z ≥ J ν ( z ) = r πz (cid:18) cos (cid:16) z − νπ − π (cid:17) + 4 ν − z sin (cid:16) z − νπ − π (cid:17) + O (cid:18) z (cid:19)(cid:19) ,H (1) ν ( z ) = r πz e i ( z − νπ − π ) (cid:18) i ν − z + O (cid:18) z (cid:19)(cid:19) , O (1 /z ) means that there exists a constant C such that the remainder is bounded by C/z forall z ≥
1. Taking the product and expanding cos χ and sin χ via the sum and difference of e iχ and e − iχ , we obtain that J ν ( z ) H (1) ν ( z ) = 1 πz (cid:18) e i ( z − νπ − π ) + i ν − z e i ( z − νπ − π ) + O (cid:18) z (cid:19)(cid:19) (A.10)for all z ≥ Remark
A.1 . One can easily deduce from (A.2), (A.3) and (A.10) that2 ρϕ ℓ ( ρ, x ) ψ ℓ ( ρ, x ) = i (cid:18) e i (cid:16) ρx − ( ℓ +1) π (cid:17) + O (cid:18) ρ (cid:19)(cid:19) , and does not converge when ρ → ∞ contrary to what is stated in [19, Remark 2.14].We refer the reader to [28] for the definition of the total variation of a function (denoted by V ( f ; R ))and the class of bounded variation vanishing at infinity functions (denoted by BV ( R )). We need thefollowing two properties of the functions from BV ( R ) class. Lemma A.2.
Let f ∈ BV ( R ) and g ∈ BV ( R ) . Then f · g ∈ BV ( R ) , and V ( f · g ; R ) ≤ V ( f ; R ) · (cid:0) V ( g ; R ) + k g k L ∞ ( R ) (cid:1) .Proof. It is well-known that the product of two functions of bounded variation is again a function ofbounded variation, see, e.g. [10], moreover V ( f · g ; R ) ≤ V ( g ; R ) · sup R | f | + V ( f ; R ) · sup R | g | . Now the statement follows observing that for BV functions one has sup R | f | ≤ V ( f ; R ). Lemma A.3 ([28, Corollary 10]) . If f ∈ BV ( R ) , then for all ω ∈ R \ { } its Fourier transform ˆ f isdefined and satisfies | ˆ f ( ω ) | ≤ V ( f ; R ) | ω | . Now we can formulate the main result of this section.
Proposition A.4.
Suppose that the potential q ∈ L (0 , ∞ ) ∩ BV [0 , ∞ ) . Then the asymptotics (2.5) holds.Proof. Due to the property F ℓ ( − ρ ) = F ℓ ( ρ ) we may assume that ρ >
0. First we assume that ℓ > − / F ℓ ( ρ ) = 1 + Z /ρ ψ ℓ ( ρ, x ) ϕ ℓ ( ρ, x ) q ( x ) dx + Z ∞ /ρ ψ ℓ ( ρ, x ) ϕ ℓ ( ρ, x ) q ( x ) dx + Z ∞ ψ ℓ ( ρ, x ) (cid:0) ϕ ℓ ( ρ, x ) − ϕ ℓ ( ρ, x ) (cid:1) q ( x ) dx =: 1 + I + I + I . The integral I can be estimated using (A.4) and noting that functions of bounded variation arebounded, | I | ≤ C Z /ρ x | q ( x ) | ρx dx ≤ C ρ Z /ρ | q ( x ) | dx ≤ C ρ .
18o estimate the integral I we utilize (A.2), (A.3) and (A.10) and obtain I = i ρ Z ∞ /ρ q ( x ) dx + ie − i ( ℓ +1) π ρ Z ∞ /ρ q ( x ) e iρx dx − ℓ ( ℓ + 1) e − i ( ℓ +1) π ρ Z ∞ /ρ q ( x ) x e iρx dx + i ρ Z ∞ /ρ q ( x ) O (cid:18) ρx ) (cid:19) dx = I + I + I + I . Now we have I = i ρ Z ∞ q ( x ) dx − i ρ Z /ρ q ( x ) dx = i ρ Z ∞ q ( x ) dx + O (cid:18) ρ (cid:19) , where we used that q is bounded. To estimate the integral in I note that it can be considered as R ∞−∞ g ( x ) e iρx dx , where g ( x ) = ˜ q ( x ) · [1 /ρ, ∞ ) ( x ), ˜ q is an extension of q to R by zero and A is thecharacteristic function of the set A . Both functions ˜ q and [1 /ρ, ∞ ) are of bounded variation and onecan easily see from Lemma A.2 that V ( g ; R ) ≤ V ( q ; [0 , ∞ )). Now applying Lemma A.3 we obtainthat | I | = 12 ρ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ g ( x ) e iρx dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ V ( q ; [0 , ∞ ))4 ρ . The estimate for the integral I is similar noting that the function x [1 /ρ, ∞ ) ( x ) ∈ BV ( R ) and V ( x [1 /ρ, ∞ ) ( x ); R ) ≤ ρ . Recalling the meaning of the O symbol, we have for I | I | ≤ C ρ Z ∞ /ρ | q ( x ) | x dx ≤ C ρ Z ∞ /ρ C x dx = CC ρ . Finally, for the integral I we utilize (A.4) and (A.6) and obtain | I | ≤ C Z ∞ x | q ( x ) | ρx (cid:18) exp (cid:18) C Z x y | q ( y ) | ρy dy (cid:19) − (cid:19) dx. Since R x y | q ( y ) | | ρ | y dy ≤ ρ R x | q ( x ) | dx ≤ ρ k q k L (0 , ∞ ) , we have (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) C Z x y | q ( y ) | ρy dy (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp (cid:18) C k q k L (0 , ∞ ) ρ (cid:19) − O (cid:18) ρ (cid:19) , hence I = O ( ρ ). Combining all the estimates we obtain the statement. Now assume that ℓ = − / I , . . . , I only the integrals I and I have to be treated differently from thecase ℓ > − /
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