STURM-LIOUVILLE THEORY Self-Adjoint Odes

PHY310 - MATHEMATICAL METHODS FOR PHYSICISTS I

Odd Term 2019 Dr. Anosh Joseph, IISER Mohali

LECTURE 32

November 1, 2019

STURM-LIOUVILLE THEORY

Topics: Sturm-Liouville theory, Self-adjoint ODEs, Eigenvalues and eigenfunctions.

Sturm-Liouville theory is an eigenvalue problem. It is the theory of the existence and asymptotic behavior of eigenvalues, the corresponding theory of eigenfunctions, and their completeness in a suitable function space. The operator acts on such functions has the property of being self-adjoint with respect to the function space where the eigenfunctions live in and the boundary conditions associated to the same space. In the previous lectures we showed that a second-order linear homogeneous ODE has two linearly independent solutions. We also proved that a third linearly independent solution does not exist. Let us now try to understand the general properties of these solutions. We can ﬁnd consider functions as vectors, function space as a vector space and linear operators as matrices. The diagonalization of a real symmetric matrix would then correspond to the solution of an ODE deﬁned by a self-adjoint operator, L in terms of its eigenfunctions. Eigenfunctions are continuous analogue of the eigenvectors.

Self-adjoint ODEs

In the previous lectures we encountered the general form of a linear, second-order, homogeneous ODE u00 + p(x)u0 + q(x)u = 0. (1)

Let us take p (x) p (x) p(x) = 1 , and q(x) = 2 , (2) p0(x) p0(x) and now consider the equation

00 0 Lu(x) = p0(x)u + p1(x)u + p2(x)u. (3) PHY310 - MATHEMATICAL METHODS FOR PHYSICISTS I Odd Term 2019

This can be interpreted as a map, in which the linear operator L mapping a function u to another function Lu.

The coeﬃcients p0(x), p1(x) and p2(x) are real functions of x over the region we are interested in, which is a ≤ x ≤ b. Note that p0(x) must not vanish in this interval a < x < b. Zeros of p0(x) are singular points. Thus, our interval [a, b] must be given so that there are no singular points in the interior of the interval. Singular points are allowed on the boundaries. Now, let us consider the following integral, with u(x) taken as a real function

hu|L|ui ≡ hu|Lui Z b = u(x)Lu(x)dx a Z b 00 0 = u p0u + p1u + p2u dx. (4) a

Integrating by parts we have

Z b Z b 00 0 u(x)Lu(x)dx = u p0u + p1u + p2u dx a a b 0 = u(x)(p1 − p0)u(x) x=a Z b d2 d + 2 [p0u] − [p1u] + p2u udx. (5) a dx dx

If we require that the integrals in Eqs. (4) and (5) be identical for all twice diﬀerentiable functions u, then the integrands have to be equal. Comparison of Eqs. (4) and (5) then yields

00 0 0 0 u(p0 − p1)u + 2u(p0 − p1)u = 0, (6) or 0 p0(x) = p1(x). (7)

Notice that this constraint will also make the boundary terms in Eq. (5) vanish. It is convenient to deﬁne the linear operator in Eq. (5) as

d2 d Lu = [p u] − [p u] + p u dx2 0 dx 1 2 00 0 0 00 0 = p0u + 2(p0 − p1)u + (p0 − p1 + p2)u, (8) as the adjoint operator L.

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Thus, if Eq. (7) is satisﬁed we have

hLu|ui = hu|Lui Z b Z b u(x)(Lu(x))dx = (L¯u(x))u(x)dx. (9) a a

We can also show more generally, following the same procedure, that

Z b Z b v(x)(Lu(x))dx = (Lv(x))u(x)dx. (10) a a

When this condition is satisﬁed we have

d Lu = Lu = p (x)u0 + p (x)u. (11) dx 0 2 and the operator L is said to be self-adjoint.

