Andrea Mantile. Fractional Integral Equations and Applications to Point Interaction Models in Quantum Mechanics TESI DI DOTTORATO DI RICERCA

Transcription

1 DOTTORATO DI RICERCA in MATEMATICA APPLICATA E INFORMATICA Ciclo XVI Consorzio tra Università di Catania, Università di Naoli Federico II, Seconda Università di Naoli, Università di Palermo, Università di Salerno SEDE AMMINISTRATIVA: UNIVERSITÀ DI NAPOLI FEDERICO II Andrea Mantile Fractional Integral Equations and Alications to Point Interaction Models in Quantum Mechanics TESI DI DOTTORATO DI RICERCA

2 Contents Contents 1 Introduction 3 1 Fractional Integral Equations Introduction to Abel integral equations Abel integral equation of the rst kind Abel integral equation of the second kind Fractional integral equation with time deendent coef- cients Small Time Asymtotics Large Time Asymtotics Case < : the L-transform analysis of the roblem Behavior on the imaginary axis at 6= Behavior at = Asymtotic behaviour of the solution in the generic case Case II: = Singularities on the imaginary axis Case III: > Point Interactions in Quantum Mechanics 3.1 Introduction Sectral Proerties Quantum Dynamics of Point Interactions The Charge Equation Solving charge equation The charge associated to the bound state The charge associated to a scattering state A Time Reversed Point Interaction Time Deendent Point Interactions The charge equation in the time deendent case

3 3 Energy Transfer Control via Point Interaction Introduction The Model The Linearized System Small Time Asymtotics for the Charge The Non Controllability Condition in the limit t! Finite Time Asymtotics for the Charge and Proof of Theorem The Nonlinear System Proof of the Main Result Remarks and Conclusions Ionization for Three Dimensional Time-deendent Point Interactions Introduction The Model Comlete Ionization in the Generic Case Further Remarks Conclusions and Persectives 74 Bibliograhy 75

4 Introduction The resent work is devoted to the study of dynamical systems described by integral equations of Abel kind. This equation - introduced by N. H. Abel in order to describe the mechanical roblem of isochrone curves - have found a wide range of alications in the framework of Continuum Mechanics (e.g. in [1]) as well as in Quantum Physics. Usually Abel, or more general fractional integral equations, are exected to aear in connection with singular erturbations of the Lalace oerator, when source or otential terms suorted by set of null measure are added to the lalacian. A classical examle is given by the the roblem of heating an in nite rod by an in ux of heat through a ointwise source of strenght (t) laced in the origin. The temerature eld u(t; x) is decribed by the equation: d dt u(t; x) w = d dx u(t; x) + (t)(x) u(; x) = u (x) which has to be intended in the weak sense. Using Dhuamel s formula, we may write the solution u(t; x) of this roblem in the form: u(t; x) = e t u + e (t s) (s)(x) ds where e t is the roagator associated to the one dimensional Lalace oerator, whose action is de ned by the relation: e t f = 1 (t) 1 Z +1 e jx By de nition of the Dirac delta, we get: 1 e t (x) = 1 e x 4t t x j 4t f(x ) dx 3

5 Then, assuming a vanishing initial temerature, we obtain: u(t; x) = 1 e t x 4(t s) s (s) ds If the interior boundary temerature: (t) = lim jxj! u(t; x) is known (for examle a result of an exerimental continuous monitoring), we may determine the corrisonding unknown in ux strenght as the solution of the following Abel equation: Z 1 t (s) ds = (t) t s In this work we will consider the energy transfer between discrete and continuous sectral comonents in nonautonomous quantum systems generated by time deendent oint interaction oerators. The roerties of these oerators, as we shall see, allow to describe energy exchanges in terms of a system of two couled integral equations of Abel kind with non constant coe cients, reducing in this way the real degree of freedom of the roblem. In the rst Chater, after a brief introduction to fractional integrals equations of Abel kind, we study the nite time and the large time asymtotic behavior of the solutions, taking into account the case of nonconstant coe - cients. Results resented there form the main contribution of this work and will be largely alied in the subsequent chaters. In the second Chater we introduce the oerators of oint interaction in Quantum Mechanics giving the main roerties of the related quantum systems. It will be shown that the state of a quantum article subjected to the action of a 3-D oint interaction is determined by the system: 8 >< >: (t; x) = e it (x) + q(t) + 4 i i (i) 3 (s)q(s) t s ds = 4 i e i x 4(t s) (t s) Z 3 t q(s)ds (e is )(x=) t s ds where is the initial state and (t) is the dynamical arameter characterizing the interaction as a function of time. It is immediate to recognize that the auxiliary variable q(t) - which will be de ned as the "charge" of the system - satis es an Abel integral equation with a nonconstant coe cient (t) in the integral kernel. Chaters three and four are devoted to the alications of Abel kind equations in the context of quantum system generated by oint interactions. In the third Chater we consider the roblem of energy-mass transfer from scattering to bound states for a one body quantum system uder the action of 4

6 a time deendent oint interaction. Under suitable assumtions on the initial state of the article and using results on the nite time asymtotic behavior for solutions of Abel equations, we rove a theorem of local controllability of this rocess. In the fourth Chater we study the time evolution of a three dimensional quantum article under the action of a time-deendent oint interaction xed at the origin. We assume that the strength of the interaction (t) is a eriodic function with an arbitrary mean. Under very weak conditions on the Fourier coe cients of (t), and making use of large time asymtotic results of Ch.1, we rove that there is comlete ionization as t! 1, starting from a bound state at time t =. 5

7 Chater 1 Fractional Integral Equations In this chater we shall consider some basic integral equations of fractional order; in articular, we will focus our attention on the asymtotic behaviour of the solutions. This analysis will be extended also to the the case of equations with time eriodic coe cients. A rather comlete exosition on the subject of integral and di erential roblems of fractional order may be found in [], [3]. For the seci c case of fractional integral equation with time deendent coe cients, we refer to [4]. 1.1 Introduction to Abel integral equations According to the Riemann-Liouville de nition, the fractional integral of order ( R + ), of a function f, is given by: J f 1 () f(s) 1 ds (1.1) (t s) where (z) denotes the Euler Gamma function. This exression may be thought as a natural generalization of the well known Cauchy formula: Z J n 1 t f(s) f = (n 1)! (t s) 1 n ds reresenting the n-fold rimitive of the function f - J n f in our notation - in terms of a convolution integral. The simlest equations involving the oerators J are the Abel equations of the rst and of the second kind: 6

