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Physics
B. L. LIVSHITS
A PERTURBATION METHOD FOR AN OPERATOR OF SIMPLE STRUCTURE
(Presented by Academician I. V. Obreimov, March 30, 1960)
In the theory of vibrations of polyatomic molecules, problems sometimes arise of finding the eigenvalues and eigenvectors of a vibration operator
\[ \hat W = \hat W_0 + \Delta \hat W , \]
where \(\Delta \hat W\) may be regarded as a small perturbation. Since the operator \(\hat W\) is, generally speaking, non-Hermitian, the application of quantum-mechanical perturbation theory in this case is not legitimate.
We shall assume that \(\hat W\) is an operator of simple structure (usually this assumption is satisfied), and by the perturbation method we shall solve the problem of the eigenvalues and eigenvectors of the operator \(\hat W=\hat W_0+\Delta\hat W\).
Let \(\hat W\) be an operator of simple structure acting on the vectors of a linear space with basis
\[ \mathbf e_1 = \begin{Vmatrix} 1\\ 0\\ \cdot\\ \cdot\\ \cdot \end{Vmatrix\} N\ \text{rows},\qquad \mathbf e_2 = \begin{Vmatrix} 0\\ 1\\ \cdot\\ \cdot\\ \cdot \end{Vmatrix}, \ldots,\quad \mathbf e_N = \begin{Vmatrix} 0\\ 0\\ \cdot\\ \cdot\\ \cdot\\ 1 \end{Vmatrix}; \]
\[ \mathbf X^{(1)} = \begin{Vmatrix} x_1^{(1)}\\ x_2^{(1)}\\ \cdot\\ \cdot\\ \cdot\\ x_N^{(1)} \end{Vmatrix}, \qquad \mathbf X^{(2)} = \begin{Vmatrix} x_1^{(2)}\\ x_2^{(2)}\\ \cdot\\ \cdot\\ \cdot\\ x_N^{(2)} \end{Vmatrix}, \ldots,\quad \mathbf X^{(N)} = \begin{Vmatrix} x_1^{(N)}\\ x_2^{(N)}\\ \cdot\\ \cdot\\ \cdot\\ x_N^{(N)} \end{Vmatrix} \]
linearly independent eigenvectors of the operator \(W\), i.e.
\[ \hat W\mathbf X^{(j)}=\lambda^{(j)}\mathbf X^{(j)},\qquad j=1,2,\ldots,N; \tag{1} \]
\[
W=\|( \mathbf e_i,\hat W\mathbf e_k )\|
\]
is the matrix of the operator \(\hat W\) in the space under consideration. The operator relation (1) corresponds to the matrix equation
\[ W X^{(j)}=\lambda^{(j)}X^{(j)},\qquad j=1,2,\ldots,N, \tag{2} \]
where \(X^{(j)}\) is the column matrix \(\|x_i^{(j)}\|_{i=1}^{N}\). The scalar product of two vectors \(Y\) and \(Z\) is defined, as is known, by the expression
\[ (Y,Z)=\widetilde Z^{*}Y=Z^{+}Y, \tag{3} \]
where \(\widetilde Z\) and \(Y\) are row and column matrices. Suppose that
\[ W = W_0 + \Delta W \tag{4} \]
and \(\{X_0^{(j)}\}\), \(\{\lambda_0^{(j)}\}\), \(j=1,2,\ldots,N\), are families of column eigenvectors and of the corresponding eigenvalues of the matrix \(W_0\), i.e.
