1. 1 Week 5 Linear operators and the Sturm–Liouville theory 1.Complex differential operators 2.Properties of self-adjoint operators 3.Sturm-Liouville theory 4.Non-homogeneous boundary-value problems

2. 2 ۞ A function f assigns to a number x another number, f(x) : 1. Complex differential operators ۞ An operator Ĥ assigns to a function f(x) another function, Ĥf(x) : Different notations can be used:

3. 3 Example 1: 1. Multiplication of a function by a given function, say x 2, is an operator: 2. Differentiation of a function is an operator:

4. 4 Example 1 (continued): 3. The Fourier and Laplace transformations, are integral operators. Observe that the ‘resulting’ functions, F [f(x)] and L [f(t)], depend on k and s, whereas the ‘original’ functions depend on x and t – i.e. integral operators change the function and its variable as well.

5. 5 ۞ A linear operator is an operator such that 1. All four operators in Example 1 are linear. (1) Example 2: 2. The following operators are non-linear: (2)

6. 6 ۞ An operator of the form where the coefficients c n (x) are complex functions of a real variable x, will be called a linear differential operator of order N.

7. 7 Comments:  The set of functions that are analytical (have infinitely many derivatives) in a closed interval [a, b] (where the “closed” means “including the endpoints”) will be denoted by C ∞ [a, b].  Unless specified otherwise, all operators considered below will satisfy the following property: This certainly holds for linear differential operators, but may not hold for integral operators (e.g. Laplace transformation) and some other types.

8. 8 We shall use the following inner product for complex functions, Observe that g*(x) is ‘conjugated’ – hence, ۞ Let S and S + be subspaces of C ∞ [a, b], and Ĥ and Ĥ + be linear operators such that Then Ĥ and Ĥ + are said to be adjoint to each other, and so are S and S +.

9. 9 Example 3: Let Then To prove the above, consider (3) “such that”

10. 10 Next, consider the l.-h.s. of the definition of adjoint operators, Then, taking into account (3), we obtain as required.

11. 11 Example 4: The proof is similar to the one in Example 3, but with two integration by parts instead of just one. Prove that Let (4)

12. 12 ۞ Let an operator Ĥ be defined in a space S, and Ĥ = Ĥ +, S = S +. Then Ĥ is said to be self-adjoint, or Hermitian in S. Example 6: Prove that the following operator: where c(x) is a given real function, is self-adjoint in both spaces defined in Examples 4 and 5. Example 5: Prove that (4) is still valid if

13. 13 ۞ A linear integral operator is given by where the kernel c(x 1, x 2 ) is a continuous complex function defined for and f(x 2 ) is a continuous function. Observe that the ‘original’ function, f(x 2 ), depends on x 2, while the ‘resulting’ one, Ĥf depends on x 1 (which makes integral operators different from differential operators).

14. 14 where x 1 = k and x 2 = x. The space in which this operator acts can be chosen as Example 7: The Fourier transform can be viewed as an integral operator with

15. 15 A proof that one equals zero Consider two equal non-zero numbers x and y, such that x = y, hence, x 2 = x y, hence, x 2 − y 2 = x y − y 2. Divide by (x − y), x + y = y. Since x = y, we see that 2 y = y. Thus, since y ≠ 0, 2 = 1, Subtract 1 from both sides, 1 = 0.