8 1.1.1 Abel integral equation of the rst kind The equation is: Z 1 t () u(s) 1 ds = f(t) (; 1) (1.) (t s) Let us solve (1.) using Lalace transform L ; rst we observe that the L-transform of the kernel function is given by: 1 1 L = 1 8 > (1.3) () t 1 from which we get the following relation for the Lalace transform of (1.): ~u() = ~ f() (1.4) where ~u and ~ f denote the transforms of u and f resectively. Next observe that there are di erent ways to erform the inverse Lalace transform of (1.4); in fact we may write this relation in the form: f() ~u() = ~ (1.5) 1 obtaining, under the regularity condition: ( f L 1 loc (; +1) f t [; 1 ) (1.6) t! the following solution: u(t) = 1 d (1 ) dt f(s) (t s) ds (1.7) Otherwise, equation (1.4) could also be set in the form: h ~ i f() f( + ) ~u() = + f(+ ) (1.8) 1 1 from which, under the condition: f L 1 loc (; +1) jf( + (1.9) )j < 1 we get an alternative form of the solution: u(t) = 1 (1 ) f (s) (t s) ds + f( + ) 1 1 (1.1) (1 ) t We notice that the exressions (1.7) and (1.1) are not eqiuvalent, and, in order to write the solution in the last form, a stronger regularity condition on the source term f is required. 7

9 1.1. Abel integral equation of the second kind The equation is: u(t) + () u(s) 1 ds = f(t) (; 1); C (1.11) (t s) Its solution may be reresented by the Picard series: 8 < u(t) = P +1 n= u n(t) u : (t) = f(t) (1.1) u n = J u n 1 The exlicit exression of u n is: and, using Lalace transform, this reads as: u n (t) = ( ) n J n f(t) (1.13) u n = ( ) n t n 1 n=1 (n) f (1.14) where indicates the convolution roduct. Then, the solution u(t) of equation (1.11) is, at least at a formal level, given by the limit: nx! u(t) = f(t)+ lim ( ) k t k 1 nx n!1 (k) f = f(t)+ lim ( ) k t k 1 f n!1 (k) k=1 k=1 (1.15) Next we observe that, due to the raid growth of the gamma function, the series P n k= ( )k t k 1 is uniformly convergent to a continuous function in (k) every bounded interval t [t ; T ] such that t >. Moreover, this sum 1 diverges as for t! +. This imlies that there exists a solution: t 1! 1X u(t) = f(t) + ( ) n t n 1 f (1.16) (n) of equation (1.11) for any function f ful lling the following regularity condition: ( f L 1 loc (; +1) f t [; ) (1.17) t! The uniqueness of this solution is a straightforward consequence of exression (1.16). 8

10 A comact form for the solution of equation (1.11) can be given in terms of Mittag-Le er functions. These functions are de ned on the comlex lane by the relation: E (z) = 1X n= z n (n + 1) > ; z C The convolution kernel in (1.16) may be exressed as the derivative of a Mittag-Le er function as follows: d dt E ( t ) = d 1X ( ) n t n dt (n + 1) = X 1 ( ) n (n) t n 1 1X ( ) n t n 1 = (n + 1) (n) n= n=1 n=1 (1.18) then, the solution (1.16) reads as: u(t) = f(t) + E ( t ) f (t) (1.19) The same solution may still be given into di erent forms; the L-transform of (1.11) reads: ~u() + ~u() = f() ~ (1.) from which we nd two alternative forms for ~u: Recalling that: we obtain: and 8 < : ~u() = h 1 ~u() = 1 + ~ i f() + h ~f() f( )i + L [E ( t )] = 1 u(t) = d dt + u(t) = f( + ) E ( t ) f(+ ) Re > jj 1 (1.1) f(t s)e ( s ) ds (1.) f (t s)e ( s ) ds (1.3) Again we notice that the exressions (1.19), (1.) and (1.3) are not equivalent because they require di erent condition of regularity on the source term f. In articular, the validity of (1.3) imlies the condition: f L 1 loc (; +1) jf( + (1.4) )j < 1 which is stronger than (1.17), needed for the validity of (1.19) and (1.). 9

11 1.1.3 Fractional integral equation with time deendent coe cients Here we shall consider a fractional integral equation of order 1 solution aears multilied by a time deendent coe cient: u(t) + where the (s) u(s) t s ds = f(t) (1.5) We will study the roerties of equation (1.5) on a nite time interval: t [; T ] under the assumtions: L 1 (; T ; C) f L 1 (1.6) (; T ; C) Firts observe that, by a simle iteration rocedure, the solution u(t) may be formally exressed using the Picard series: 8 < : u(t) = P +1 n= u n(t) u (t) = f(t) u n (t) = R t (s) u n 1 (s) t s ds (1.7) which admit the following estimate: " +1X n= ju n (t)j kf(t)k L 1 (;T ) X n=1 kk n L 1 (;T ) A n n t n # (1.8) with: 8 < A n = : 1 ( n )! ( n + 1 ) ( 1 ) 1 n!! n even n odd (1.9) Thus (1.7) de nes a sum of continuous functions which is, by virtue of our assumtions (1.6) and of the estimates (1.8), uniformely convergent into [; T ]. It remains roved the following result: Theorem 1 Let and f satisfy the assumtions (1.6). Then equation (1.5) admits an unique continuous solution on the interval t [; T ], given by the Picard series (1.7), (1.9), for which holds the estimate: " # kuk 1 kfk L 1 (;T ) X n=1 kk n L 1 (;T ) A n n T n (1.3) 1

12 Proosition The homogeneous equations associated to (1.5) with L 1 (R) has no non-zero solution in L loc (R+ ), 1 1. Proof The roof (see e.g. [6]) exloits the fact that, due to the estimate (1.8), the homogeneous equation associated to (1.5) with L 1 (R) has a null solution in any L (; T n ) with T n increasing to in nity for increasing n. 1. Small Time Asymtotics Our aim is to study the small time behaviour of the solution u(t) of the equation (1.11) with = 1 : u(t) + ( 1 ) u(s) t s ds = f(t) C (1.31) under the following hyothesis on the source term f: f(t) = u in E 1 (t 1 ) + g(t) (1.3) g(t) = a m t m+ 1 + o(t m+ 3 ); am 6= (1.33) with: o(t m+ 3 ) c t m+ 3 ; c > (1.34) for t [; ), R +, m N. First we notice that the solution of (1.31) may be exressed as the sum of the solutions u and u g of the equations: u (t) + ( 1 ) u (s) ds = u in t s E 1 (t 1 ) (1.35) u g (t) + ( 1 ) Using the Lalace transform (1.35) become: ~u () = u in u g (s) t s ds = g(t) (1.36) ) ~u () = uin 11