Relabelling p0 → p and p2 → q, we have

d Lu = Lu = p(x)u0 + q(x)u. (12) dx

Note that a given operator is not inherently self-adjoint. Its self-adjointness property depends on the properties of the function in which it acts and also on the boundary conditions. 0 We can convert Eq. (3), with p0 6= p1 into the required self-adjoint form. Let us multiply L by 1 Z x p (t) exp 1 dt , p0(x) p0(t) we obtain

1 Z x p (t) d p (t) exp 1 dt Lu(x) = exp 1 dt u0(x) p0(x) p0(t) dx p0(t) p (x) Z x p (t) + 1 exp 1 dt u, (13) p0(x) p0(t) which is clearly self-adjoint, see Eq. (12). All second-order linear ODEs can be recast in the above form by multiplying both sides of the equation by an appropriate integrating factor.

Notice that we have p0(x) in the denominator. That is why we require p0(x) 6= 0, a < x < b. From now on we will assume that L has been put into self-adjoint form.

Eigenfunctions and eigenvalues

We have the Schroedinger equation Hψ(x) = Eψ(x). (14)

The diﬀerential operator L here is deﬁned by the Hamiltonian H and this operator may no

3 / 5 PHY310 - MATHEMATICAL METHODS FOR PHYSICISTS I Odd Term 2019 longer be real. The eigenvalue becomes the total energy E of the system. The eigenfunction ψ(x) may be complex and it is usually called the wave function. Sometimes an eigenvalue equation takes the more general Sturm-Liouville or self-adjoint form

Lu + λw(x)u = 0, (15) where λ is a constant, and w(x) is a known weight or density function. We have w(x) > 0 except possibly at isolated points at which w(x) = 0.

For a given choice of the parameter λ, a function uλ(x), which satisfies Eq. (15) and the imposed boundary conditions is called an eigenfunction corresponding to λ. The constant λ is then called an eigenvalue. The value of λ is not specified in Eq. (15). Finding the values of λ for which there exists a non-trivial solution of Eq. (15) satisfying the boundary conditions is part of the Sturm-Liouville problem. One could try to find the eigenvalues λ1, λ2, ··· , and the corresponding eigenvectors u1, u2, ··· , of the L operator. The inner product of two functions is defined as

Z b hv|ui ≡ v∗(x)w(x)u(x)dx (16) a depends on the weight function. The weight function also modiﬁes the deﬁnition of orthogonality of two eigenfunctions. They are orthogonal if their inner product

huλ0 |uλi = 0. (17)

In Table 1 we provide some of the common second order linear ODEs occurring in mathematical physics that can be recast in Sturm-Liouville form. The coefficient p(x) is the coefficient of the second derivative of the eigenfunction. The eigenvalue λ is the parameter that is available in a term of the form λw(x)u(x); any x dependence apart from the eigenfunction becomes the weighting function w(x). If there is another term containing the eigenfunction (not the derivatives), the coefficient of the eigenfunction in this additional term is identified as q(x). If no such term is present, q(x) is zero.

Worked example: Legendre’s equation

We have the Legendre’s equation

(1 − x2)y00 − 2xy0 + l(l + 1)y = 0, − 1 ≤ x ≤ 1. (18)

Comparing this with

00 0 p0(x)u + p1(x)u + p2(x)u + λw(x)u(x) = 0, (19)

4 / 5 PHY310 - MATHEMATICAL METHODS FOR PHYSICISTS I Odd Term 2019 gives

2 p0(x) = 1 − x = p 0 p1(x) = −2x = p

p2(x) = 0 = q, w(x) = 1, λ = l(l + 1).

Recall that our series solutions of Legendre’s equation diverged unless l was restricted to one of the integers. This represents a quantization of the eigenvalue λ.

Equation p(x) q(x) λ w(x) Legendre 1 − x2 0 l(l + 1) 1 2 m2 Associated Legendre 1 − x − (1−x2) l(l + 1) 1 n2 2 Bessel, 0 ≤ x ≤ a x − x a x Hermite, 0 ≤ x < ∞ e−x2 0 2α e−x2 Simple harmonic oscillator 1 0 ω2 1

Table 1: Some common second order linear ODEs occurring in mathematical physics that can be recast in Sturm-Liouville form.

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