\[ W_0 X_0^{(j)} = \lambda_0^{(j)} X_0^{(j)}, \qquad j=1,2,\ldots,N. \tag{1^0} \]
The column vector \(X^{(i)}\) can be expanded in the eigenvectors of the matrix \(W_0\):
\[ X^{(i)} = a_i^{(i)} X_0^{(i)} + \sum_{j\ne i}' a_i^{(j)} X_0^{(j)} . \tag{5} \]
Substituting (5) into (2) and using (4), we obtain
\[ (W_0+\Delta W)\left(a_i^{(i)}X_0^{(i)}+\sum_{j\ne i}' a_i^{(j)}X_0^{(j)}\right) = \lambda^{(i)}\left(a_i^{(i)}X_0^{(i)}+\sum_{j\ne i}' a_i^{(j)}X_0^{(j)}\right), \tag{6} \]
and, further, representing \(\lambda^{(i)}\) in the form
\[ \lambda^{(i)} = \lambda_0^{(i)} + \Delta\lambda^{(i)}, \tag{7} \]
expanding the parentheses and taking ( \(1^0\) ) into account, we transform (6) into
\[ a_i^{(i)}\Delta W X_0^{(i)} + \sum_{j\ne i}' a_i^{(j)}\lambda_0^{(j)} X_0^{(j)} + \sum_{j\ne i}' a_i^{(j)}\Delta W X_0^{(j)} = \]
\[ = a_i^{(i)}\Delta\lambda^{(i)}X_0^{(i)} + \lambda_0^{(i)}\sum_{j\ne i}' a_i^{(j)}X_0^{(j)} + \Delta\lambda^{(i)}\sum_{j\ne i}' a_i^{(j)}X_0^{(j)} . \tag{8} \]
Equality (8) denotes a column-vector equality; therefore scalar multiplication of the column vectors in the left- and right-hand sides of (8) by one and the same column vector does not violate this equality.
In the perturbation theory of Hermitian operators, a relation of type (8) is multiplied scalarly by a column vector from the family \(\{X_0^{(j)}\}\), using their pairwise orthogonality:
\[ (X_0^{(j)},X_0^{(j')})=\delta_{jj'}=1,\quad j=j'; \qquad (X_0^{(j)},X_0^{(j')})=\delta_{jj'}=0,\quad j\ne j'. \tag{9} \]
However, in the case of an operator of simple structure, the column vectors \(X_0^{(j)}\), \(j=1,2,\ldots,N\), are, generally speaking, not orthogonal to one another; therefore it is not rational to multiply equality (8) scalarly by them.
But to each matrix equation (2) one can put into correspondence a matrix equation containing the Hermitian conjugate matrix \(W^+\):
\[ W^+ P^{(j')} = \mu^{(j')} P^{(j')}, \qquad j'=1,2,\ldots,N. \tag{10} \]
In matrix theory \({}^{(1)}\) it is proved that if \(\{X^{(j)}\}\) are \(N\) linearly independent vectors, then \(\{P^{(j')}\}\) also forms a system of \(N\) linearly independent vectors, i.e. \(W^+\) has simple structure, and the systems of vectors \(\{X^{(j)}\}\) and \(\{P^{(j')}\}\) are biorthogonal and can be bionormalized:
\[ P^{(j')+}X^{(j)}=\delta_{jj'}. \tag{11} \]
Relation (11) shows that the column-vector equality (8) should be multiplied scalarly by the column vectors from the family \(\{P_0^{(j')}\}\), \(j'=1,2,\ldots,N\). Carrying this out, we obtain
\[ a_i^{(i)} P_0^{(i)+}\Delta W X_0^{(i)} + \sum_{j\ne i}' a_i^{(j)} P_0^{(i)+}\Delta W X_0^{(j)} = \Delta\lambda^{(i)} a_i^{(i)}, \tag{12} \]
\[ a_i^{(i)} P_0^{(j')+}\Delta W X_0^{(i)} + \lambda_0^{(j')} a_i^{(j')} + \sum_{j\ne i}' a_i^{(j)} P_0^{(j')+}\Delta W X_0^{(j)} = \lambda_0^{(i)} a_i^{(j')} + \Delta\lambda a_i^{(j')},\quad j'\ne j. \tag{12'} \]
Taking into account that \(\left|a_i^{(j')}\right| \ll \left|a_i^{(i)}\right|\) and neglecting the second term in the left-hand side of (12), and also taking (11) into consideration, we have
\[ \Delta\lambda_{(1)}^{(i)}=P_0^{(i)+}\Delta W X_0^{(i)}; \tag{13} \]
this is the first correction to \(\lambda_0^{(i)}\).
