8.7 Eigenvalues and Eigenfunctions of Operator Equations
491
It follows from Theorem 8.7.2 that the characteristic value problem of finding
bounded below operator is in the Hilbert space to find out the element that makes
the functional
(T u,u)
u
2 take the minimum. If u is the element in the orthogonal system,
namely if there exists the additional condition (u, u) = u
2
= 1, the problem of
finding the minimal characteristic value is to find the element that makes the functional (T u, u) take the minimum. Besides the important properties the characteristic
value and characteristic function given by Theorems 8.7.1 and 8.7.2, for the Sturm–
Liouville equation (8.7.2), the minimal characteristic value is equal to the minimum
of the functional (8.7.4) under the boundary conditions (8.7.3) and the isoperimetric
condition (8.7.5), this is also an important property of characteristic value.
Theorem 8.7.3 Let λ 1 ≤ λ 2 ≤ · · · ≤ λ n−1 be the former n − 1 characteristic values
of the bounded below symmetric operator T, u 1 , u 2 , …, u n−1 are the corresponding
orthonormal characteristic functions. If there exists the function u n = 0, such that
(u i , u n ) = 0(i = 1, 2, . . . , n − 1), λ n =
(T u n ,u n )
u n
2
obtains the minimum, then λ m is
the next characteristic value of λ n−1 , u n is the characteristic function corresponding
to λ n .
Proof Let v be an arbitrary element in the domain D(T ) of the operator T, η
is the remainder term η = v −
n−1
k=1 (v, u k )u k of the Fourier series of v about
(u 1 , u 2 , . . . , u n−1 ), then η satisfies the orthogonal condition. In fact, making the
inner product
(η, u i ) = (v, u i ) −
n−1
k=1
(v, u k )(u k , u i )
where, i = 1, 2, . . . , n − 1. Because the characteristic functions of the operator T
are orthogonal, and according to the condition of theorem, u k are the orthonormal
characteristic functions, there are
(u k , u i ) = δ ki
Thus
(η, u i ) = (v, u i ) − (v, u i ) = 0
Let t be an arbitrary real number, then the product tη like η also satisfies the
orthogonal condition, the linear combination u n + tη ∈ D(T ) also satisfies the
orthogonal condition. Similar to Theorem 8.7.2, making the auxiliary function ψ(t),
change u 0 into u n , it can be obtained
(T u n − λ n u n , η) = 0
Substituting the expression of η into the above expression, we get
491
It follows from Theorem 8.7.2 that the characteristic value problem of finding
bounded below operator is in the Hilbert space to find out the element that makes
the functional
(T u,u)
u
2 take the minimum. If u is the element in the orthogonal system,
namely if there exists the additional condition (u, u) = u
2
= 1, the problem of
finding the minimal characteristic value is to find the element that makes the functional (T u, u) take the minimum. Besides the important properties the characteristic
value and characteristic function given by Theorems 8.7.1 and 8.7.2, for the Sturm–
Liouville equation (8.7.2), the minimal characteristic value is equal to the minimum
of the functional (8.7.4) under the boundary conditions (8.7.3) and the isoperimetric
condition (8.7.5), this is also an important property of characteristic value.
Theorem 8.7.3 Let λ 1 ≤ λ 2 ≤ · · · ≤ λ n−1 be the former n − 1 characteristic values
of the bounded below symmetric operator T, u 1 , u 2 , …, u n−1 are the corresponding
orthonormal characteristic functions. If there exists the function u n = 0, such that
(u i , u n ) = 0(i = 1, 2, . . . , n − 1), λ n =
(T u n ,u n )
u n
2
obtains the minimum, then λ m is
the next characteristic value of λ n−1 , u n is the characteristic function corresponding
to λ n .
Proof Let v be an arbitrary element in the domain D(T ) of the operator T, η
is the remainder term η = v −
n−1
k=1 (v, u k )u k of the Fourier series of v about
(u 1 , u 2 , . . . , u n−1 ), then η satisfies the orthogonal condition. In fact, making the
inner product
(η, u i ) = (v, u i ) −
n−1
k=1
(v, u k )(u k , u i )
where, i = 1, 2, . . . , n − 1. Because the characteristic functions of the operator T
are orthogonal, and according to the condition of theorem, u k are the orthonormal
characteristic functions, there are
(u k , u i ) = δ ki
Thus
(η, u i ) = (v, u i ) − (v, u i ) = 0
Let t be an arbitrary real number, then the product tη like η also satisfies the
orthogonal condition, the linear combination u n + tη ∈ D(T ) also satisfies the
orthogonal condition. Similar to Theorem 8.7.2, making the auxiliary function ψ(t),
change u 0 into u n , it can be obtained
(T u n − λ n u n , η) = 0
Substituting the expression of η into the above expression, we get