On multi-variable Zassenhaus formula
aa r X i v : . [ m a t h . QA ] M a r ON MULTI-VARIABLE ZASSENHAUS FORMULA
LINSONG WANG, YUN GAO, AND NAIHUAN JING † Abstract.
In this paper, we give a recursive algorithm to compute the multi-variable Zassenhaus formula e X + X + ··· + X n = e X e X · · · e X n ∞ Y k =2 e W k and derive an effective recursion formula of W k . Introduction
The celebrated Baker-Campbell-Hausdorff (BCH) is a fundamental identity in Lietheory [1, 2, 3] connecting Lie algebra with Lie group. The BCH says that for anylinear operators
X, Y in a bounded Hilbert space one has the formula e X e Y = e X + Y + P ∞ k =2 Z k ( X,Y ) , (1.1)where exp is defined in the usual sense and Z k ( X, Y ) is a degree k homogeneous Liepolynomial in the noncommutative variables X and Y . The first few terms are Z = 12 [ X, Y ] , Z = 112 ([ X, [ X, Y ]] − [ Y, [ X, Y ]]) ,Z = 124 [ X, [ Y, [ Y, X ]]] . and the general expressions of Z k ( X, Y ) can be explicitly computed by combinatorialformulas.The dual form of the BCH is the famous Zassenhaus formula which establishesthat the exponential e X + Y can be uniquely decomposed as e X + Y = e X e Y ∞ Y m =2 e W m ( X,Y ) (1.2)= e X e Y e W ( X,Y ) e W ( X,Y ) · · · e W k ( X,Y ) · · · , where W k ( X, Y ) is a homogeneous Lie polynomial in X and Y of degree k [4]. Thefirst few terms are W = −
12 [
X, Y ] , W = 13 [ Y, [ X, Y ]] + 16 [ X, [ X, Y ]] ,W = −
124 [ X, [ X, [ X, Y ]]] −
18 ([ Y, [ X, [ X, Y ]]] + [ Y, [ Y, [ X, Y ]]]) . † Corresponding author: [email protected] (2010): Primary: 16W25; Secondary: 22E05, 16S20Keywords: Baker-Campbell-Hausdorff formula, Zassenhaus formula . here are several methods to compute W k [5, 6, 7, 8]. In particular, a recursivealgorithm has been proposed in [9] to express directly W k with the minimum numberof independent commutators required at each degree k .Similar to the BCH formula, the Zassenhaus formula is useful in many differ-ent fields: q -analysis in quantum groups [10], quantum nonlinear optics [13], theSchr¨odinger equation in the semiclassical regime [12], and splitting methods in nu-merical analysis [11], etc.We now consider the multivariate BCH and Zassenhaus formulas. It is easy toobtain the multivariable BCH formula by repeatedly using the usual BCH: e X e X · · · e X n = e X + X + ··· + X n + P ∞ m =2 Z m ( X ,X , ··· ,X n ) , (1.3)where Z m is a Lie polynomial in the X i of degree m . On the hand, we also have themultivariable Zassenhaus formula e X + X + ··· + X n = e X e X · · · e X n ∞ Y k =2 e W k (1.4)where the product is ordered and W k is a homogeneous Lie polynomial in the X i ofdegree k . However, it is more complicated to express W k in terms of X ′ i s .The existence of the formula (1.4) is a consequence of Eq. (1.3). In fact, it isclear that e − X e X + X + ··· + X n = e X + X + ··· + X n + D , where D involves Lie polynomialsof degree >
1. Then e − X e X + X + ··· + X n + D = e X + X + ··· + X n + W ′ + D , where D is aninfinite Lie power series in the X i with minimum degree >
2. Note that W ′ = W ,we need to repeat the process ( n −
1) times to determine W , i.e. e − X n − e X n − + X n + W ( n − + W ( n − + ··· + W ′ n − + D n − = e X n + W + W ( n − + ··· + W ′′ n − + W ′ n − + D n − where D n − involves Lie polynomials of degree > ( n − W k in (1.4). Ourmethod is inspired by the recent algorithm in [9], and our formula is based on a newformula for f ,k using compositions of integers.The paper is organized as follows. In Section 2, we give our recursive algorithmand a concrete procedure to compute W k , k = 1 , , , ,
5. In Section 3. we establisha combinatorial formula of f ,k (see Theorem 3.1). We will show that our formulacan give a slightly better recursion formula of W k when k > f ,k are used to derive Lie polynomialformulas of W k in terms of the operators X , . . . , X n . The latter set of formulas areexpected be useful in the quantum control problem.2. Multivariable Zassenhaus terms
A recurrence.
