Mathematical Physics Vol 1

5.9 Orthogonal and normalized functions

267

Principal values. Principal functions

From relations (5.229) and (5.230) it can be seen that for every λ , there exists a trivial solution y ≡ 0, i.e. y ( x )= 0 for ∀ x from the observed interval. Definition If there exists a value λ , for which the problem (5.229), (5.230) has a non-trivial solution ( y̸ ≡ 0), this value is called the principal value of the problem.

Definition The non-trivial solution of the problem (5.229), (5.230), which corresponds to the principal value λ is called the principal function .

We will give some properties of the previously introduced concepts in the form of two (following) theorems.

Theorem16 Assume that the functions p , q , r i r ′ in equation (5.229) are real an continuous on the in terval a ≤ x ≤ b . Let y m ( x ) and y n ( x ) be two principal functions of the Sturm–Liouville problem (5.229), (5.230), which correspond to two different principal values λ m and λ n , respectively. Then y m and y n are orthogonal functions on the observed interval, with respect to the weight function p . Proof Given that y m and y n are solutions of the observed problem, they satisfy the following relations ry ′ m ′ +( q + λ m p ) y m = 0 , ry ′ n ′ +( q + λ n p ) y n = 0 . If the first relation is multiplied by y n , and the second relation by − y m , and then the two resulting relations are added, we obtain ( λ m − λ n ) py m y n = y m ry ′ n ′ − y n ry ′ m ′ = ry ′ n y m − ry ′ m y n ′ . (5.231) This expression represents a continuous function on the interval a ≤ x ≤ b , because r and r ′ are continuous functions according to the initial assumption, and y m and y n are also continuous functions, as solutions of the problem. Thus, we can integrate the observed expression (5.231), which yields

( λ m − λ n ) b Z a py m y n d y = r y ′ n y m − y ′ m y n b a = = r ( b ) y ′ n ( b ) y m ( b ) − y ′ m ( b ) y n ( b ) − r ( a ) y ′ n ( a ) y m ( a ) − y ′ m ( a ) y n ( a ) .

(5.232)