On the Weyl symbol of the resolvent of the harmonic oscillator

Transcription

1 On the Weyl symbol of the resolvent of the harmonic oscillator Jan Dereziński, Maciej Karczmarczyk Department of Mathematical Methods in Physics, Faculty of Physics University of Warsaw, Pasteura 5, -93, Warszawa, Poland January 5, 7 Abstract We compute the Weyl symbol of the resolvent of the harmonic oscillator and study its properties. Keywords: Harmonic oscillator, Weyl quantization. Introduction Throughout our paper, by the quantum harmonic oscillator, we mean the selfadjoint operator on L R d defined as H := + x, where is the Laplacian and x = d i= x i. The spectrum of H is {d, d +, d + 4,... }. The central object of our paper is the resolvent of H, that is, the operator H z defined for z outside of the spectrum of H. Another central concept of our paper is the Weyl symbol of an operator. Weyl symbol is a very natural parametrization of operators on L R d, extensively used in PDE s. It is plays also an important role in foundations of quantum mechanics, where it is usually called the Wigner function, and can be used to express the The financial support of the National Science Center, Poland, under the grant UMO- 4/5/B/ST/6, is gratefully acknowledged.

2 semiclassical limit. For an introduction to the Weyl quantization we refer the reader to Chap. 5 of [], Chap. XVIII of [7], [8], [] or Sec. 8.3 of [4]. It is a well-known fact that the Weyl symbol of a function of H is an another function or distribution of x + p. This fact was e.g. shown and discussed in [3], as well as in the recent paper []. Therefore, the Weyl symbol of H z can be written as F d,z x + p for some F d,z. Our paper is devoted to a study of the properties of F d,z. In particular, we give a few explicit expressions for F d,z we present an integral formula, a power series expansion and an expression in terms of confluent-type functions. We also provide some estimates on the derivatives of F d,z. Some of our formulas simplify in the case z =, that is, for the inverse of the harmonic oscillator. In particular, the Weyl symbol of the inverse can be expressed in terms of Bessel-type functions. We find it interesting and potentially useful that the Weyl symbol of the resolvent of the harmonic oscillator has an explicit description. With our formulas we are able to study its properties, deriving in particular rather precise bounds on its derivatives. In the literature we have not seen a study of the Weyl symbol of the resolvent of the harmonic oscillator except for a recent paper [], devoted to the inverse of the harmonic oscillator H. [] contains a formula for the Weyl symbol of H in terms of a power series. It also proves that its derivatives satisfy some estimates. The authors of [] call them Gelfand-Shilov bounds. The results of our paper are stronger than those of []. First, we consider the more general case of the resolvent H z, whereas [] is restricted to z =. Second, our explicit representation in terms of Bessel-type function and in terms of an integral representation is absent in []. Third, our bounds on the derivatives easily imply those proven in []. An interesting discussion of Weyl quantization of spherically symmetric symbols is contained in a recent paper of Unterberger []. That paper contains in particular a formula for the symbol of the nth spectral projection of the harmonic oscillator. We give an alternative derivation of Unterberger s formula, using our results about the resolvent as the starting point. Weyl quantization Let us recall the definition of the Weyl quantization, following e.g. [4], [] or []. If a is a distribution in S R d R d, then its Weyl quantization is defined to be the operator Opa from SR d to S R d such that x + y OpaΦx = π d a, p e ipx y Φy dpdy. The distribution a is then called the symbol of the operator Opa. Let us remark that in the literature one can find various classes of symbols. S R d R d, that we use in our definition, is broad enough for our purposes. As

3 noted in [], one can define the Weyl quantization on more general class of symbols: e.g. on the dual of the so-called Gelfand-Shilov space. The usual product of operators corresponds on the level of symbols to the so-called star product sometimes called the Moyal star. That means, if a bx, p := e i x p p x ax, p bx, p x:=x =x, 3 p:=p =p then OpaOpb = Opa b. 3 The symbol of the resolvent For z C outside of the spectrum of H, a d,z will denote the symbol of the harmonic oscillator, that is, Opa d,z = H z. 4 As discussed in the introduction, we can then define F d,z by F d,z x + p = a d,z x, p. 5 Some properties of F d,z can be derived in an easy way with use of the integral representation given by Theorem. For Re z < d, the following formula holds: F d,z = s d z + s d+z e s ds. 6 Proof. It is well known that e th = Op cosh t d e tgh t x +p. 7 see e.g. [9, 4]. Hence, H z = e th+tz dt 8 = Op cosh t d e tgh t x +p e tz dt 9 = Op s d e sx +p e z artgh s ds, where at the end we made the substitution tgh t = s. Using artgh s = the linearity of the quantization, we obtain the final result. 3 ln +s s and