For the operators X , . . . , X n we consider the following func-tion of t : e t ( X + X + ··· + X n ) = e tX e tX · · · e tX n e t W e t W · · · . (2.1) here the W k can be determined by differential equations step by step, and it is easyto see that W k is a polynomial of degree k in the X i . Note that the multivariableZassenhaus formula (1.4) is the case when t = 1.First we consider the iterated system of equations R ( t ) = e − tX n · · · e − tX e − tX e t ( X + X + ··· + X n ) , (2.2) R m ( t ) = e − t m W m R m − ( t ) , m ≥ . (2.3)It follows from (2.3) that R m ( t ) = e t m +1 W m +1 e t m +2 W m +2 · · · , m ≥ . (2.4)We then take the logarithmic differentiation F m ( t ) = R ′ m ( t ) R m ( t ) − m ≥ . (2.5)For m = 1, we have that F ( t ) = − X n − e − tad Xn X n − − e − tad Xn e − tad Xn − X n − − · · · − e − tad Xn · · · e − tad X X + e − tad Xn e − tad Xn − · · · e − tad X ( X + X + · · · + X n )= e − tad Xn ( e − tad Xn − · · · e − tad X e − tad X − I ) X n + e − tad Xn e − tad Xn − ( e − tad Xn − · · · e − tad X e − tad X − I ) X n − + e − tad Xn e − tad Xn − e − tad Xn − ( e − tad Xn − · · · e − tad X e − tad X − I ) X n − + · · · + e − tad Xn e − tad Xn − · · · e − tad X ( e − tad X e − tad X − I ) X + e − tad Xn e − tad Xn − · · · e − tad X ( e − tad X − I ) X = ∞ X k =1 ( − t ) k n X i =2 X j + ··· + j i − ≥ j + ··· + j n = k ad j n X n · · · ad j X ad j X j ! j ! · · · j n ! X i , where ad A B = [ A, B ] and we have used the well-known formula e A Be − A = e ad A B = X n ≥ n ! ad nA B, as well as the fact that e ad X X = X . Write F ( t ) = ∞ X k =1 f ,k t k , (2.6)then f ,k = ( − k n X i =2 X j + ··· + j i − ≥ j + ··· + j n = k ad j n X n · · · ad j X ad j X j ! j ! · · · j n ! X i . (2.7) similar expansion can be obtained for F m ( t ), m ≥
2, by using R m ( t ) in (2.3).More specifically, F m ( t ) = − mW m t m − + e − t m W m R ′ m − ( t ) R − m − ( t ) e t m W m = − mW m t m − + e − t m ad Wm F m − ( t )= e − t m ad Wm ( F m − ( t ) − mW m t m − ) . Writing F m ( t ) = P ∞ k = m f m,k t k , we immediately get that f m,k = [ km ] − X j = o ( − j j ! ad jW m f m − ,k − mj , k ≥ m (2.8)where [ km ] denotes the integer part of km .On the other hand, if we take the logarithmic derivative of R m ( t ) using the ex-pression (2.4), we arrive at F m ( t ) = ( m + 1) W m +1 t m + ∞ X j = m +2 jt j − e t m +1 ad Wm +1 · · · e t j − ad Wj − W j . (2.9)Comparing the coefficients of the terms t , t , t and t in (2.6) and (2.9) for F ( t ),we get that f , = 2 W , f , = 3 W , f , = 4 W , f , = 5 W + 3[ W , W ] , so that W = 12 f , , W = 13 f , , W = 14 f , , W = 15 f , −
