Generating Function and a Rodrigues Formula for the Polynomials in d Dimensional Semiclassical Wave Packets

Transcription

1 Generating Function and a Rodrigues Formula for the Polynomials in d Dimensional Semiclassical Wave Packets George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia , U.S.A. E mail: hagedorn@math.vt.edu September 6, 05 Abstract We present a simple formula for the generating function for the polynomials in the d dimensional semiclassical wave packets. We then use this formula to prove the associated Rodrigues formula. Introduction The generating function for dimensional semiclassical wave packets is presented in formula.47) of []. In this paper, we present and prove the d dimensional analog. As an application of this formula, we also prove the associated multi dimensional) Rodrigues formula. These results have also been proven from a completely different point of view by Helge Dietert, Johannes Keller, and Stephanie Troppmann. See Lemma 3 and Section 3 particularly Proposition 6) and formula 3) of []. See also [3]. We have also received a conjecture Partially Supported by National Science Foundation Grant DMS 098.

2 from Tomoki Ohsawa [4] that the generating function result could be proved abstractly by using the formula for products of Hermite polynomials and the action of the metaplectic group. Acknowledgements It is a pleasure to thank Raoul Bourquin and Vasile Gradinaru for motivating this work. It is also a pleasure to thank Johannes Keller, Tomoki Ohsawa, Sam Robinson, and Leonardo Mihalcea for their enthusiasm and numerous comments. It is also a pleasure to thank the referee whose comments improved this paper significantly. The author was supported in part by National Science Foundation grant DMS 098. Semiclassical Wave Packets The semiclassical wave packets depend on two invertible d d complex matrices A and B that are always assumed to satisfy A B + B A I and A t B B t A 0. They also depend on a phase space point a, η) that plays no role in the present work. After chosing a branch of the square root, we define ϕ 0 A, B,, a, η, x) π d/4 d/4 det A) / x a), B A x a) + i ) η, x a). Here, and for the rest of this paper, we regard R d as being embedded in C d, and for any two vectors a C d and b C d, we use the notation a, b j For l d, we define the l th raising operator a j b j. R l A l A, B,, a, η) B e l, x a) i i η) ). Then recursively, for any multi-index k, we define ϕ k+el A, B,, a, η, x) kl + R lϕ k A, B,, a, η))x).

3 For fixed A, B,, a, η, these wave packets form an orthonormal basis indexed by k. It is easy to see that ϕ k A, B,, a, η, x) k / k!) / P k A,, x a)) ϕ 0 A, B,, a, η, x), where P k A,, x a)) is a polynomial of degree k in x a), although from this definition, it is not immediately obvious that P k A,, x a)) is independent of B. Since they play no interesting role in what we are doing here, we henceforth assume a 0 and η 0. 3 The Generating Function Our main result for the generating function is the following: Theorem 3. The generating function for the family of polynomials P k A,, x) is Gx, z) z, A A z + z, A x ). I.e., Gx, z) k P k A,, x) zk k!. Remark We make the unconventional definition A A A. By our conditions on the matrices A and B, this forces A to be real symmetric and strictly positive. We also have the polar decomposition A A U A, where U A is unitary. With this notation, we can write Gx, z) U A z, U A z + UA z, A x ). This equivalent formula is the one we shall actually prove. We begin the proof of Theorem 3. with a lemma that provides an alternative formula for R l. From this formula and an induction on k, one can easily prove that P k A,, x) is independent of B, because ϕ 0 A, B,, 0, 0, x) ϕ 0 A, B,, 0, 0, x) π d/ d/ det A 3 x, ) A x.

4 Lemma 3. For any ψ S, R l ψ)x) ϕ 0 A, B,, 0, 0, x) ϕ 0 A, B,, 0, 0, x) ψx)). Proof: The gradient on the right hand side of the equation in the lemma can act either on the ϕ 0 or on the ψ. So, we get two terms when we compute this: j ) e j e j, B A x + x, B A e j ψx) ) ψ)x). The second term here is precisely the second term i i ) ψx) ), in the ression for R l ψ)x). So, we need only show the first term here equals the first term, B e l, x ψx), in the ression for R l ψ)x). To do this, we begin by noting that the first term here equals j ) e j e j, B A x + x, B A e j ψx) j ) e j e j, B A x + B A e j, x ψx) j ) e j e j, B A x + B A e j, x ψx) j e j e j, B A x + e j, A ) ) B x ψx) B A + A ) B x ψx) Because of the relations satisfied by A and B, B A is real symmetric) + i real symmetric). So, its conjugate, B A has this same form. Thus, B A equals its transpose, which 4

5 is A ) B. So, the quantity of interest here equals A el, A ) B x ψx) el, A A ) B x ψx) e l, B x ψx) B e l, x ψx), which is what we had to show. Proof of Theorem 3.: p k in d variables with the following properties: By an induction on k, we prove there is a k th order polynomial P k A,, x) p k A x/ ). Gx, z) p k A x/ U A z) Gx, z). z The result then follows by setting z 0. We never compute the polynomial p k because it may be complicated. We also use the notation p k A,, x) p k A x/ ). +el We define p 0. Below, we inductively compute Gx, z). For our second z condition above to hold, p k+el must be defined via the sum of formulas 3.) and 3.). This uniquely defines the polynomial p k+el. l d: For k 0, the result is trivial since P 0 A,, x). For the induction step, it is sufficient to do the following for an arbitrary positive integer Assuming we have already proved these for some k, we prove them for the multi index k +e l. To do this, we begin by noting that ϕ k A, B,, 0, 0, x) k! R k ϕ 0 A, B,, 0, 0))x). 5

