Mathematical Methods for Physics and Engineering : A Comprehensive Guide

17.2 ADJOINT, SELF-ADJOINT AND HERMITIAN OPERATORS

17.1.1 Some useful inequalities

Since for a Hilbert space〈f|f〉≥0, the inequalities discussed in subsection 8.1.3

hold. The proofs are not repeated here, but the relationships are listed for

completeness.

(i) The Schwarz inequality states that

|〈f|g〉| ≤ 〈f|f〉^1 /^2 〈g|g〉^1 /^2 , (17.12)

where the equality holds whenf(x) is a scalar multiple ofg(x), i.e. when

they are linearly dependent.

(ii) The triangle inequality states that

‖f+g‖≤‖f‖+‖g‖, (17.13)

where again equality holds whenf(x) is a scalar multiple ofg(x).

(iii) Bessel’s inequality requires the introduction of anorthonormalbasisφˆn(x)

so that any functionf(x) can be written as

f(x)=

∑∞

n=0

cnφˆn(x),

wherecn=〈φˆn|f〉. Bessel’s inequality then states that

〈f|f〉≥

∑

n

|cn|^2. (17.14)

The equality holds if the summation is over all the basis functions. If some

values ofnare omitted from the sum then the inequality results (unless,

of course, thecnhappen to be zero for all values ofnomitted, in which

case the equality remains).

17.2 Adjoint, self-adjoint and Hermitian operators

Having discussed general sets of functions, we now return to the discussion of

eigenfunctions of linear operators. We begin by introducing theadjointof an

operatorL, denoted byL†, which is defined by
∫b

a

f∗(x)[Lg(x)]dx=

∫b

a

[L†f(x)]∗g(x)dx+ boundary terms,

(17.15)

where the boundary terms are evaluated at the end-points of the interval [a, b].

Thus, for any given linear differential operatorL, the adjoint operatorL†can be

found by repeated integration by parts.

An operator is said to beself-adjointifL†=L. If, in addition, certain boundary

conditions are met by the functionsfandgon which a self-adjoint operator acts,