13 then we have for u (t) an exonential solution: u (t) = u in e t For the second equation we have: ~u g () 1 + = ~g() ) ~u 1 g () = ~g() = ~g() whose solution of reads as: u g (t) = (1.37) E 1 ( t 1 )g(t s) ds (1.38) Thus, the solution of equation (1.31) admits the following reresentation: u(t) = u in e t + E 1 ( t 1 )g(t s) ds (1.39) Recalling that, from de nition (1.18), the function E 1 ( t 1 ) may be exressed as: E 1 ( t 1 1 ) = t ( 1) + + o(t 1 ) (1.4) with: o(t 1 ) c1 t 1 ; c1 > (1.41) in a suitable right neighbourhood of the origin, by substitution in (1.39), we have: u(t) e t = = a m s m ( 1) t u in E 1 ( t 1 )g(t s) ds = ds+ a m s m ds+a m s m o((t s) 1 )ds s ( 1) + o(s m+1 )ds + which, after exlicit calculations, becomes: u(t) o((t s) 1 )o(s m+1 )ds u in e t = a m m+1 m! 1 ( 1) (m + 1)!! tm+ + t m+1 a m m o(tm+ ) (1.4) o(s m+1 ) ds+ t s with: o(t m+ 3 ) c t m+ 3 ; c > (1.43) for any t in a neighbourhood [; 1 ). 1

14 1.3 Large Time Asymtotics In this section we will erform the analysis of the long time behaviour of the solution of an (1.5)-tye equation of order 1 : q(t) + ( 1 ) (s) q(s) t s ds = f(t) (1.44) under the following assumtions for the source term and the coe cients: = 4 i (1.45) f(t) = 4 j()j E 1 ( () t 1 ) (1.46) In the hysical alications - as we shall see - the meaningful arameter of the system is the negative lower bound of (t). Hence we require that: () < (1.47) Moreover we shall assume that (t) is a real eriodic continuous function of eriod T. The continuity of (t) guarantees that it can be decomosed in a Fourier series, for each t R +, and the series converges uniformly on every comact subset of the real line. In terms of the Fourier coe cients (t) is given by: (t) = P n e i!nt ; f n g `1 (Z) ;! = T nz (1.48) n = n In order to justy e our use of Lalace transform in the analysis of equation (1.44), our next task is to rove the following: Lemma 3 Let q(t) be the solution of equation (1.5) with ; f L 1 (R). Then, the lalace transform Lq() exists and is analytic at least in the oen half lane of C de ned by the condition: C : Re > jj kk 1 (1.49) Proof Consider the function: q (t) = e t q(t) with C and Re >, it satis es the equation: q (t) + (s) e (t s) q(s) ( 1) ds = e t f(t) = '(t) (1.5) t s where, from our hyothesis, we have '(t) L 1 (; +1). Then, alying the Young s inequality: kf gk 1 kfk 1 kgk 1 13

15 to the convolution in equation (1.5), we obtain the following estimate: kq jj k 1 1 ( 1) e t 1 kk t 1 k'k 1 which rovide an e ective bound for the norm kq k 1 if the coe cient jj e 1 t 1 ( 1 ) t kk 1 is ositive. Recalling that for Re > : e t r 1 = t Re we get the condition: 1 > jj r ( 1) Re kk 1 ) Re > jj kk 1 (1.51) Following the same line, it s easy to rove that the artial derivatives of the function q w.r.t Re and Im Re q Im q = te t q(t) are bounded by measurable functions of t if the condition (1.51) is ful lled: te t q(t) 1 te (16 kk +iy) 1 t q(t) < 1 1 Then e t q(t) is C 1 measurable w.r.t. t [; +1) for any in the domain (1.49) and, in the same hyothesis, its artial derivatives w.r.t. are bounded by measurable functions of t. This allows us to conclude that the Lalace integral: Lq() = Z +1 q(t)e t dt de nes a C 1 class function for in the domain (1.49) Lemma 3 allow us to say that, under the hyothesis (1.45) and (1.46), the Lalace transform of the solution q(t) exist analytic at least for: C : < () > 16 kk 1 In the notation: Lf = ~ f 14

16 the Lalace transform to equation (1.44), reads as: s i X ~q() = 4 k ~q( + i!k) + f() ~ (1.5) kz where ~f() 4 h j()j L E 1 ( () t 1 ) i () = 4 j()j 1 1 4() i with the choice of the branch cut for the square root along the negative real line: if = % e i#, = % e i#= (1.53) with < #. In the following sections we shall erform the asymtotic analysis of system (1.44): we shall rove that, under generic conditions on (t), the solution of equation (1.44) goes asymtoticaly to zero with a olynomial ower law. Although the result does not deend on the sign of the mean of (t), we have to discuss searately the case < and, because of the slightly di erent form of equation (1.5). 1.4 Case < : the L-transform analysis of the roblem In what follows we introduce the analysis of equation (1.44) using as main tool the Lalace transform. Since () <, we will assume that (t) satis es the normalization condition: X n = 1 (1.54) 4 nz In the framework of alications, this condition reresents a change in the energy scale of a hysical system; on the other hand, it rovide a simli cation of the Lalace transform calculus, but does not e ects the asymtotic behaviour of the solution. Moreover we introduce another condition we shall use later on: let T the right shift oerator on `1(N), i.e. (T a) n a n+1 (1.55) 15

17 we say that = f n g `1(Z) is generic with resect to T, if ~ f n g n> `1(N) satis es the following condition e 1 = (1; ; ; : : :) 1_ T n ~ (1.56) For a detailed discussion of genericity condition see [14]. If (1.54) holds, equation (1.5) becomes (at least for <() > ) 4 ~q() = 4 + X i k ~q( + i!k) i i kz k6= n= i 1 + i (1.57) Setting q n () ~q( + i!n), we obtain a sequence of functions on the stri I = f C; =() <!g. Setting q() fq n ()g nz equation (1.57) can be rewritten in the form: where (Bq) n () and g() = fg n ()g nz with g n () q() = B()q() + g() (1.58) 4 4 +!n i 4 +!n X k q n+k () (1.59) i kz k6= 1!n i i 1 + i!n (1.6) We observe that (1.58) de nes an equation in the Hilbert sace H = `(Z) for any I. From the exlicit exression of the oerator (1.59) and (1.6), it is clear that the coe cients of the equation fails to be analytic on the imaginary axis at = ((4 )!n)i, for some n Z and then the solution may be singular there. Since =() [;!), one has (4 ) 1 < n (4 ) (1.61)!! and then the singularity aears at most in the equation for q n (there is only one integer 1 which satis es the revious inequality) at = ((4 )!n)i. For instance, if! > (4 ), the ole may be at = (4 ) i in the equation for q. The next Proosition shows the roerties of oerator B: 1 In fact n must be non negative. 16

