On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices

Transcription

1 On the approximation of real poers of sparse, infinite, bounded and Hermitian matrices Roman Werpachoski Center for Theoretical Physics, Al. Lotnikó 32/ Warszaa, Poland Abstract We describe a method to approximate the matrix elements of a real poer α of a positive (for α R) or non-negative (for α 0), infinite, bounded, sparse and Hermitian matrix W. The approximation uses only a finite part of the matrix W. Key ords: linear operator, infinite matrix, approximation 1 Introduction The motivation for this paper has been the research, carried out by Jerzy Kijoski [1 3] and, for the last year, the author, on the quantum field theory on a lattice. This physical theory arises from the discretization of the classical field Lagrangian and its subsequent quantization. As the number of degrees of freedom is infinite, Stone von Neumann s theorem [4,5], hich proves the uniqueness (up to a unitary transformation) of the representation of the algebra of field operators, does not apply. The problem of finding such a representation has been customarily solved by resorting to global symmetries, such as the translational invariance. Our research has been concerned, hoever, ith the quantum field theory on the background of non-homogeneous gravitational field. This eliminates global symmetries and renders traditional ays of finding the physical representation obsolete. An attempt as made to find a representation of the field operators on a Hilbert space via a limit of the representations on finite lattices, hich are, as proven by von Neumann, uniquely defined up to a unitary transformation. address: roman@cft.edu.pl (Roman Werpachoski). Preprint submitted to Elsevier 30 April 2007

2 When applied to a quadratic Hamiltonian on a d-dimensional rectangular lattice: 1 ˆπ n 2 + W mn ˆφm ˆφn, 2 n Z d m Z d n Z d (here n is a vector of integer indices numbering the lattice sites, ˆπ n is the (real) momentum operator for the site n, ˆφn is the field operator, also for the site n, and W an infinite, positive, real and symmetric matrix), it turns out that the existence of the limit of representations is connected ith the existence of the folloing limit: for each finite lattice T Z d, e define a finite matrix W T, hich is a finite part of W corresponding to the lattice T. In order to make the problem ell-posed physically, e need to impose some boundary conditions on the elements of W T corresponding to the edges of T. To ensure the existence of the groundstate, e choose non-negative boundary conditions. It turns out that the existence of the limit of finite-dimensional representations is guaranteed by the convergence of the sequence of matrix elements (W ±1/2 T ) mn ith increasing lattices T. Thus, the problem is a subset of a more general problem: for a real poer of an infinite, positive and bounded Hermitian matrix W, e try to approximate its matrix elements ith the corresponding elements of its finite parts, ith possible boundary conditions added. This is the topic of this paper. 2 Main result An infinite matrix W is also a linear operator on the Hilbert space l 2 of square-summable infinite sequences. The matrix elements W mn are equal to the scalar product (W e n e m ), here the e n are the canonical basis vectors of l 2. Conversely, any bounded linear operator on l 2 corresponds to a certain infinite matrix. The infinite matrix W α is, like W, a positive, bounded, linear and Hermitian operator on l 2 hich, hen acting on any eigenvector v of W ith an eigenvalue λ, fulfills the equation W α v = λ α v. For a general theory of fractional poers of operators, see [6]. The core result of this paper is Theorem 1 Consider an infinite Hermitian matrix W ith the folloing properties: (1) W <, (2) each ro of W has at most k non-zero elements, (3) inf σ(w ) 0, here σ(w ) is the spectrum of W as an operator on l 2. 2

3 For α R (for inf σ(w ) > 0) or α 0 (for inf σ(w ) = 0), e have for a fixed pair of indices m, n Z lim (W P α Q) mn = Wmn α (pointise). (1) P,Q here W P Q is a (P + Q + 1) (P + Q + 1) matrix ith indices in the range [ P, Q] hich is defined as follos: (W P Q ) mn = W mn, P < m < Q P < n < Q (2) and (W P Q ) mn = W mn + D mn, m, n { P, Q}. (3) The Hermitian matrix D fulfills the folloing conditions: for any P, Q > 0, e have W P Q inf σ(w ) (4) (i.e. the loest eigenvalue of W P Q is greater or equal to inf σ(w )) and for d 0. W P Q W + d, (5) 3 Physical example Consider an infinite matrix M hich generates the folloing quadratic form for x l 2 : x Mx = a n= x n 2 + b n= x n x n 1 2, a > 0 b > 0. It arises from the homogeneous theory of scalar field on a one-dimensional lattice. The finite matrix M P Q is chosen in such a ay that it generates the folloing quadratic form for x C P +Q+1 : x M P Q x = a Q x= P k=0 x n 2 + b Q n=1 P x n x n b x Q x P 2, an example of the so-called periodic boundary conditions in physics. It turns out that for P m, n Q and α R e have [ ( )] α P +Q (MP α a + b sin 2 πk ( ) P +Q+1 2πk(m n) Q) mn = cos. P + Q + 1 P + Q + 1 On the other hand, M α mn = 1 0 (a + b sin 2 πk) α cos[2πk(m n)]dk. 3

