Key to "Consider the Regular Sturm Liouville(RSLP) problem(p(x)y′)′+(q(x)+λr(x))y=0"

Reuben Brennan

Answered question

2022-03-28

Consider the Regular Sturm Liouville(RSLP) problem

{(p (x) y^{'})}^{'} + (q (x) + λ r (x)) y = 0

Nathanial Carey

Beginner2022-03-29Added 12 answers

Step 1
If your boundary conditions are

y (a) = y (b)

and

y^{'} (a) = y^{'} (b)

with

p (a) = p (b)

then
1) there are infinitely many eigenvalues

- \infty < λ_{0} < λ_{1} \leq λ_{2} \leq \dots

2) For

λ_{0}

there is unique eigenfuntion

ϕ_{0}

λ_{2 i + 1} < λ_{2 i + 2}

then there are unique eigenfunctions

ϕ_{2 i + 1}, ϕ_{2 i + 2}

3) If

λ_{2 i + 1} = λ_{2 i + 2}

then there are two independent eigenfunctions

ϕ_{2 i + 1}, ϕ_{2 i + 2}

.
4)If

ϕ_{i}

and

ϕ_{j}

correspond to distinct

λ_{i}

and

λ_{j}

then they are orthogonal.
5) The set of all eigenfunctions forms a basis of corresponding space.
6)Furthermore,

ϕ_{0}

has no roots in

[a, b], ϕ_{2 i + 1}, ϕ_{2 i + 2}

have exactly

2 i + 1

roots in

[a, b)

.
Proof is given (as well as some more general cases) in Coddington and Levinson, Theory of ODE, in Chapter 8 and exercises.

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