18 Proosition 4 For I, <() =, 6= ;, B() is an analytic oeratorvalued function and L() is a comact oerator on `(Z). Proof: Analyticity on the imaginary axis for 6= ; easily follows from the exlicit exression of the oerator. Moreover B() can be written where b() is the oerator B() = b() X kz k6= k T n+k (b q) n () b n () q n () = 4q n () 4 +!n i and T is the right shift oerator on `(Z). Since kt k = 1, the series converges strongly to a bounded oerator. Moreover b() is a comact oerator on the imaginary axis for 6= ; : b() is the norm limit of a sequence of nite rank oerators, because lim n!1 b n () =. Hence the result follows for examle from Theorem VI.1 and VI.13 of [7]. Let us rst consider the behavior of the solution for Re > : Proosition 5 There exists a unique solution q n () `(Z) of (1.58) and it is analytic for Re >. Proof: The key oint will be the alication of the analytic Fredholm theorem to the oerator L() (Theorem VI.14 of [7]), in order to rove that (I B()) 1 exists for : Re >. Since there is no non-zero solution in L loc (R+ ) of the homogeneous equation associated to (1.44) (see the Proosition ), then the homogeneous equation associated to (1.58) has only the trivial solution in `(Z). Moreover the oerator B is comact and thus analytic Fredholm theorem alies. The result easily follows, because g() `(Z) and each g n () is analytic for : Re >. In the following subsections we shall extend the equation (1.57) above to the imaginary axis and study the behavior of the solution there. 17

19 1.4.1 Behavior on the imaginary axis at 6= Actually we have to distinguish the so called (see [14]) resonant case, i.e. when (4 ) = N! for some N N, because in that case we can have a ole only at = and then the solution is immediately seen to be analytic on the whole imaginary axis excet at most for =. At rst we consider the behavior of the solution on the imaginary axis for 6= ; 6=. We are going to rove that the solution is in fact analytic there. Proosition 6 There exists a unique solution q n () `(Z) of (1.58) and it is analytic on the imaginary axis for 6= ;. Proof: The Proof of Proosition 5 still alies because each g n () is analytic for 6= ; on the imaginary axis. We can now study the equation (1.58) in a neighborhood of (if 6= ). An imortant reliminary result is the following Lemma 7 Let (1.48) and the genericity condition (1.56) be satis ed by f n g. The system of equations 8 9 >< >= r n = 4 4 +!n i >: X kz k6=n;n k n r k + h n () >; (1.6) has a unique solution fr n g `(Znfng) in a ure imaginary neighborhood of, where n Z and I, <() =, are de ned by (1.61), for every h n () such that h h n () n() 4 +!n i belongs to `(Z n fng). Moreover, if h n () is analytic in a neighborhood of, the solution is analytic in the same neighborhood. 18

20 Proof: Equation (1.6) is of the form r = B r + h where h fh ng belongs to `(Z n fng) and B is a comact oerator (see Proosition 4). In order to aly analytic Fredholm theorem to the oerator B, we need to rove that there is no non-zero solution in a neighborhood of of the homogeneous equation. Suose that the contrary is true, so that fr n g `(Z n fng) is a non-zero solution of R n = 4 4 +!n i X kz k6=n;n k n R n Multilying both sides of equation above by Rn and summing over n Z n fng, one has X!n i jrn j = 4 X R n k n R k nz n6=n n;kz n;k6=n and, since the right hand side is real, =[ X!n i jrn j ] = nz n6=n for = i, < <!, and then R n = for n <. Now suose that R 6= and let n N be such that R n =, n < n, and R n 6= (hence n ). Fixing R n =, for each n < n the homogeneous equation gives 1X or, setting k = n 1 + k, for n, k=n k n R k = 1X k +nr n 1+k = k =1 which imlies (see (1.48)), for each n, (R ; T n )`(N) = where R n = R n 1+n and (; ) stands for the standard scalar roduct on `(N). 19

21 If f n g satis es the genericity condition (1.56), R has to be orthogonal also to e 1 and then R n =, which is a contradiction. Therefore R =. The rst art of the Lemma then follows from analyticity of B () and analytic Fredholm theorem. Moreover if fh n ()g is analytic in a neighborhood of, analyticity of the solution is a straightforward consequence. Proosition 8 If f n g satis es (1.48) and the genericity condition with resect to T (1.56), the unique solution fq n g `(Z) of (1.58) in analytic on the imaginary axis excet at most for =. Proof: If (4 ) = N! for some N N (resonant case) there is nothing to rove, since the coe cients of (1.58) fails to be analytic only at =. On the other hand, in the non resonant case, Proosition 6 guarantees analyticity on imaginary axis for 6= ;. Therefore it is su cient to study the behavior of the solution in a neighborhood of, where the coe cients of (1.58) have a singularity. We are going to rove that in fact the solution is analytic at. The strategy of the roof is to analyze searately the terms q n, n 6= n, n being de ned in (1.61), and then rove that also q n is analytic in a neighborhood of. By Lemma 7 there is a unique solution of the system t n = r n + t n q n = 4 4 +!n i 4 4 +!n X kz k6=n;n i k n t k 4 n n 4 +!n Setting q n = r n + t n q n, n 6= n, on (1.58), one has 8 >< r n = 4 4 +!n i 4 +!n i >: >: n nq n + X kz k6=n;n 1!n i i 1 + i!n and therefore the equation for fr n g, n 6= n, becomes 8 >< X kz k6=; n i 9 >= k n (r k + t k q n ) >; + (1.63) 9 k r n+k + i 1!n i >= 1 + i!n >; (1.64)

22 while q n satis es the equation >< q n = 4 + X k n (r k + t k q n ) + i 1!n i >=!n i >: 1 + i!n >; kz k6=n or !n i + 4 X kz k6=n k n t k q n = 4 X i k n r k 1 +!n i kz k6=n Since the last term is analytic in a neighborhood of and ft n g, fr n g `(Z n fng) are both analytic, as it follows alying Lemma 7 above to (1.63) and (1.64), it is su cient to rove that X k n ~t k 6= where kz k6=n ~t n t n ()j = Assume that the contrary is true: from equation (1.63) we obtain X 4 + ~ X!n i t n = 4 ~t n k n ~t k 4 X n n t ~ n = nz nz n6=n n6=n = 4 X n;kz n;k6=n;n6=k n;kz n;k6=n;n6=k ~t n k n ~t k where we have used the second condition in (1.48). The revious equation imlies (the right hand side is real) ~t n =, 8n < N = i and then, since! 1 < N <, ~t n =, 8n <. Hence from (1.63) we have, 8n <, X k n ~t k + n n = k k6=n Now suosing without loss of generality that ~t 6= and setting T n = ~t n 1, n 6= n + 1, and T n+1 = 1, we obtain, 8n, 1X k+n T k = k=1 1