4 Theorem. The function F d,z is entire analytic. It can be written as k F d,z = c k k!, where c k = where F a, b; c; z = k= Γk + Γ d z Γk + + d z F d + z, k + ; k + + d z ;, k= a k b k c k z k k! is the hypergeometric function. Proof. Entire analyticity of F d,z is obvious from 6. Of course, c k = k F d,z. Differentiating the integral representation 6 and putting = results in c k = s d z + s d+z s k ds. 3 Then we apply Euler s formula, see e.g. [5], Subsection 3.7.6, ΓbΓc b F a, b; c; z 4 Γc = s b s c b zs a ds, for Re c > Re b >. 5 Theorem 3. For, we have F d,z = + O. 6 More generally, there exist d :=, d, d 3,... such that for any n Proof. For < a <, we write F d,z = n + F d,z = n a a n j= d j + O. 7 j n s d z + s d+z n s e s ds 8 s d z + s d+z e s ds. 9 9 is Oe a. We integrate 8 n times by parts. The boundary terms at s = have the form d j and the boundary terms at s = a are O e a, j =,..., n. The j j remaining integral is O. n 4

5 4 Confluent-type functions and the harmonic oscillator The harmonic oscillator is closely related to the confluent equation. We devote this section to this relationship. First, recall that the confluent differential operator is defined as x x + c x x a, for a, c C. Among functions annihilated by, two are distinguished: The confluent, F or Kummer s function: Ma, c; x := F a; c; x = k= a k c k x k k!. It is the only solution behaving as in the vicinity of. Tricomi s function: Ua, c; x := x a F a; a c + ; ; x. Tricomi s function is the only solution having the asymptotic behaviour Ua, c; z z a at infinity. If a is a nonpositive integer, then both Kummer s and Tricomi s functions are proportional to Laguerre polynomials: L α nx = α + n M n; α + ; x n! 3 = n U n; α + ; x. n! 4 We will need Green s function of the confluent operator, that is, the integral kernel of its inverse Ra; b; x, y. It should satisfy x x + c x x a Ra; b; x, y = δx y, 5 Ra; b; x, y y y + c y y a = δx y. 6 It can be checked by a straightforward calculation, using that Γc Γa y c e y is the Wronskian of M and U, that Ra; b; x, y = Γa Γc yc e y { Ma, c; xua, c; y for x < y, Ma, c; yua, c; x for y < x. 7 5

6 We can transform the confluent operator as follows: 4 x x e = 4 x x + c x x a e x 8 x + c x x + c a x 4 9 = + c + c a, 3 where we substituted x =. Let Ra; c,, η denote the resolvent of 3. Then, using the fact that the kernel of an operator is a halfdensity in both variables, we obtain Ra; c;, η 3 = e Ra; c;, ηe η η { 3 = Γa Γc c η c e η Ma, c; Ua, c; η for < η, Ma, c; ηua, c; for η <. 33 As we will see in the theorem below, the symbol of the resolvent of harmonic oscillator can be expressed by the Green s function of the confluent operator: Theorem 4. The symbol F d,z has the representation d z d Γ F d,z = d z d! e M, d; d z + U, d; d z η d e η M, d; η dη d z η d e η U, d; η dη. 34 Proof. We would like to find the solution of l = H zopa d,z. formula 3 for the Moyal star, we obtain = x + p z F d,z x + p = x + p z 4 x + p F d,z x + p. Using the 35 Substitution of = x + p leads us to the equation d z + F d,z =. 36 6