110 [ f , , f , ] . (2.10)Similarly, comparing (2.8) and (2.9), we get f m,m = ( m + 1) W m +1 , therefore W m +1 = 1 m + 1 f m,m = 1 m + 1 f [ m ] ,m , m ≥ , i.e. W m = 1 m f [ m − ] ,m − , m ≥ Examples of W k . When k = 1 in the expression (2.7), the summation of thefirst i − f , = − n X i =2 X j + ··· + j i − =1 ad j i − X i − · · · ad j X ad j X j ! j ! · · · j i − ! X i = X ≤ i
12 [[ X i X j ] X j ]= X ≤ i To reveal the explicit rule for f ,k ( k ≥ 1) based on the computations we gave inSection 2, we recall the definition of partitions and compositions [14]. .1. Formulation in terms of partitions. A partition of a positive integer m isan integral unordered decomposition m = λ + · · · + λ l such that λ ≥ · · · ≥ λ l > λ = ( λ λ . . . λ l ) and λ ⊢ m . Here λ i are called the parts and l is thelength of the partition. A composition is an ordered integral decomposition of m : m = λ + · · · + λ l such that λ i > λ (cid:15) m , in another words,compositions of m are obtained by permuting the unequal parts of the associatedpartition of m . The set of partitions of m is denoted by P ( m ) and the cardinalityis denoted by p ( m ).For example, the partitions of 4 are:( λ ) = (4) , ( λ ) = (3 , , ( λ ) = (2 , , ( λ ) = (2 , , , ( λ ) = (1 , , , . Therefore, p (4) = 5. The associated compositions are distinct permutations of thepartitions: (4) , (3 , , (1 , , (2 , , (2 , , , (1 , , , (1 , , , (1 , , , f ,k go as follows. For f , , p (1) = 1,( λ ) = (1) : X ≤ i 1) : X ≤ i 1) : 12! X ≤ i 1) : X ≤ i 1) : 13! X ≤ i 2) : 12!2! X ≤ i 1) : 12! X ≤ i 1) : X ≤ i For each k , the following formula holds f ,k = X ( k ··· k l ) | = k k ! k ! · · · k l ! X ≤ i Theorem 3.2. For each k ≥ the exponents W m in the multi-variable Zassenhausformula (1.4) for m = 6 k + i , where i = 0 , , , , , are given by W k = 16 k ( f k − , k − − ad W k − f k − , k + 12! ad W k − f k − , k +1 − ad W k f k − , k − + ad W k ad W k − f k − , k − ad W k +1 f k − , k − + ad W k +1 ad W k − f k − , k − − ad W k +2 f k − , k − − ad W k +3 f k − , k − − · · · − ad W k − f k − , k ) . (3.7) W k +1 = 16 k + 1 ( f k − , k − ad W k f k − , k + 12! ad W k f k − , k − ad W k +1 f k − , k − − ad W k +2 f k − , k − − ad W k +3 f k − , k − − · · · − ad W k f k − , k ) . (3.8) W k +2 = 16 k + 2 ( f k − , k +1 − ad W k f k − , k +1 + 12! ad W k f k − , k +1 − ad W k +1 f k − , k + ad W k +1 ad W k f k − , k − ad W k +2 f k − , k − − ad W k +3 f k − , k − − ad W k +4 f k − , k − − · · · − ad W k f k − , k +1 ) . (3.9) W k +3 = 16 k + 3 ( f k − , k +2 − ad W k f k − , k +2 + 12! ad W k f k − , k +2 − ad W k +1 f k − , k +1 + ad W k +1 ad W k f k − , k +1 − ad W k +2 f k − , k + ad W k +2 ad W k f k − , k − ad W k +3 f k − , k − − ad W k +4 f k − , k − − · · · − ad W k +1 f k − , k +1 ) . (3.10) k +4 = 16 k + 4 ( f k, k +3 − ad W k +1 f k, k +2 + 12! ad W k +1 f k, k +1 − ad W k +2 f k, k +1 − ad W k +3 f k, k − ad W k +4 f k, k − − · · · − ad W k +1 f k, k +2 ) . (3.11) W k +5 = 16 k + 5 ( f k, k +4 − ad W k +1 f k, k +3 + 12! ad W k +1 f k, k +2 − ad W k +2 f k, k +2 + ad W k +2 ad W k +1 f k, k +1 − ad W k +3 f k, k +1 − ad W k +4 f k, k − ad W k +5 f k, k − − · · · − ad W k +2 f k, k +2 ) . (3.12) Proof. As we know that W m = m f [ m − ] ,m − , m ≥ m intoeven and odd integers.When m = 2 a + 1 , a ≥ f [ m − ] ,m − = f a,m − ([ m − a ] = 2)= f a − ,m − − ad W a f a − ,a , if a = 2, we stop the computation since we reach f ,k . Otherwise,[ m − a − a − ( , a = 3;2 , a ≥ . [ aa − a − , a ≥ . so that if a = 3, f [ m − ] ,m − = f a − ,m − − ad W a − f a − ,a +1 + 12! ad W a − f a − , − ad W a f a − ,a , we stop the computation. If a ≥ f [ m − ] ,m − = f a − ,m − − ad W a − f a − ,a +1 − ad W a f a − ,a . Repeating the procedure, we obtain (3.3) and (3.5) as well as (3.8), (3.10), (3.12) inthe theorem by using induction.Similarly, when m = 2 a, a ≥ f [ m − ] ,m − = f a − ,m − ([ m − a − a − , a ≥ f a − ,m − − ad W a − f a − ,a , if a = 3, we stop the computation. Otherwise,[ m − a − a − , a = 4;3 , a = 5;2 , a ≥ . [ aa − a − ( , a = 4;1 , a ≥ . o that if a = 4, f [ m − ] ,m − = f a − ,m − − ad W a − f a − ,a +1 + 12! ad W a − f a − , − ad W a − f a − ,a + ad W a − ad W a − f a − , , we stop the computation. If a = 5, f [ m − ] ,m − = f a − ,m − − ad W a − f a − ,a +1 + 12! ad W a − f a − , − ad W a − f a − ,a . If a ≥ f [ m − ] ,m − = f a − ,m − − ad W a − f a − ,a +1 − ad W a − f a − ,a . Repeating the procedure, we obtain (3.2), (3.4) and (3.6) as well as (3.7), (3.9),(3.11) in Theorem 3.2 using induction. (cid:3) According to Theorem 3.2, we know that W m ( m ≥ 5) can be expressed as a linearcombination of f ,k ( k ≥ 1) in the end, then we use f ,k ( k ≥ 1) given in Theorem 3.1to obtain W m ( m ≥ W , W according to (2.10) and (3.2): W = 15 ( f , − ad W f , )= 1120 X ≤ i N. Jing’s work was partially supported by the National Natural Science Founda-tion of China (Grant No.11531004) and Simons Foundation (Grant No. 523868). References [1] Baker H F. Alternants and continuous groups. 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New York: Addison-Wesley Publishing, 1976 School of Mathematics, South China University of Technology, Guangzhou,Guangdong 510640, China E-mail address : [email protected] Department of Mathematics and Statistics, York University, Toronto E-mail address : [email protected] School of Mathematics, South China University of Technology, Guangzhou,Guangdong 510640, ChinaDepartment of Mathematics, North Carolina State University, Raleigh, NC27695, USA E-mail address : [email protected]@math.ncsu.edu