23 and using the genericity condition (1.56) (as in the roof of Lemma 7) we get T 1 = t =, which is a contradiction. In conclusion q n is analytic in a neighborhood of : analyticity of q n, n 6= n is then a straightforward consequence of analyticity of fr n g, ft n g and decomosition q n = r n + t n q n. The roof is then comleted, since r n and t n belong to `(Z n fng) in a neighborhood of = Behavior at = We shall now study the behavior of the solution of (1.58) on the imaginary axis at the origin. With the choice (1.53) for the branch cut of the square root, it is clear that we must exect branch oints of ~q(), solution of (1.57), at = i!n, n Z, which should imly a branch oint at = for each q n in (1.58). We are going to show that q n, n Z has a branch oint at =. The non-resonant case and the resonant one will be treated searately. Proosition 9 (non resonant case) If (4 ) 6= N!, 8N N and f n g satis es (1.48) and (1.56) (genericity condition), the solution of equation (1.58) has the form q n () = c n () + d n (), n Z, in an imaginary neighborhood of =, where the functions c n () and d n () are analytic at =. Proof: Setting q n = r n + t n q, n 6= and choosing a solution ft n g `(Z n fg) of the system of equations (1.63) with n =, we obtain that fr n g must satisfy (1.64). It is easy to see that the result of Lemma 7 holds also in a neighborhood of = with n =, so that fr n g, ft n g `(Z n fg) are unique and analytic at =. Thus it is su cient to rove that q, which is solution of i + 4 X kz k6= 7 k t k 5 q = 4 X kz k6= has the required behavior near =. First, setting t n = t n ( = ), we have to rove that X k t k 6= kz k6= i (1 k r k 1 + i i)

24 but, assuming that the contrary is true and multilying both sides of equation (1.63) by t n and summing over n Z, n 6=, one has X!n jt n j = 4 X t n k n t k + 4 nz n;kz n;k6= and then, because of genericity condition (1.56), ft ng =, 8n Z n fg, which is imossible, since ft n g solves (1.63). Now, calling F 4 X k t k kz k6= and we have and h 4 + G 4 X kz k6= k r k i i + F q = G + i (1 1 + i q = F + G where F is analytic in a neighborhood of =, because of analyticity of F and G, and i) G i i (4 + F + 1) + i (1 + i) G (1 + i)[(4 + F ) + i] (1.65) The resonant case, i.e. 4 =!N for some N N, is not so di erent from the non-resonant one and we shall rove that the solution has the same behavior at the origin. The roof is slightly di erent because we need to show the absence of a ole at = : from (1.58) one has >< X q N () = k q n+k () + i 1!N i >=!N!N i >: 1 + i!n >; kz k6= and the coe cients have a singularity at =. We are going to rove that in fact the solution has no ole at the origin: roceeding as in the roof of Proosition 8, let us begin with a reliminary result, which take the lace of Lemma 7: 3

25 Lemma 1 Let (1.48) and the genericity condition (1.56) be satis ed by f n g. The system of equations 8 9 >< >= r n = 4!N!n i >: X kz k6=; n k r n+k + h n () >; (1.66) has a unique solution fr n g `(Z n fng) in a ure imaginary neighborhood of =, for every h n () such that h n() h n ()!N!n i belongs to `(Z n fng). Moreover, if h n () is analytic in a neighborhood of =, the solution is analytic in the same neighborhood. Proof: We shall roceed as in the roof of Proosition 8, searating the contribution of r N, which may be singular: setting r n = u n + v n r N, n 6= ; N, on (1.66), one has u n +v n r N = 8 4 ><!N!n i >: N nr N + X kz k6=; n;n n i 1!n i +!N!n i 1 + i!n and requiring that fv n g, n 6= ; N, solves v n = u n = 4 X!N!n i 4!N!n i kz k6=; n;n n the equation for fu n g, n 6= ; N, becomes 8 >< X >: kz k6=; n;n n 9 >= k (u n+k + v n+k r N ) >; + 4 N n k v n+k + (1.67)!N!n i k u n+k + 9 i : 1!n i >= 1 + i!n >; (1.68) 4

26 Moreover r N satis es r N = 4!N!N i or 8 >< >: X kz k6=; N 6 X 4!N!N i 4 9 k (u k + v k r N ) + i 1!n i >= 1 + i!n >; kz k6=;n k N v k r N = = 4 X k N u k + i 1!n i kz 1 + i!n k6=;n Alying the discussion contained in the roof of Lemma 7, it is not di cult to see that the solutions of equations (1.68) and (1.67) are analytic in a neighborhood of the origin and belong to `(Znf; Ng). Therefore it remains to rove that (setting vn = v n ( = )) X k N vk 6= kz k6=;n but the argument in the roof of Proosition 8 excludes this ossibility, if f n g satis es the genericity condition. The roof is then comleted, because analyticity of r N imlies analyticity of all r n, n 6= ; N. Proosition 11 (resonant case) If (4 ) = N!, for some N N and f n g satis es (1.48) and (1.56) (genericity condition), the solution of equation (1.58) has the form q n () = c n () + d n (), n Z, in an imaginary neighborhood of =, where the functions c n () and d n () are analytic at =. Proof: See the roof of Proosition 9 and Lemma 1 above. 5