4 It is clear that lim(mp α P,Q Q) mn = Mmn α. 4 Proof of Theorem 1 For α = 0, the theorem is trivial. Since every ro of W has at most k non-zero elements, any matrix element of any positive and integer poer of W can be calculated as a finite sum of finite products of the matrix elements of W : here W 2 mn = W 3 mn = m 1 r(m) W 4 mn =..., W mm1 W m1 n m 1 r(m) m 2 r(m 1 ) W mm1 W m1 m 2 W m2 n r(m) := {n W mn 0}. Because of (2) and (6), for any indices m, n and for every non-negative and integer poer j one can choose such finite P j and Q j that This proves (1) for α N. P Pj,Q Q j W j mn = (W j P Q) mn. Because of (2), (3), (4) and (5), e have the folloing bounds for the spectrum of W P Q as an operator on C P Q, the space of all complex finite sequences indexed from P to Q: σ P Q (W P Q ) [inf σ(w ), W + d]. We make the distinction beteen σ P Q (spectrum of an operator on C P Q ) and σ (spectrum of an operator on l 2 ), because W P Q, as an operator on l 2, has infinitely many eigenvectors ith the zero eigenvalue: W P Q e n = 0, n < P n > Q, (6) hich means that σ(w P Q ) = σ P Q (W P Q ) {0}. The loer and upper bound ill be denoted by c and, respectively: c := inf σ(w ), c 0 := W + d, c <. 4

5 Since W, W α can be expressed [7] as the folloing poer series: W α = α ( ) (W α I j ) j, (7) here I is the infinite identity matrix, I mn = δ m,n. We kno that W I c. Hence, W I / 1 and the series (7) is eakly convergent (in the case of W I / = 1 the series is eakly convergent for α 0 only). Weak convergence of (7) implies the convergence of the folloing series: (W α ) mn = α ( ) α (W I) j mn, (8) here (W α ) mn is defined as a matrix element of the linear operator W α beteen to canonical basis vector of l 2, e m and e n. Since every ro of W has at most k non-zero elements, any matrix element of any positive and integer poer of W I can be calculated as a finite sum of finite products of the matrix elements of W I: (W I) 2 mn = m 1 r (m) (W I) 3 mn = (W I) 4 mn =..., here (W I) mm1 (W I) m1 n m 1 r (m) m 2 r (m 1 ) (W I) mm1 (W I) m1 m 2 (W I) m2 n r (m) := {n (W I) mn 0}. Because of (2) and (9), e can define a non-negative integer j P Q such that j < j P Q (W I) j mn = (W P Q I P Q ) j mn, (10) here I P Q is the (P + Q + 1) (P + Q + 1) identity matrix, and (9) lim j P Q =. (11) P,Q (W α P Q) mn, for W P Q being an operator on C P Q, can be expressed as a series similar to (8): (W α P Q) mn = α ( ) α (WP Q I P Q ) j mn. 5

6 We split this series ith the help of (10): j P Q 1 (WP α Q) mn = α Using (8) and (12), e get ( ) α (W I) j mn + α ( ) α (WP Q I P Q ) j mn. (12) (WP α Q) mn (W α ) mn = ( ) = α α (WP Q I P Q ) j mn ( ) α (W I) j mn. The difference can be estimated as follos: (WP α Q) mn (W α ) mn = ( ) α α (WP Q I P Q ) j mn α ( ) α j (W P Q I P Q ) j mn j + ( ) α (W I) j mn ( ) α (W I) j mn. (13) j j It is a ell-knon fact that, for every Hermitian operator A on a Hilbert space H hich has a countable orthonormal basis {e n }, every matrix element of A fulfills the inequality A mn A. Indeed, using the Cauchy-Scharz inequality e obtain A mn = (Ae n e m ) e m Ae n A e m e n = A. Since l 2 has a countable orthonormal basis and W I is Hermitian, e have Ditto (W I) j mn ( c) j. (WP Q I P Q) j mn ( c) j, since W P Q I P Q is a Hermitian matrix. We insert these bounds into (13) and obtain ( ) (WP α Q) mn (W α ) mn < 2 α α ( ) c j j. (14) The series ( ) α ( ) c j (15) j 6