In the same approximation, from \((12')\) it follows that
\[ a_i^{(j')}=a_i^{(i)} \frac{P_0^{(j')+}\Delta W X_0^{(i)}}{\lambda_0^{(i)}-\lambda_0^{(j')}} , \qquad j'\ne i, \tag{14} \]
provided that there is no degeneracy. Substituting (14) into (5), we find \(X^{(i)}\) in the first approximation:
\[ X_{(1)}^{(i)}=a_i^{(i)} \left[ \sum_{j\ne i}^{\prime} \frac{P_0^{(j)+}\Delta W X_0^{(i)}}{\lambda_0^{(i)}-\lambda_0^{(j)}}\,X_0^{(j)} +X_0^{(i)} \right]. \tag{15} \]
If, using (14), we take into account the second term of the left-hand side of (12), then
\(\Delta\lambda^{(i)}=\Delta\lambda_{(1)}^{(i)}+\Delta\lambda_{(2)}^{(i)}\), where
\[ \Delta\lambda_{(2)}^{(i)} = \sum_{j\ne i}^{\prime} \frac{ P_0^{(i)+}\Delta W X_0^{(j)}\cdot P_0^{(j)+}\Delta W X_0^{(i)} }{ \lambda_0^{(i)}-\lambda_0^{(j)} }. \tag{16} \]
Carrying out further, on the basis of \((12')\), a refinement of \(a_i^{(j')}\) and substituting the refined \(a_i^{(j')}\) into (12), we obtain \(\Delta\lambda_3^{(i)}\), etc.
Putting \(a_i^{(i)}=1\), we obtain that, to terms of second order of smallness,
\[ P_{(1)}^{(i)+}X_{(1)}^{(i)}=P_0^{(i)+}X_0^{(i)}=1, \tag{17} \]
i.e., the choice \(a_i^{(i)}=1\) ensures preservation of the normalization condition (11).
If the eigenvalue under consideration is \(s\)-fold degenerate, then it is convenient to represent the perturbed eigen-column vector in the form
\[ X^{(i)}=\sum_{n=1}^{s} a_i^{(i_n)}X_0^{(i_n)} +\sum_{j\ne i}^{\prime} a_i^{(j)}X_0^{(j)}, \tag{18} \]
where the first sum is the component of \(X^{(i)}\) in the \(i\)-th proper invariant subspace of the matrix \(W_0\),
\[ W_0X_0^{(i_n)}=\lambda_0^{(i)}X_0^{(i_n)},\qquad n=1,2,\ldots,s. \tag{19} \]
Substituting (18) into (2) and taking (19) into account, we obtain a relation analogous to \((8')\):
\[ \sum_{n=1}^{s} a_i^{(i_n)}\Delta W X_0^{(i_n)} +\sum_{j\ne i}^{\prime} a_i^{(j)}\lambda_0^{(j)}X_0^{(j)} +\sum_{j\ne i}^{\prime} a_i^{(j)}\Delta W X_0^{(j)} = \]
\[ = \Delta\lambda^{(i)} \sum_{n=1}^{s} a_i^{(i_n)}X_0^{(i_n)} +\lambda_0^{(i)}\sum_{j\ne i}^{\prime} a_i^{(j)}X_0^{(j)} +\Delta\lambda^{(i)}\sum_{j\ne i}^{\prime} a_i^{(j)}X_0^{(j)}. \tag{8'} \]
Multiplying (8′) scalarly first by the column vectors from the family \(\{P_0^{(i_n)}\}\), and then by \(P_0^{(j')}\), \(j'\ne i\), and taking into account (11) and (1⁰), we have:
\[ \sum_{n=1}^{s} a_i^{(i_n)} P_0^{(i_n)+}\Delta W X_0^{(i_n)} +\sum_{j\ne i}^{\prime} a_i^{(j)} P_0^{(i_n)+}\Delta W X_0^{(j)} =\Delta\lambda^{(i)}a_i^{(i_{n'})},\quad n'=1,2,\ldots,s; \tag{12a} \]
\[ \sum_{n=1}^{s} a_i^{(i_n)} P_0^{(j')+}\Delta W X_0^{(i_n)} +\lambda_0^{(j')}a_i^{(j')} +\sum_{j\ne i}^{\prime} a_i^{(j)}P_0^{(j')+}\Delta W X_0^{(j)} =\lambda_0^{(i)}a_i^{(j')}+\Delta\lambda^{(i)}a_i^{(j')}. \tag{12′a} \]