7 This leads us to d z F d,z = R ; d;, η dη 37 η d z d Γ = d z d z d! e M, d; η d e η U, d; η dη 38 d z + U, d; d z η d e η M, d; η dη. 39 Recall that M is analytic and its asymptotic behaviour of M around zero is Ma, c; z. However, for integer c the function Ua; c; z is not analytic at z =. It can be written as c Ua, c; z = logzma, c; z + Da, c; z, 4 Γa c + Γc where Da, c; is a meromorphic function of z with a pole of order c. Thus, around the function U d z, d; x diverges polynomially and logarithmically, see 4. Hence, 34 suggests that F d,z may have a logarithmic singularity at. However, we already know that F d,z is analytic. Let us check that 34 implies the analyticy of F d,z. Let us insert the representation 4 of U into 34 and split the integration range: d d+ Γd Γ d+z Γ d z F d,z =e M d z, d; e η η d log ηm d z, d; η dη + e M d z, d; e η η d D d z, d; η dη + d ΓdΓ d + z e M d z, d; η d e η U d z, d; η dη + e log M d z, d; η d e η M d z, d; η dη + e D d z, d; η d e η M d z, d; η dη. 4 7

8 After integrating the first integral by parts, the logarithms cancel out and we get the following result: d d+ Γd Γ d+z Γ d z F d,z = e M d z, d; η e r r d M d z η, d; r dr dη + e M d z, d; e η η d D d z, d; η dη + d ΓdΓ d + z e M d z, d; η d e η U d z, d; η dη + e D d z, d; η d e η M d z, d; η dη. 4 Now it is easy to convince ourselves that F d,z is analytic in the region. Let us start with analysing the first summand. The inner integral behaves at least like η and so the outer integral does not give elements proportional to logarithm or worse. The second summand consists of analytic functions times an integral behaving at zero at least like a constant. The third one is an analytic function times a finite number the integral does not depend on. The fourth summand is an integral behaving around zero like d, multiplied by analytic functions and d+, hence it is analytic as well. 5 The Weyl symbol of spectral projections Recall that the eigenvalues of H are E := d, E := d +, E := d + 4,... Let P n denote the spectral projection of H onto E n. In this section we will compute the Weyl symbol of P n. We will see that it has a simple expression in terms of the Laguerre polynomial L d n. Thus we will reproduce a recent result of Unterberger [], using the formula 34 as the main tool. Theorem 5. The Weyl symbol of P n is p n x + p, where p n = d n e L d n. 43 Proof. The resolvent z H has a pole at E n and the corresponding residue is P n, that is P n = Resz H z=en. 44 8

9 Therefore, p n = ResF z,d. 45 z=en All terms in 34 are analytic in z around E n except for Γ d z, which has a st order pole. By the well-known properties of the Gamma function Therefore, ResΓ d z ResF d,z = d n z=en n!d! e M n, d; + U n, d; Then we use 3 and 4 which lead us to z=en = n. 46 n! η d e η M n, d; η dη η d e η U n, d; η dη. p n = d n! d + n! e L d n ν d e ν Ln d νdν. The calculation that is elementary after recalling that ν d e ν L d n d + n! n νdν = n! 47 L α nx = n! ex x α n x e x x n+α The Weyl symbol of the inverse The inverse to harmonic oscillator is just the resolvent computed at the point z =. Nevertheless, various properties of its symbol are easier than in the general case, which is the reason why we devote a separate section to the inverse. Theorem 6. Putting z = in 6 we see that the function F d, can be represented by F d, = t d e t dt. 49 9

10 The integral representation 49 easily implies the series representation of the symbol F d, : Theorem 7. F d, can be represented by the following series: F d, = d!! [ α d where α = π k= Proof. Write F d, = k!! k + d!!k! k for odd d and α = for even d. n= k= c n n. By 49 we see that n! c n = n F d, = n = = n k!! ] k + d!!k +! k+, 5 t d t n dt 5 s d s n+ ds = n Γ d n+ Γ Γ d+n+, 5 where we set t = s and applied the integral formula for Euler s Beta function. The relation Γx+Γy = x ΓxΓy leads to an easy recurrence, which gives for Γx+y+ x+y Γx+y k N c k+ = k!!d!! k + d!! c k = Γ d = d!! + d k!! k + d!!, 53 Γ d k!!d!! Γ Γ d k + d!! Γ +d = { π k!! k+d!! k!! k+d!! for d odd, for d even. 54 Note that the representation 5 can be found in [], proven by a different method. 7 Bessel-type functions and the harmonic oscillator The modified Bessel function I m and the MacDonald function K m, both solutions of the modified Bessel equation, can be easily represented by confluent functions M and U: I m z = zm e z Γ + m Mm +, m + ; z, K m z = π z m e z Um + 55, m + ; z.

