412
7 Variational Principles
(λT x, y) = λ(T x, y) = λ(x, T y) = (x, λT y) = (x, λT y)
[(T − λI )x, y] =(T x, y) − λ(x, y) = (x, T y) − (x, λy)
= (x, T y) − (x, λy) = [x, (T − λI )y]
Quod erat demonstrandum.$$ (\lambda Tx,y) = \lambda (Tx,y) = \lambda (x,Ty)
= (x,\overline{\lambda } Ty) = (x,\lambda Ty) $$
If the set composed of differentiable functions, through the differential operation,
can be changed into a set of the other functions, then the differential operation is a kind
of operator on the differentiable functions, which is called the differential operator.
Simply speaking, the operator with derivative symbol or differential symbol is called
the differential operator. The differential operator can also be defined like this: Let
T be the mapping (operator) from the function space F 1 to the function space F 2 ,
T u = f , where u ∈ F 1 , f ∈ F 2 , if the value f (x) at every point x of the image f is
decided by the value at point x of the preimage u and the derives of finite u s, then
T is called the differential operator.
Let X be the linear space that consists of all of the polynomials in the interval
[a, b], T is the operator in X , if for any x ∈ X , there is T x(t) =
n
k=1
d
k
dt k x(t), then
T is called the nth linear differential operator, it is called the linear differential
operator for short.
The following give a few examples of the differential operator.
(1) Let y(x) ∈ C
2 , T y = y
+ y, then T =
d
2
dx 2 + 1 is a second-order differential
operator.
(2) T y = [p(x)y
]
+ q(x)y, T =
d
dx
p(x)
d
dx
+ q(x) is a general second-order
differential operator.
(3) u =
∂
2 u
∂ x 2 +
∂
2 u
∂ y 2 +
∂
2 u
∂z 2 , =
∂
2
∂ x 2 +
∂
2
∂ y 2 +
∂
2
∂z 2 is a partial differential operator of
second order, it is usually called the three-dimensional Laplacian operator or
three-dimensional Laplace operator. If there is not term z, then it is called the
two-dimensional Laplacian operator or two-dimensional Laplace operator.
(4) ∇φ =
∂φ
∂ x
i+
∂φ
∂ y
j+
∂φ
∂z
k, ∇ =
∂
∂ x
i+
∂
∂ y
j+
∂
∂z
k is a vector differential operator, it is
called the Hamiltonian, Hamilton operator, Hamiltonian operator, gradient
operator, nabla operator, del operator or potential operator.
(5) Let F(x, y, y
, · · · , y
(n)
) ∈ C
2n , T F =
n
k=0 (−1)
k d
k
dx k
∂ F
∂ y (k) , then T =
n
k=0 (−1)
k d
k
dx k
∂
∂ y (k) is a differential operator.
(6)
F yy −
d
dx
F yy
u −
d
dx
F yy
du
dx
= 0, J = F yy −
d
dx
F yy −
d
dx
F yy
d
dx
is a
differential operator, it is called the Jacobi operator.
Example 7.4.2 Let y(x) ∈ C
2
[x 0 , x 1 ], p
0 (x), p
1 (x), p 2 (x) ∈ C[x 0 , x 1 ], and
p 0 (x) = 0. For second-order linear differential operator T , there is
T y = p 0 (x)y
+ p 1 (x)y
+ p 2 (x)y
(1)