Mathematical Methods for Physics and Engineering - Matematica.NET

17.2 ADJOINT, SELF-ADJOINT AND HERMITIAN OPERATORS17.1.1 Some useful inequalitiesSince for a Hilbert space 〈f|f〉 ≥0, the inequalities discussed in subsection 8.1.3hold. The proofs are not repeated here, but the relationships are listed forcompleteness.(i) The Schwarz inequality states that|〈f|g〉| ≤ 〈f|f〉 1/2 〈g|g〉 1/2 , (17.12)where the equality holds when f(x) is a scalar multiple of g(x), i.e. whenthey are linearly dependent.(ii) The triangle inequality states that‖f + g‖ ≤‖f‖ + ‖g‖, (17.13)where again equality holds when f(x) is a scalar multiple of g(x).(iii) Bessel’s inequality requires the introduction of an orthonormal basis ˆφ n (x)so that any function f(x) can be written as∞∑f(x) = c n ˆφn (x),n=0where c n = 〈 ˆφ n |f〉. Bessel’s inequality then states that〈f|f〉 ≥ ∑ n|c n | 2 . (17.14)The equality holds if the summation is over all the basis functions. If somevalues of n are omitted from the sum then the inequality results (unless,of course, the c n happen to be zero for all values of n omitted, in whichcase the equality remains).17.2 Adjoint, self-adjoint and Hermitian operatorsHaving discussed general sets of functions, we now return to the discussion ofeigenfunctions of linear operators. We begin by introducing the adjoint of anoperator L, denoted by L † , which is defined by∫ baf ∗ (x) [Lg(x)] dx =∫ ba[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 operator L, the adjoint operator L † can befound by repeated integration by parts.An operator is said to be self-adjoint if L † = L. If, in addition, certain boundaryconditions are met by the functions f and g on which a self-adjoint operator acts,559