11 Setting z = in 34 and using Γ d = π d, we can express the inverse of the Γd Γ d+ harmonic oscillator in terms of Bessel-type functions. Theorem 8. Set m = d. The function F d, can be represented as F d, = K m m η m I m η dη + I m η m K m η dη. 56 Note that the authors of [] ask the question whether it is possible to express the Weyl symbol of the inverse of the harmonic oscillator in terms of known special functions. 56 is our answer to this question. 8 Representation by elementary functions for even dimensions It is well-known that Bessel functions of odd parameters can be expressed in terms of elementary functions. For instance, I x = sinh x and K πx x = π x e x. Therefore, F, = e, 57 which has also been calculated in [] see equation.3 in there. An even easier derivation of this fact is to put d = in 49 and integrate. More generally, for even d the symbol of the inverse, F d,, can be represented by elementary functions. Theorem 9. For d = p, p =,,..., we get where p j t = j p p F d, = k k! k p k e, 58 k l= Proof. We compute: F d, = k= and then we use Lemma. t l l! is the partial Taylor expansion of e t. p p t p e t dt = k k k Lemma. For n =,,..., we have k= v k e v dv 59 v n e v dv = n n! p n e. 6

12 Proof. For n =, 6 is immediate. Then we use induction wrt n: v n e v dv = n e + n v n e v dv was proven by a different method in []. 9 Bounds on derivatives of the symbol The symbol of the resolvent is a very well behaved smooth function. Its derivatives satisfy bounds, which we describe in the theorem below. They follow easily from the integral representation 6. Theorem. Let Re z, n =,,,... and let α be a multi-index. Let s. Then the following bounds are true: n F d,z n! s n + s sn+, d ; 6 n F,z Cn! s n + s sn+. 63 where C is a constant independent of n. Proof. The case d is very easy. From 6 we get n F d,z t d t n e t dt t n e t dt = n! n. 64 Next we estimate n F d,z t d t n e t dt Then we interpolate between 64 and 65 to obtain 6. t n dt = n +. 65

13 The case d = is slightly more complicated. n F,z Next, consider < ɛ < : n F,z Setting ɛ := n+ we obtain t t n e t dt 66 t t n e t dt + t n e t dt + t t n e t dt 67 t e dt 68 = C n! n + C e Cn! n. 69 t t n dt 7 ɛ ɛ ɛ t n dt + t dt 7 ɛ C ɛn + + C ɛ. 7 n F,z C n Interpolating between 69 and 73 we obtain 63. Note that the main result of [] are the estimates.6,.7 and.8 for the symbol of the inverse of the harmonic oscillator a d, x, p. The authors call them bounds of Gelfand Shilov type. Our Thm easily implies the bounds.6,.7 and.8 of []. References [] F. A. Berezin, M. A. Shubin: Schrödinger Equation, Editions of Moscow University 983 Russian [] M. Cappiello, L. Rodino and J. Toft, On the inverse to harmonic oscillator, Communications in Partial Differential Equations 46 5,

14 [3] J. Dereziński, Some remarks on Weyl pseudodifferentail calculus, Journées Équations aux dérivées partielles 993, 4. [4] J. Dereziński and Ch. Gérard, Mathematics of Quantization and Quantum Fields, Cambridge Monographs in Mathematical Physics, Cambridge University Press, 3. [5] J. Dereziński, Hypergeometric type functions and their symmetries, Annales Hendri Poincaré 5 4, [6] G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Princeton University Press, 989. [7] L. Hörmander, The analysis of linear partial differential operators III, Grundlehren der mathematischen Wissenschaften 56, Springer, 985. [8] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer [9] A. Unterberger, Oscillateur harmonique et operatéurs pseudodifférrentiels, Annales de l Institut Fourier 3 979,. [] A. Unterberger, Which pseudodifferential operators with radial symbols are non-negative?, J. Pseudo-Differ. Oper. Appl. 6 7, p. 67 9, DOI.7/s [] M. Zworski, Semiclassical Analysis, AMS 4