27 1.4.3 Asymtotic behaviour of the solution in the generic case Summing u the results about the behavior of the Lalace transform ~q() of q(t) we can state the following Theorem 1 If f n g satis es (1.48) and the genericity condition (1.56) with resect to T, as t! 1, jq(t)j A t 3 + R(t) (1.69) where A R and R(t) has an exonential decay, R(t) Ce Bt for some B >. Proof: Proositions 6, 8 and 9 guarantee that ~q() is analytic on the closed right half lane, excet branch oint singularities on the imaginary axis at = i!n, n Z. Therefore we can chose a integration ath for the inverse of Lalace transform of ~q(q) along the imaginary axis like in [14]. Proosition 9 imlies that the contribution of the branch oint at = is given by the integral i Z 1 d G ( ) e t where G, de ned in (1.65), is a bounded analytic function on the negative real line: from exlicit exression of F and G and equations (1.64) and (1.63), it is clear that G is analytic and lim!1 G ( ) = on the real line. So that the corresonding asymtotic behavior as t! 1 is Z 1 d Z 1 G ( ) e t C d e t = A t 3 Let us consider now the contribution of branch oints at = i!n, n 6= : from Proositions 9 and 11 it follows that, in a neighborhood of =, q n () = c n () + d n () where c n () and d n () are analytic at =. Moreover using the decomosition q n = r n + t n q, n 6=, as in the roof of Proosition 9 and 11, and studying the equation (1.63) for t n, we immediately obtain fd n g `1(Znfg), because of condition in (1.48). Since q n () = ~q( + i!n), the contribution of singularities at = i!n, n 6=, is then given by X nz n6= Z i!n i!n 1 d d n ( i!n) i!n e t = 6

28 and the series = i Z 1 df X nz n6= X d n ( nz n6= d n ( ) e i!nt g e t = ) e i!nt converges uniformly to a bounded function of t, because fd n g `1(Z n fg). Adding u the contributions of every branch cut, one obtain the required leading term in the asymtotic behavior. Indeed the rest function R(t) is given by the contribution of oles outside the imaginary axis and then shows an exonential decay as t! Case II: = If (t) = = does not deend on time, the roblem has a simle solution: the sectrum of H (t) is absolutely continuous and equal to the ositive real line, with a resonance at the origin; hence there is no bound state and the system shows comlete ionization indeendently on the initial datum. On the other hand if (t) is a zero mean function, we shall see that the genericity condition (1.56) is still needed to have comlete ionization. So let us assume that =, the normalization (1.54) holds and the initial datum is given by (4.4): equation (1.5) then becomes ~q() = s i X 4 k ~q( + i!k) kz k6= i s i 1 i 1 + i (1.7) with the choice (1.53) for the branch cut of. By Proosition 5 the solution is analytic on the oen right half lane. In the following section we shall study the singularities on the imaginary axis Singularities on the imaginary axis Setting q n () ~q( + i!n), I = [;!), as in Section 3.1, equation (1.7) assumes the form (1.58), q() = M()q() + o() (1.71) 7

29 with and o() = fo n ()g nz, (Mq) n () 4!n X k q n+k () (1.7) i kz k6= o n () i!n i (1 +!n i) (1.73) Proosition 13 For I, <() =, 6=, M() is an analytic oeratorvalued function and M() is a comact oerator on `(Z). Proof: See the roof of Proosition 4. Proosition 14 There exists a unique solution q n () `(Z) of (1.71) and it is analytic on the imaginary axis for 6=. Proof: See the roof of Proosition 6. Proosition 15 If f n g satis es (1.48) and the genericity condition (1.56), the solution of equation (1.71) has the form q n () = c n () + d n (), n Z, in a neighborhood of =, where the functions c n () and d n () are analytic at =. Proof: Let us roceed as in the roof of Proosition 9: setting q n = r n + t n q, n Z n fg, where ft n g is the solution of t n = 4!n i X kz k6=; n k t n+k 4 n!n i (1.74) A slightly di erent version of Lemma 7 guarantees that the solution ft n g `(Z n fg) is unique and analytic at =. By means of this substitution we obtain r n = 4!n i X kz k6=; n k r n+k i!n i (1 +!n i) (1.75) 8

30 and or q = 4 X i kz k6= k (r k + t k q ) i + F q = G where (like in the roof of Proosition 9) i i (1 + i) 1 + i and F 4 X kz k6= G k t k 4 X k r k kz k6= Moreover F () 6=, because of genericity condition (1.56) (see the roof of Proosition 9), F and G are analytic in a neighborhood of = (see Lemma 7), so that q = F + G where F and G are analytic and G i(f + 1) i(1 + i)g (1 + i)(f + i) As in section 4 of this Chater, we can now state the main result: Theorem 16 If f n g satis es (1.48) and the genericity condition (1.56) with resect to T, as t! 1, jq(t)j A t 3 + R(t) (1.76) where A R and R(t) has an exonential decay, R(t) Ce Bt for some B >. Proof: See the roof of Theorem 1. 9

31 1.6 Case III: > To comlete the analysis of the roblem, we shall consider the case of mean greater than : taking the normalization (1.54) and the initial condition (4.4), (1.5) assumes the form (1.57): ~q() = X k ~q( + i!k) i kz k6= i i i 1 + i (1.77) Analyticity of the solution on the oen right half lane is a consequence of Proosition 5. Moreover, following the discussion contained in section 4 and setting q n () ~q( + i!n), =() [;!), the equation assumes the form (1.58). Let us now consider the behavior on the imaginary axis: singularities for <() = are associated to zeros of 4 +!n + s, s [;!), but, since >, it is clear that the exression can not have zeros on the imaginary axis. Hence the roof of Proosition 6 can be extended to the closed right half lane excet the origin: Proosition 17 If f n g satis es (1.48), the solution ~q() of (1.77) is unique and analytic for <(), 6= i!n, n Z. Proof: See the roof of Proosition 6, Proositions 4 and and the revious discussion. Moreover the behavior at the origin is described by the following Proosition 18 If f n g satis es (1.48) and the genericity condition with resect to T (1.56), then, in an imaginary neighborhood of = i!n, n Z, the solution of equation (1.77) has the form ~q() = c n () + d n () i!n, where the functions c n () and d n () are analytic at = i!n. Proof: The roof of Proosition 9 still alies with only one di erence: since, indeendently on!, the solution can not have a ole on the imaginary axis, we need not to distinguish between the resonant case and the nonresonant one. We can now rove the asymtotic result: 3

32 Theorem 19 If f n g satis es (1.48) and the genericity condition (1.56) with resect to T, as t! 1, jq(t)j A t 3 + R(t) (1.78) where A R and R(t) has an exonential decay, R(t) Ce Bt for some B >. Proof: See the roof of Theorem 1. Remark: If (t), 8t R +, Proosition 18 holds without the genericity condition on the Fourier coe cients of (t): for instance the genericity condition enters (see the roof of Proosition 9) in the roof of absence of non-zero solutions of the homogeneous equation t n = 4 4 +!n + s X kz k6=; n k t n+k where s [;!). Let us suose that there exists a non-zero solution ft n g `(Z). Multilying both sides of the equation by Tn, one has X!n + s jtn j = 4 X Tn k n T k nz n6= n;kz n;k6= Since the right hand side is real, T n =, 8n <. Moreover, xing T = and setting T (t) X T n e i!nt nz it follows that 4 X n;kz T n k n T k = 4 (T (t); (t)t (t)) L ([;T ]) because (t), 8t [; T ], but the left hand side is ositive and then Q n =, 8n Z. 31

