Computation of the coefficients in perturbation ...

Bull. Grad. Sch. Sci. Tech. Hirosaki Univ. 2, １―14（2007）

Computation of the coefficients in perturbation expansions Nobuo KATAOKA and Hiroshi NAKAZATO （Received April 18， 2008）

Abstract In this note we express the coefficients of the perturbation expansion by the matrix entries of the perturbation matrix and the perturbed matrix. We also present computer programs to perform the computation of the coefficients.

1. Introduction Perturbation theory of linear operators provides an efficient method to analyze eigenvalues and eigenvectors of linear operators and matrices. The book [5] is a standard textbook of perturbation theory. The first author of this note is studying this subject based on this book. The second author applied well known formulas in perturbation theory to some mathematical problems in [2,3]. Some standard textbooks in quantum mechanics treat the perturbation formula (cf. [7]). Many active physicists use perturbation methods (cf. [1]). The theme of this note is rather elementary. It is to write down some famous formulas by using matrix entries. The authors think that it is worthwhile to present some elementary formulas on perturbation theory in such forms because of its possibility of applications to numerical analysis of eigenvalues of large size matrices. This note is based on the master thesis [4] of the first author and the papers [2,3].

2. Perturbation of Hermitian matrices Suppose that H is an n×n Hermitian matrix and T is another n×n Hermitian matrix. We consider a 1-parameter family of Hermitian matrices {H + κT : κ ∈ R}. It is known that there are n analytic functions λi (κ) satisfying det(λIn − (H + κT )) =

n 

(λ − λi (κ))

j=1

for every κ ∈ R. In the case λi = λi (0) is a non-repeated eigenvalue of H, the implicit function theorem implies that the analytic function λi (κ) near κ = 0 is determined by the condition det(λi (κ) − (H + κT )) = 0.

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N. KATAOKA, H. NAKAZATO

By its analytic dependence on κ, the function λi (κ) is represented by a Taylor series λi (κ) = λi +

∞ 

λ(j) κj ,

j=1

for a sufficiently small |κ|. In a celebrated textbook of Kato, the formula to compute the coefficients λ(j) is provided by using the traces of some matrices. The aim of this note is to write down such formulas by using matrix entries of H and T . We treat only the coefficients λ(1) , λ(2) , λ(3) and λ(4) . We assume that the eigenvalue problem for H is already solved and we know the complete information on the eivenvalues and eigenvectors of H. Corresponding to this assumption, later on we assume that H is a real diagonal matrix. For column vectors f = (f1 , f2 , · · · , fn )t ∈ Cn , g = (g1 , g2 , · · · , gn )t ∈ Cn its inner product is defined by f, g = f1 g 1 + f2 g 2 + · · · + fn g n .

(1)

For an n × n complex matrix A = (aij ){1≤i,j≤n} , its trace tr(A) is defined by tr(A) = a11 + a22 + · · · + ann .

(2)

An n × n complex matrix T = (tij ) is said to be Hermitian if its entries satisfy tij = tji , (1 ≤ i, j ≤ n), where tji is the complex conjugate of tji . We consider a real diagonal matrix ⎛ ⎜ ⎜ H=⎜ ⎜ ⎝

⎞

λ1

⎟ ⎟ ⎟, ⎟ ⎠

λ2 ..

. λn

where we assume that every off-diagonal entry of H vanishes. We take a standard orthonormal basis ⎛ ⎞ ⎛ ⎞ 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎟ , · · · , ξn = ⎜ . ⎟ ξ1 = ⎜ . ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ ⎝ . ⎠ 0 1 of the n-dimensional vector space Cn . Then the following equation holds Hξi = λi ξi , 1 ≤ i ≤ n.