6 Also, So, ϕ k A, B,, 0, 0, x) k / k!) / P k A,, x) ϕ 0 A, B,, 0, 0, x). R k ϕ 0 A, B,, 0, 0))x) k / P k A,, x) ϕ 0 A, B,, 0, 0, x). Thus, when we apply the l th raising operator, the polynomial P k A,, x) gets changed to P k+el A,, x). Assuming the induction hypothesis, when we differentiate k G z with respect to z l, the z k l derivative can act on the Gx, z) or it can act on the p k A x/ U A z). When it acts on the Gx, z), we obtain U A x/ ) U A z p k A x/ U A z) Gx, z). 3.) From the induction hypotheses, this is a polynomial of degree k + evaluated at the argument A x/ U A z. Note that this result depends on the following calculation, with Gx, z) written with the polar decomposition of A: G z l x, z) U U A z U A z, U A e l + ) U A x U A x/ ) U A z Gx, z). Gx, z) When the k. z l acts on the polynomial, we either get zero or a polynomial of degree p k ) A x/ U A z), U A e l U p k ) A x/ U A z) Gx, z) Gx, z). 3.) Recall that R l ψ)x) ϕ 0 A, B,, 0, 0, x) 6 ) ϕ 0 A, B,, 0, 0, x) ψx),

7 and that from our induction hypothesis, ϕ 0 A, B,, 0, 0, x) ϕ k A, B,, 0, 0, x) π d/ d/ k / k!) / det A p k A,, x) { x, } A x. When computing R l ϕ k by this formula, the gradient in R l can act on the onential or the p k A,, x). When it acts on the onential, we get k / k!) / p k A,, x) k +)/ k l + k + e l )!) / A el, A x ϕ 0 A, B,, 0, 0, x) U A x/ p k A,, x) ϕ 0 A, B,, 0, 0, x). 3.3) When the gradient in R l acts on the p k A,, x), in R l ϕ k, we get the term k / k!) / x p k A,, x)) ϕ 0 A, B,, 0, 0, x) k +)/ k!) / e j, p k )A,, x) A e j ϕ 0 A, B,, 0, 0, x) j k +)/ k!) / A p k )A,, x) ϕ 0 A, B,, 0, 0, x) k +)/ k l + k + e l )!) / U p k )A,, x) ϕ 0 A, B,, 0, 0, x). 3.4) From 3.) and 3.) with z 0, we obtain U A x/ p k A,, x) U p k ) A x/ ). From 3.3) and 3.4) and taking into account the factor of R l ϕ k ) k l + ϕ k+el, we obtain kl + in P k+el A,, x) U A x/ p k A,, x) U p k ) A x/ ). 7

8 The quantities of interest contain the same polynomial evaluated at the appropriate arguments, and P k+el A,, x) p k+el A,, x). Since l is arbitrary, with l d, the result is true for all multi-indices with order k +, and the induction can proceed. 4 The Rodrigues Formula As an application of the Generating Function formula, we prove a multi dimensional Rodrigues formula for these polynomials. The result is Theorem 4. In d dimensions, with the convention that A is the positive square root of A A, ) A x P k A,, x) ) ) k A x A x. Remark By scaling, it is sufficient to prove this for. We begin the proof of this result with a lemma that embodies a special case of the chain rule for high order derivatives in d dimensions. Lemma 4. Assume F : R d R d is C m and that M : R d R d is linear. Then viewing gradients as column vectors and using multi index notation, x ) m F M x) M t y ) m F )ym x One proves this by induction on m, proving that each component is correct. Proof of Theorem 4. Using this technical result and the generating function, P k A,, x) z U A z) t U A z) + U A z) t A x) ) z0 z U A z A x) t U A z A x) ) A x) t A x) ) z0 A x) t A x) ) z U A z A x) t U A z A x) ) z0 8

9 A x) t A x) ) U A ) t w w A x) t w A x) ) w0 A x) t A x) ) U A ) t w w u) t w u) ) w 0 u A x A x) t A x) ) U A ) t u w u) t w u) ) w 0 u A x A x) t A x) ) U A ) t A t x w A x) t w A x) ) w0 A x) t A x) ) U A ) t A t x A x) t A x) ). However, A t A is real, and with our unusual convention, A A U A. So, U A ) t A A. This proves the theorem. References [] Dietert, H., Keller, J., and Troppmann, S.: An Invariant Class of Hermite Type Multivariate Polynomials for the Wigner Transform. 05 preprint, arxiv: ). [] Hagedorn, G.A.: Raising and Lowering Operators for Semiclassical Wave Packets. Ann. Phys. 69, ). [3] Keller, J.F.: Quantum Dynamics on Potential Energy Surfaces. Doctoral Thesis, Technische Universität München 05). [4] Ohsawa, T.: private communication 05). 9