Question

Show that a Sturm-Liouville operator with periodic boundary condion \([a, b]\) is self-adjoint if and only if \(p(a)=p(b)\). [Recall that periodic dary conditions are given as \(u(a)=u(b)\) and \(\left.u^{\prime}(a)=u^{\prime}(b) .\right]\).

Short answer

A Sturm-Liouville operator with periodic boundary conditions on \([a, b]\) is self-adjoint if and only if \(p(a) = p(b)\).

Step-by-step solution

Recall the definitions

A differential operator \(L\) is said to be self-adjoint if its adjoint \(L^{*}\) is the same as \(L\). For an operator to be a Sturm-Liouville operator, it has to be of the form \(Ly = -(p(x)y')' + q(x)y\) for some functions \(p(x)\) and \(q(x)\). The operator \(L\) is said to satisfy periodic boundary conditions if \(y(a) = y(b)\) and \(y'(a) = y'(b)\).

Prove self-adjointness

Consider the inner product \(\langle u, Lv \rangle = \langle Lu, v \rangle\) which is implicit in the definition of a self-adjoint operator. Using integration by parts, the left-hand side can be written as \(\int_{a}^{b}u(x)(-(p(x)v'(x))' + q(x)v(x))dx = -[u(x)p(x)v'(x)]_{a}^{b} + \int_{a}^{b} u'(x)p(x)v'(x) + u(x)q(x)v(x)dx\). The right-hand side can be rewritten as \(\int_{a}^{b}(-u'(x)p(x)v'(x) + u(x)q(x)v(x))dx + [u'(x)p(x)v(x)]_{a}^{b}\). The two sides are equal if and only if the boundary terms cancel each other out, which holds if \(p(a) = p(b)\) and the periodic boundary conditions are satisfied.

Conclusion

A Sturm-Liouville operator with periodic boundary conditions on \([a, b]\) is self-adjoint if and only if \(p(a) = p(b)\).

Key concepts

Self-Adjoint Operators

In mathematics, especially in the study of differential equations, operators play a crucial role. A self-adjoint operator is one of the most significant concepts here. What makes an operator self-adjoint is its behavior regarding inner products. This means an operator \(L\) is self-adjoint if it satisfies the condition \(\langle u, Lv \rangle = \langle Lu, v \rangle\) for functions \(u\) and \(v\). This occurs when the effects of applying the operator to \(v\), and the effects of applying the operator to \(u\) are symmetric in terms of integration over a certain interval.

Self-adjoint operators ensure real eigenvalues, which is very useful in solving physical problems. In the context of a Sturm-Liouville problem, these operators appear prominently, especially when searching for solutions to differential equations that model real-world phenomena. The symmetry ensures that mathematical solutions make sense in terms of real numbers, contributing to stability and predictability in applications.

Periodic Boundary Conditions

When solving differential equations, boundary conditions help define the limits within which a solution is valid. Periodic boundary conditions are a specific type where the function and its derivatives are equal at the endpoints of the interval. This means \(u(a) = u(b)\) and \(u'(a) = u'(b)\).

These conditions often model phenomena that repeat periodically, such as waves or oscillations. By stating that the function and its slope are the same at both ends of an interval, it creates a smooth transition, effectively connecting the endpoint to the start point. This is essential in problems where the domain simulates a circle, like vibrations in a circular drum or temperature distribution along a circular wire.

This natural cyclicity simplifies the mathematical handling of problems, ensuring the conditions are met everywhere within the domain. It's not just about starting and ending points, but about maintaining continuous, seamless behavior across the entire interval.

Differential Operators

Differential operators are fundamental in calculus and differential equations. They represent the action of taking derivatives and are used extensively to express physical phenomena. In a Sturm-Liouville problem, the typical differential operator takes the form \(Ly = -(p(x)y')' + q(x)y\). Here, the functions \(p(x)\) and \(q(x)\) are coefficients that can vary with the independent variable \(x\).

The purpose of these operators is to act on a function \(y(x)\), producing another function that retains key characteristics of \(y(x)\), such as continuity and differentiability.

• **- (p(x)y')'**: This part denotes a transformation involving the second derivative of \(y\), adjusted by \(p(x)\).
• **q(x)y**: This simply scales the function \(y\) itself by \(q(x)\).

Understanding differential operators is essential for solving many boundary value problems, as they form the foundational mechanics for modeling dynamical systems and processes.