Mathematical Methods for Physics and Engineering - Matematica.NET

22.7 ESTIMATION OF EIGENVALUES AND EIGENFUNCTIONSλ max is finite, or λ max = ∞ and λ min is finite. Notice that here we have departedfrom direct consideration of the minimising problem and made a statement abouta calculation in which no actual minimisation is necessary.Thus, as an example, for an equation with a finite lowest eigenvalue λ 0 anyevaluation of I/J provides an upper bound on λ 0 . Further, we will now show thatthe estimate λ obtained is a better estimate of λ 0 than the estimated (guessed)function y is of y 0 , the true eigenfunction corresponding to λ 0 . The sense in which‘better’ is used here will be clear from the final result.Firstly, we expand the estimated or trial function y in terms of the completeset y i :y = y 0 + c 1 y 1 + c 2 y 2 + ··· ,where, if a good trial function has been guessed, the c i will be small. Using (22.25)we have immediately that J =1+ ∑ i |c i| 2 . The other required integral isI =∫ ba[ (p y 0 ′ + ∑ i2 (c i y i) ′ − q y 0 + ∑ i) ] 2c i y i dx.On multiplying out the squared terms, all the cross terms vanish because of(22.27) to leaveλ = I J= λ 0 + ∑ i |c i| 2 λ i1+ ∑ j |c j| 2= λ 0 + ∑ i|c i | 2 (λ i − λ 0 )+O(c 4 ).Hence λ differs from λ 0 by a term second order in the c i , even though y differedfrom y 0 by a term first order in the c i ; this is what we aimed to show. We noticeincidentally that, since λ 0