From relations (12a) and (12′a), under known assumptions concerning the moduli of the coefficients \(P_0^{(i_n)+}\Delta W X_0^{(i_n)}\), \(P_0^{(i_n)+}\Delta W X_0^{(j)}\), \(P_0^{(j')+}\Delta W X_0^{(i_n)}\), \(P_0^{(j')+}\Delta W X_0^{(j)}\), one can obtain successive approximations for \(\Delta\lambda^{(i)}\) and \(a_i^{(i_n)}\), \(a_i^{(j)}\), \(j\ne i\). Thus, if these coefficients are assumed to be of the same order, then in the first approximation it follows from (12a) that
\[ \sum_{n=1}^{s}\left(P_0^{(i_n)+}\Delta W X_0^{(i_n)}\right)a_i^{i_n} =\Delta\lambda^{(i)}a_i^{i_{n'}},\quad n'=1,2,\ldots,s. \tag{20} \]
System (20) leads to the secular equation of order \(s\) for \(\Delta\lambda^{(i)}\)
\[ \left\|P_0^{(i_{n'})+}\Delta W X_0^{(i_n)}-\Delta\lambda^{(i)}\delta_{n'n}\right\|=0. \tag{13′} \]
If, however, \(\left|P_0^{(i_n)+}\Delta W X_0^{(i_n)}\right|\) is of the same order as \(\left|P_0^{(j')+}\Delta W X_0^{(j')}\right|\) and is one order smaller than \(\left|P_0^{(i_n)+}\Delta W X_0^{(j)}\right|\) and \(\left|P_0^{(j')+}\Delta W X_0^{(i_n)}\right|\), then, neglecting the third term in the left-hand side of (12′a), we obtain (2):
\[ a_i^{(j')}= \sum_{n=1}^{s} P_0^{(j')+}\Delta W X_0^{(i_n)} \frac{a_i^{(i_n)}}{\left(\lambda_0^{(i)}-\lambda_0^{(j')}\right) \left(1+\dfrac{\Delta\lambda^{(i)}}{\lambda_0^{(i)}-\lambda_0^{(j')}}\right)} = \tag{21} \]
\[ = \frac{1}{\lambda_0^{(i)}-\lambda_0^{(j')}} \sum_{n=1}^{s} P_0^{(j')+}\Delta W X_0^{(i_n)}a_i^{(i_n)} - \frac{\Delta\lambda^{(i)}}{\left[\lambda_0^{(i)}-\lambda_0^{(j')}\right]^2} \sum_{n=1}^{s} P_0^{(j')+}\Delta W X_0^{(i_n)}a_i^{(i_n)} +\cdots . \]
Restricting ourselves to the first term in this expansion \(\left(\Delta\lambda^{(i)}\ll \lambda_0^{(i)}-\lambda_0^{(j')}\right)\) and substituting (21) into (12a), we find
\[ \sum\left( P_0^{(i_{n'})+}\Delta W X_0^{(i_n)} +\sum_{j\ne i}^{\prime} \frac{P_0^{(i_{n'})+}\Delta W X_0^{(j)}\cdot P_0^{(j)+}\Delta W X_0^{(i_n)}}{\lambda_0^{(i)}-\lambda_0^{(j)}} \right)a_i^{(i_n)} = \]
\[ =\Delta\lambda^{(i)}a_i^{(i_{n'})},\quad n'=1,2,\ldots,s. \tag{22} \]
From (22) it follows that
\[ \left\| P_0^{(i_{n'})+}\Delta W X_0^{(i_n)} +\sum_{j\ne i}^{\prime} \frac{P_0^{(i_{n'})+}\Delta W X_0^{(j)}\cdot P_0^{(j)+}\Delta W X_0^{(i_n)}}{\lambda_0^{(i)}-\lambda_0^{(j)}} -\Delta\lambda^{(i)}\delta_{nn'} \right\|=0. \tag{23} \]
The \(\Delta\lambda^{(i)}\) found from (13′) and (23), after substitution respectively into systems (20) and (22), give the zeroth approximation for \(a_i^{(i_n)}\).
Other assumptions concerning the orders of the coefficients in (12a) and (12′a) lead to more complicated results.
In conclusion I express my gratitude to Academician I. V. Obreimov for his attention to and interest in this work.
Institute of Organoelement Compounds
Academy of Sciences of the USSR
Received
30 III 1960
REFERENCES
- F. R. Gantmakher, Theory of Matrices, Moscow, 1959, p. 218.
- M. H. L. Pryce, Proc. Phys. Soc., 63, A, 25 (1950).