7 (i.e. the same as above ith j P Q set to 0) converges (as ith (7), for c = 0 it converges only for α 0). For negative α and c > 0, ( ) α ( ) c j ( ) (c ) α j j = = (c/) α, α < 0. (16) j For positive and non-integer α, e have ([α] denotes the integer part of α): = ( ) α ( ) c j j = [α]+1 ) ( c) j + ( 1) [α] (c ) j ( α j j +( 1) [α]+1 (c/) α, α > 0 α / Z. (17) Convergence of (15), proven in (16) and (17), shos that the sum of the elements of the series (13) from j = j P Q to goes to zero for j P Q : lim j P Q 2α ( ) α j ( ) c j = 0. (18) Using (11), (14) and (18), e calculate: lim (WP α Q) mn (W α ) mn < P,Q hich proves (1). ( ) α lim P,Q 2α j ( ) = lim j P Q 2α α j ) j ( c ( ) c j = 0, 5 Accuracy and applicability of the approximation Inequality (14) and results (16) and (17) can be used to estimate the accuracy of our approximation. For a given unknon matrix element W α mn, e are interested in ho fast the right side of (14) diminishes ith increasing P and Q. This involves to problems: (1) the relation beteen P, Q and j P Q, hich is determined by the structure of W near the chosen matrix element, and (2) the speed of convergence of the series (15), hich appears in (14). Since the series (15) has only positive elements, its convergence is determined by ho fast they vanish ith increasing j. This in turn is determined by the 7

8 poer factor (1 c/) j (for c > 0) and the binomial coefficient ( ) α j. The smaller is 1 c/, the faster ill (15) converge. The dependence of ( ) α j on j is such that (15) ill converge more sloly hen α 1, than in the opposite case. The relation beteen P, Q and j P Q (i.e. ho fast j P Q diverges hen P, Q ) cannot be determined ithout additional assumptions about W. It is obvious that j P Q = 1 (the minimal value it can take) for P > min(m, n) or Q < max(m, n), here m, n are indices of the matrix element that e ant to compute. Suppose, for example, that W is k-diagonal for k = 2l + 1, so that W mn = 0 for m n > l and is non-zero otherise. We then have j P Q = 1 + [min(min(m, n) + P 1, Q max(m, n) 1)/l] for P min(m, n) 1 and Q max(m, n) + 1. Asymptotically, j P Q gros linearly ith min(p, Q). Since W is sparse, positive integer poers may be easily computed ithout resorting to any approximations, using (6). Our approximation can be used to approximate locally the solutions to infinite matrix equations W x = f (i.e. approximate the solutions of linear operator equations, see [8,9]), for sequences f hich are non-zero only for a finite subset of indices. Additionally, the approximation might be applied also to calculate numerically the matrix element of some poer of a finite, but very large matrix hich fulfills the conditions of our theorem. 6 Summary and generalization We have shon that one can use a finite part of the infinite, non-negative, Hermitian, sparse bounded matrix W to approximate a chosen matrix element of a poer W α for any real or positive α (for inf W > 0 or inf W = 0, respectively). The accuracy of the approximation depends on the bounds of the matrix spectrum, the value of α and on the structure of W near the chosen element. Possible applications of the approximation have been discussed briefly. It is apparent that the result presented in this paper can be easily generalized to the case of not only poer functions, but also all functions hich are finite and possess a poer series expansion absolutely convergent on the hole spectrum of W. 8

9 7 Acknoledgements The author ishes to thank prof. Jerzy Kijoski for inspiring this ork and dr Andrzej Wakulicz for encouragment to the riting of this paper. The author also acknoledges the support of the Polish Ministry of Science and Higher Education under the grant Quantum Information and Quantum Engineering No. PBZ-MIN-008/P03/2003. References [1] J. Kijoski, A. Thielmann, Quantum electrodynamics on a space-time lattice, J. Geom. Phys. 19 (1996) [2] J. Kijoski, G. Rudolph, A. Thielmann, Algebra of observables and charge superselection sectors for QED on the lattice, Comm. Math. Phys. 188 (3) (1997) [3] J. Kijoski, G. Rudolph, C. Ślia, Charge superselection sectors for scalar QED on the lattice, Ann. Henri Poincaré 4 (6) (2003) [4] M. H. Stone, Linear transformations in Hilbert space, III: Operational methods and group theory, Proc. Nat. Acad. Sci. U.S.A. 16 (1930) [5] J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931) [6] C. M. Carracedo, M. S. Alix, The theory of fractional poers of operators, Elsevier, [7] K. Maurin, Methods of Hilbert Spaces, PWN, Warsa, [8] A. B. Bakušinskij, M. Y. Kokurin, Iterative methods for approximate solution of inverse problems, Springer, [9] M. Chen, Z. Chen, G. Chen, Approximate solutions of operator equations, World Scientific,