(3)

Computation of the coefficients in perturbation expansions

3

Suppose that T is an arbitrary n × n Hermitian matrix expressed as ⎛ ⎞ t11 · · · · · · t1n ⎟ ⎜ ⎜ t21 · · · · · · t2n ⎟ T =⎜ .. ⎟ ⎜ .. ⎟. . ⎠ ⎝ . tn1 · · · · · · tnn Each entry tij satisfies the equation tij = T ξj , ξi  = ξj , T ξi  = T ξi , ξj . Suppose that λi (κ) is the eigenvalue of the Hermitian matrix H + κT which converges to λi as κ → 0, where we assume that the multiplicity of the eigenvalue λi of H is 1, that is, λi is a non-degenerate (or non-repeated) eigenvalue of H. By the implicit function theorem, the eigenvalue λi (κ) is expressed as λi (κ) = λi + κλ(1) + κ2 λ(2) + κ3 λ(3) + κ4 λ(4) + O(κ5 ).

(4)

The formula to obtain the coefficients λ(1) , λ(2) , λ(3) , λ(4) was given in [5]. The aim of this paper is a more explicit presentation of these coefficients. In [5], each coefficient λ(n) is expressed as the trace of some (non commutative) polynomial in S, Pi and T , where these operators on Cn are defined in the following. We define a 1-dimensional orthogonal projection Pj on Cn for 1 ≤ j ≤ n as the following Pj ξ = ξ, ξj  ξj . (5) In the sequel we choose i ∈ {1, 2, . . . , n} and fix it. We define an operator S on Cn as the following  1 S= Pj . (6) λ j − λi 1≤j ≤n j = i

Since we frequently use the operator Pi for the fixed i, we denote it by P . We shall express the coefficients λ(n) for n = 1, 2, 3, 4 explicitly. Firstly we compute the coefficient λ(1) .

λ(1) =tr (T P ) = tr (P T P ) =

n 

P T P ξj , ξj 

j=1

=

n  j=1

ξj , ξi  T ξi , ξi  ξi , ξj  =

n  j=1

ξj , ξi  T ξi , ξi  ξi , ξj  . 

(7)

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N. KATAOKA, H. NAKAZATO

Hence we obtain a well known formula λ(1) = T ξi , ξi  .

(8)

Secondly we compute the coefficient λ(2) . We have λ(2) =tr (P T ST P ) = − T ST ξi , ξi  = − ST ξi , T ξi  .

(9)

It follows that 

ST ξi , T ξi  =

1≤j ≤n j = i



=

1≤j ≤n j = i



=

1≤j ≤n j = i

T ξi , ξj  ξj , T ξi  λj − λi T ξi , ξj  T ξi , ξj  λj − λi |T ξi , ξj |2 . λj − λi

Hence we obtain a well known formula λ(2) =

 1≤j ≤n j = i

|T ξi , ξj |2 . λ i − λj

(10)

The above formulas (8) and (10) are popular and written in many standard textbooks. We shall compute the coefficient λ(3) .

 λ(3) =tr (P T ST ST P ) − tr P T S 2 T P T P

= T ST ST ξi , ξi  − T S 2 T P T ξi , ξi

= T ST ξi , ST ξi  − T P T ξi , S 2 T ξi .

(11)

We refer to [5], page 30 for this formula (11). We shall write down the coefficient (11) by using the matrix entries λj ’s and tij ’s. We shall use the following expressions T ST ξi , ST ξi  ,

− T P T ξ i , S 2 T ξi .

The coefficient λ(3) is the sum of (12) and (13)．

(12) (13)

Computation of the coefficients in perturbation expansions

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The expression (12) is computed as T ST ξi , ST ξi  =



T ξi , ξj  T ST ξi , ξj  λ j − λi

1≤j ≤n j = i

=

=





1≤j ≤n j = i

1≤k≤n k = i



|T ξi , ξj |2 + (λj − λi )

1≤j ≤n j = i

Here we use the symbol hold:

T ξi , ξj  T ξi , ξk  ξk , T ξj  (λj − λi ) (λk − λi ) 



1≤j ≤n j = i

1≤k≤n k = i

T ξi , ξj  T ξi , ξk  T ξk , ξj  . (λj − λi ) (λk − λi )

T ξi , ξj  T ξi , ξk  T ξk , ξj  = Sjk . Then the following equations (λj − λi ) (λk − λi ) Skj = Sjk , Skj + Sjk = Sjk + Sjk = 2Re(Sjk ).

Therefore we have 

|T ξi , ξj |2 T ξj , ξj  2

1≤j ≤n j = i

(λj − λi )



+2

1≤j