33 Chater Point Interactions in Quantum Mechanics.1 Introduction A erturbation of the lalacian suorted by a nite set of oints fy i g n i=1 in R d - with d 3 - de nes a secial case of singular erturbation referred to as oint interaction. At a formal level, the associated Schrödinger oerator can be written as 1 : nx H = + i (x y i ) (.1) i=1 A oint interaction hamiltonian, then, is intended as the selfadjoint realization in L (R d ) of the formal exression (.1) [5]. These oerators aeared rst in Theoretical Physics during the 3 s. They were introduced in order to realize a model for the interaction of articles in nuclei [6]. After then, they became a natural tool to describe short range forces or "small" obstacles for scattering of waves and articles. In the alications ersective, the main reason of interest of this subject rests uon the fact that oint ineractions often lead to models which are exlicitely solvables. It turns out that the sectral characteristics (eigenvalues and eigenfunctions) of oerators (.1), and then all the hysical relevant quantities related to, can be exlicitely comuted [7]. This circumstance motivates an increasing attention on this subject in the alication of mathematics in various sciences, e.g. in hysics, chemistry, biology, and in technology. 1 Note that this exression is always to be intended in a weak sense when working in dimension greater than one. 3

34 The rigorous de nition of a one oint interaction - due to F.A. Berezin and L.D. Fadeev [5] - is based on the theory of selfadjoint extensions of symmetric oerators. Let us consider the set of all selfadjoint extentions of the oerator: H = 4 D(H) = C 1 (R 3 = fg) it may be exressed in the form: 8 H = H >< D(H ) = L (R 3 ) >: = + q G ; H (R 3 ); lim = q ( + ) 4 x! (H + ) = ( + ) (.) where G is the Green function of the lalacian with resect to > : whose exlicit exression is: ( )G = (x) G (x) = e jxj 4 jxj (.3) while is a real arameter. From de nition (.) results that every function in the domain of H is then comosed of a regular art in H and a singular art given by G. Moreover it can be shown that the domain D(H ) doesn t deend on the choice of. In articular for <, we may choose the arameter such that: + =, giving rise to the following domain s 4 reresentation: D(H ) = L (R 3 ) = ' + q ; ' H (R 3 ); '() = ; q C (.4) which will be extensively used in this work. By rojecting the action of H on a sace of test functions we get : ((H + ) ; ') = (( + ) ; ') = then we have: = + + q G q G ; ' = = + + q G q q G ; ' = ( + q ; ') (H ; ') = ( q ; ') 8' sace of test functions (.5) Here (; ) denotes the usual L scalar roduct 33

35 This relation shows that, in a weak sense, we may consider H as the oerator associated to a delta shaed otential laced in x =. We will refer to H as a oint interaction Schrödinger oerator. In the following some basic roerties of oint interactions oerators are resumed.. Sectral Proerties Assume that the coule R and D(H ) satis es the eigenvalues equation for the oerator H. (H + ) = By de nition (.), we have: ( + ) = H (R 3 ) whose unique solution in H (R 3 ) is: =. Then, the eigenfunction related to should be roortional to the Green function G : = q G and the boundary condition in (.) imlies that: 8 < = q ( + 4 ) = () ) or : q = 4 (.6) Relation (.6) imlies that the discrete sectrum of H is emty if, othervise, if <, it consists of one eigenvalue = 16. The corrisondent normalized eigenfunction is: jje 4 jxj = jxj (.7) The risolvent oerator associated to H, may be exressed as follows: Z 1 H + z '(x) = Rz (x; y)'(y) dy (.8) R 3 where the integral kernel R z (x; y) - obtained by an alication of Krein s formula to H (e.g. [7])- is given exlicitely by the relation: R z (x; y) = G z (x y) + 4 z + 4 G z (x)g z (y) (.9) 34

36 where the Green function G z (x) -with the usual de nition- rovide the integral kernel for the lalacian resolvent. From (.9) easily follows that 1 +z the continuous sectrum of H is urely continuous and coincides with the interval equal to [; +1). The generalized eigenfunctions of H are de ned by the relation: ' k (x) = e ik x 1 ijkj 4 ijkj jxj e jxj with k R 3. Then all scattering states may be exressed in the integral form: 8 D(H ) : (; ) = ) 9 ~ L (R 3 ) ) (x) = 1 Z! i jkj jxj (k) ~ e ik x 1 e dk (.1) () 3 R 3 i jkj jxj 4.3 Quantum Dynamics of Point Interactions The dynamics of a quantum article subjected to the action of a oint interaction H is described by the evolution of a state function which satis es the equation: i d = H dt (.11) () = D(H ) The existence of the dynamics for Schrödinger oerator of tye H is a well established fact (e.g. []); this imlies that H is the generator of a strongly continuous unitary grou of oerators e i t H acting on the sace D(H ). The time evolution of system (.11) is determined by the relation: (t) = e i t H An exlicit exression of the time roagator may be achieved setting equation (.11) in the weak form: From (.5) and (.1) we get: (i@ t ; ') = (H ; ') 8 ' C 1 c (R 3 ) (.1) (i@ t ; ') = ( q ; ') 8 ' C 1 c (R 3 ) (.13) Let us denote with e it the unitary grou generated by the lalacian oerator. Its action is well de ned on the sace H (R 3 ) and may be reresented in the following integral form: e it '(x) = 1 Z e i jx xj (t) 3 4t '(x ) dx (.14) R 3 35

37 In what follows we will refer to the integral kernel of (.14) also as U(t; x). The same letter will be used also to indicate the oerator itselfs: e it ' = U t ' Setting: e it ~ =, and taking into account the relation: we get: i@ t e it ~ = e it ~ + e it i@ t ~ i@ t ~ ; ' = q e i t ; ' 8 ' Cc 1 (R 3 ) (.15) Observing that ~ and have the same initial conditions, a direct integration of (.15) gives the following solution: ~ (x) w = (x) + i q (s) e i s (x) ds (.16) where the time deendece of the arameter q is connected with the time variation of the regular art of the state via the boundary condition: lim = q ( + 4 ) (.17) x! From (.16) we obtain the equation: ( ; ') = e it + i q (s) e i (t s) ds; ' from which we get the weak solution of (.11) 3 : (t; x) = e it (x) + i = e it (x) + i = e it (x) + i e i(t s) q (s) (x) ds = Z 8 ' C 1 c (R 3 ) R n U(t s; x x ) q (s) (x ) ds = U(t s; x) q (s) ds (.18) which, due to the existence of the dynamics for the system (.11), is a strong solution as well. 3 Here the function U is the kernel of the free roagator in n dimensions: 1 U(t; x) = (4it) n e i jxj 4t 36

