PHYS 326 Problem Set #5
Problem 1 : Show that the equation y′′ + a δ(x) y + λ y = 0 , with y(±π) = 0 , and a real, has a set of positive
eigenvalues λ satisfying tan(π λ ) = 2 λ .
a
 Asin(νx ) + Bcos (νx ) for – π  x < 0 ,

[Hint : The solutions on the two sides of x = 0 are y(x) = 

C sin(νx ) + Dcos (νx ) for 0  x < π ,
where ν = λ . Then use the continuity of y at x = 0 and the discontinuity in y′ (because of the Dirac
delta function) , together with the boundary conditions. Find the final form of y(x) . ]
Problem 2 : Consider the set of functions, { f (x)}, of the real variable x , defined in the interval −∞ < x < ∞ ,
that  0 at least as quickly as x −1 as x  ±∞ . For unit weight function, determine whether each of the
following linear operators is Hermitian when acting upon { f (x)} :
(i) d + x ; (ii) − i d + x2 ; (iii) ix d .
dx
dx
dx
Problem 3 : Consider the following two approaches to constructing a Green’s function:
2
(i) Find those eigenfunctions yn(x) of the self-adjoint linear differential operator d 2 that satisfy the
dx
boundary conditions yn(0) = yn(π) = 0 , and hence construct its Green’s function G(x , z) . (Take the weight
function as ρ = 1 )
(ii) Construct the same Green’s function using a method based on the complementary function of the
differential equation G′′(x , z) = δ(x − z) and the boundary conditions to be satisfied at the position of the δ

 x(z – π )/π for 0  x  z ,
function, showing that it is G(x , z) = 

 z(x – π )/π for z  x  π .
[Hint : The answer you will find in part (i) of the problem is the Fourier-sine expansion of the answer found
in part (ii) .]
dy
Problem 4 : The differential operator  is defined by  y = − d (e x
) − 1 ex y .
dx
dx
4
Determine the eigenvalues λn of the problem,  yn = λn e x yn , 0 < x < 1
dy
+ 1 y = 0 at x = 1 .
2
dx
(i) Find the corresponding unnormalized yn . Note that with respect to the weight function ρ(x) = e x the yn
are orthogonal. Hence, select a suitable normalization for the yn .
with boundary conditions : y(0) = 0 , and
(ii) By making an eigenfunction expansion, solve the equation  y = − e x/2 ,
0<x<1,
subject to the same boundary conditions as previously.
[Answers : λn = (n + 1 )2π 2 . (i) yn(x) = Bn e − x/2 sin[(n + 1 )πx] , Normalization gives: Bn = 2 .
2
2

2
e − x/2 sin[(n + 1 )πx] . NOTE: You are not allowed to use the answers given here
2
1
3
3
n =0 (n + ) π
2
in your solution of the problem. You are supposed to obtain these solutions.]
Problem 5 : (i) Show that the linear operator
2
 ≡ 1 (1 + x 2 )2 d 2 + 1 x (1 + x 2 ) d + a ,
2
dx
4
dx
acting upon functions defined in −1 ≤ x ≤ 1 and vanishing at the end-points of the interval, is Hermitian
with respect to the weight function ρ(x) = 1/(1 + x 2 ) .
(ii) By making the change of variable x = tan(θ/2), find two even eigenfunctions, y1(x) and y2(x) , of the
(ii) yn(x) = −

2
2 3
2
differential equation u = λu . [Answers : y1(x) = A( 1 – x 2 ) , y2(x) = B[4( 1 – x 2 ) − 3( 1 – x 2 )] .
1+ x
1+ x
1+ x
NOTE: You are not allowed to use the answers given here in your solution of the problem. You are supposed
to obtain them.]