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By E.C. Titchmarsh

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17) also holds for k = l + 1. 1 the general solution for the (l + 2)nd fundamental equation is given by V l +2 = V˜ C l +2 − A†0 (A1V l +1 + · · · + Al +2 V0 ), where C l +2 is some coefficient matrix. 17) for Vk+1 , k = 0, . . 17) for k = l + 1. 17) hold for any integer k. The next theorem provides a recursive formula for the computation of the coefficients Vk , k = 0, 1, . . 3. Let the matrix A( ) be a regular perturbation of A0 . 8). Moreover, the coefficients of the power series for V ( ) can be calculated recursively by the formula Vk = −A†0 k j =1 k = 1, 2, .

In the next theorem it is important to observe that the reduced system has the same form as the original, but the number of matrix equations is decreased by one and the coefficients are reduced in size to matrices in p× p , where p is the dimension of N (C0 ) or, equivalently, the number of redundant equations defined by the matrix coefficient C0 . Typically, the dimension of the null space N (C0 ) is significantly smaller than m. 14. Let {Ck }k=0 ⊆ m×m and {Rk }k=0 ⊆ the system of t + 1 matrix equations m×n , with m ≤ n, and suppose that k i =0 Ci Vk−i = Rk , k = 0, .

In fact, it is terminated after exactly s steps, where s is the order of the pole of the Laurent series for A−1 (z). In the generic case of the first order pole, we can expand (A¯22 + z B¯22 )−1 as follows: B¯ A¯−1 + . . 45) U ∗. 46) 22 ✐ ✐ ✐ ✐ ✐ ✐ ✐ 26 book2013 2013/10/3 page 26 ✐ Chapter 2. 5). 9487 ⎦ . 6325 ⎦ . 3162 ¯ Thus, we have The factor Q corresponds to U , and the factor R corresponds to B. 1619 ⎦ . 25 ⎦ . 25 Then, the subsequent regular coefficients Y1 , Y2 , . . 38). 7 The case of polynomial perturbation Here we treat the polynomial perturbation, which is yet another special case of the analytic perturbation, namely, A(z) = A0 + zA1 + · · · + z p A p .

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Eigenfunction expansions associated with second-order differential equations. Part 1 by E.C. Titchmarsh

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