38 .4 The Charge Equation As shown in equation (.18), the dynamics of a quantum article, subjected the action of the Schrödinger oerator H, is determined by the time evolution of the boundary condition (.17). From the decomosition roerties of the domain D(H ) we see that relation (.17) may also be written in the form: lim x! lim x! q G = q ( + 4 ) (.19) Acting on (.19) with the Lalace transform oerator L and making use of (.18), we get the equation for the coe cient q : Le it + il U(t s; x) q (s) ds Lq G A = Lq + 4 Lq from which follows: L e it () + lim x! ilu(t; x) ~q () q ()G (x) = L [q ] + 4 ~q () (.) where ~q () de nes the Lalace trensform of q (t). Observing that the transform of the free roagator kernel U is: LU(t; x)() = i e i jxj 4 jxj the limit at rst member of (.) is exlicitely given by: lim x! ilu(t; x) ~q () ~q ()G (x) = ~q () lim x! i jxj e 4 jxj e! jxj = ~q r () 4 jxj 4 ( + i ) then, by substitution into (.) we get: 1 L e it ~q () () + 4 i = 1 L [q ] (.1) Alying the inverse transform and observing that L 1 = 1 t, an equation for the arameter q(t) is obtained: 4 i e is () ds q(t) = 4 i t s q(s) t s ds (.) 37

39 In what follows, the comlex scalar eld q(t) will be referred to as the charge associated to the oint interaction H and to the initial state. It is worthwile notice that dynamics of q does not on the articular choice of in the domain reresentation: The charge evolution in time is de ned by an Abel integral equation of II kind; the existence and uniqueness of solutions for this kind of roblem - deending from the regularity of the nonhomogeneous term - have been analyzed in Chater 1. As an aside we notice that by a simle iteration it is ossible to regularize the integral kernel of the charge equation: q(t) + 4 i ) q(t)+4 i q(s) ds = 4 i t s 1 44 i t s Z s e is () t s ds ) q(s ) s s ds + 4 i = 4 i Z s e is () t s e is () t s ds Alying Dirichlet formula to the double integral we obtain the equation: q(t)+16 i q(s) ds = 16 i Zt e is () ds+4 i 3 ds 5 ds = e is () t s ds (.3) which describes the motion of a forced armonic obscillator in the comlex lane..4.1 Solving charge equation The charge equation (.) may be considered as a articular case of the following roblem: q(t) + 4 i q(s) ds = 4 i t s Oarating a Lalace transform of (.4), we have: ~q() f(s) t s ds (.4)! i = 4 i ~ 4 i f() ) ~q() = f() ~ + 4 i 38

40 and taking into account the relation: L i = 1 t (4 i)e i(4)t erfc(4 it) K (t) (.5) an exlicit exression for the solution q(t) is obtained: q(t) = 4 i K (t s) f(s) ds (.6) From (.6) we get an exlicit exressions of the charge q in terms of the initial state function : q(t) = 4 i K (t s) U s ()ds (.7).4. The charge associated to the bound state If the article is in the bound state (.7) at t =, the nonhomogeneous term in (.3) may be exlicitely comuted. The Fourier integral of the function e i t evaluated in the oint x = is: e i t () = 1 Z F () 3 (k) e i k t dk (.8) R 3 where we denoted with F the Fourier transform of. As already mentioned, the bound state is roortional to the Green function (.3) with = 16, whose Fourier transform is given by: G 16 (k) = 1 () 3 1 k + 16 By substitution into (.8) and taking into account (.7) we get: e i t () = 4 jj () 3 Z 1 R k e i k t dk (.9) where C is a suitable constant. Passing to the Lalace Transform L w.r.t. time and using Fubini s theorem to change the order of sace and time integrations, equation (.9) reads as: L e i t () () = 4 jj () 3 39 Z 1 1 R k ik + dk

41 and, after some calculation, the following Lalace transform is deduced: L e i t () () = 4 jj () 3 i i 4 i (.3) Exression (.3) may be used to evaluate the nonhomogeneous term in equation (.) when = ; recalling that: L 1 r = t the Lalace transform of the source nonhomogeneous term is: L 44 3 e is () i ds5 () = 4 1 jj t s 1 4 i from which we have: 4 i (.31) e is () ds = 4 jj E 1 (4 i t 1 ) (.3) t s Then, the charge equation associated to the bound state reads as: q(t) + 4 i ( 1 ) whose solution is given by (1.37) with = 4 i : q(s) ds = q () E 1 (4 i t 1 ) (.33) t s q(t) = q () e i16 t (.34) We stress out that relation (.34) could also be recovered directely from the Schrodinger equation (.11). In fact, once assumed the initial condition =, the state function at time t is: e ith = e i t with = 16. Then from the oerator domain reresentation, (.4), it is immediate to conclude once more that the charge associated to the bound state is given by (.34). Another relation, which will be useful for further calculations, is obtained alying the general formula (.7) to the case we re taking into account: q(t) = 4 i from which, using (.34), we deduce: 4 i K (t K (t s) U s ()ds s) U s ()ds = e i 16 t 4 (.35)

42 .4.3 The charge associated to a scattering state From general roerties of generator set of L (R 3 ), we know that any function L (R 3 ) may be reresented as the sum of a radial art s (r) L (R + ; r ), r = jxj, lus a term given by a linear combination of sherical harmonics: (x) = which are both in L (R 3 ): +1X l=1 l6= lx m= l f lm (r)yl m (#; ') (x) = S (r) + P (x) (.36) This imlies also that the charge q associated to the hamiltonian H and to the initial state, can be slitted in two contributions: q = q S + q P which satis e the equations 4 : 8 >< >: q S (t) + 4 R t i q P (t) + 4 i R t q S (s) t s ds = 4 R t i q P (s) t s ds = 4 i R t U s S () t s ds U s P () t s ds (.37) Next we observe that, due to the ortogonality relation: Z d Yl m (#; ') Ym l (#; ') = ll mm the free roagator of P is always null when evaluated in the oint x =. From this follows that the source term in the second of equations (.37) is zero. By the uniqueness of solution of equation (.), we conclude that the charge associated to any state of kind is zero: q = (.38) As an aside we notice that from (.38) and from the de nition (3.4) follows that a article, initially laced in a state,, and subjected to the action 4 Here we follows the already introduced notation: for the free roagator at time t U t = e it 41