For a self-adjoint Sturm-Liouville problem with Dirichlet conditions

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Function spaces and operator theory · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

For a self-adjoint Sturm-Liouville problem with Dirichlet conditions, write

N[y]=\int_a^b(p\,y'^2+q\,y^2)\,\mathrm dx

and

D[y]=\int_a^bw\,y^2\,\mathrm dx

for real

y

, so that

R[y]=N[y]/D[y]

. ((a)) Let

y_k

be any eigenfunction,

\mathcal Ly_k=\lambda_kwy_k

, and let

\eta

be any real function meeting the boundary conditions. Starting from the integrals, show that

R[y_k+\epsilon\eta]=\lambda_k+O(\epsilon^2)

, so that

R

is stationary at every eigenfunction. ((b)) For

-y''=\lambda y

[0,1]

with

y(0)=y(1)=0

, evaluate

R

exactly for

y=\sin2\pi x+\epsilon\sin\pi x

and for

y=\sin2\pi x+\epsilon\sin3\pi x

, and find the sign of

R-\lambda_2

in each case. ((c)) Say what part (b) shows about the shape of

R

near

y_2

, and why it does not contradict

R[y]\ge\lambda_1

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