Question

In this problem we indicate a proof that the eigenfunctions of the Sturm- Liouville problem \((1),(2)\) are real. (a) Let \(\lambda\) be an eigenvalue and \(\phi\) a corresponding eigenfunction. Let \(\phi(x)=U(x)+\) \(i V(x),\) and show that \(U\) and \(V\) are also eigenfunctions corresponding to \(\lambda .\) (b) Using Theorem \(11.2 .3,\) or the result of Problem \(20,\) show that \(U\) and \(V\) are linearly dependent. (c) Show that \(\phi\) must be real, apart from an arbitrary multiplicative constant that may be complex.

Short answer

Question: Prove that eigenfunctions of the Sturm-Liouville problem are real, apart from a possible arbitrary complex multiplicative constant. Answer: The eigenfunction φ must be real, apart from an arbitrary multiplicative constant that may be complex, as we showed in the step-by-step solution that U(x) and V(x) are linearly dependent. This means that φ can be represented as a linear combination of the real eigenfunction U(x), with the only complex part being the constant (1 + iC).

Step-by-step solution

(a) Show that U and V are also eigenfunctions corresponding to λ

To show that \(U(x)\) and \(V(x)\) are eigenfunctions corresponding to λ, let's first rewrite the eigenfunction \(\phi(x)\) as the sum of its real and imaginary parts: $$\phi(x) = U(x) + iV(x)$$ Now, substitute this into the Sturm-Liouville problem equation (1): $$-\frac{d}{dx} \left[p(x)\frac{d}{dx} (U(x) + iV(x))\right] + q(x)(U(x) + iV(x)) = \lambda w(x)(U(x) + iV(x))$$ Now, distribute and group the terms with \(U\) and \(V\): $$-\frac{d}{dx} \left[p(x)\frac{dU(x)}{dx}\right] - i\frac{d}{dx} \left[p(x)\frac{dV(x)}{dx}\right] + q(x)U(x) + iq(x)V(x) = \lambda w(x)U(x) + i\lambda w(x)V(x)$$ Since the equations must be valid for both the real and imaginary parts, we can separate the equations to get: $$ \begin{cases} -\frac{d}{dx} \left[p(x)\frac{dU(x)}{dx}\right] + q(x)U(x) = \lambda w(x)U(x) \\ -\frac{d}{dx} \left[p(x)\frac{dV(x)}{dx}\right] + q(x)V(x) = \lambda w(x)V(x) \end{cases} $$ These two equations show that both \(U(x)\) and \(V(x)\) are eigenfunctions corresponding to eigenvalue λ.

(b) Show that U and V are linearly dependent using Theorem 11.2.3

According to Theorem 11.2.3, if we have two eigenfunctions \(U(x)\) and \(V(x)\) corresponding to the same eigenvalue, then they are linearly dependent. From part (a), we showed that \(U(x)\) and \(V(x)\) are both eigenfunctions corresponding to the eigenvalue λ. Therefore, \(U(x)\) and \(V(x)\) are linearly dependent.

(c) Conclude that eigenfunction φ must be real

Since \(U(x)\) and \(V(x)\) are linearly dependent, it means that we can express one of them as a linear combination of the other. Suppose that there is a constant \(C\) such that $$V(x) = CU(x)$$ Substitute this expression into the original representation of φ: $$\phi(x) = U(x) + iV(x) = U(x) + i(CU(x)) = (1 + iC)U(x)$$ We can see that the eigenfunction φ can be represented as a linear combination of the real eigenfunction \(U(x)\), where the only complex part comes from the constant \((1 + iC)\). Therefore, we can conclude that the eigenfunction φ must be real, apart from an arbitrary multiplicative constant that may be complex.

Key concepts

Eigenfunction

An eigenfunction is a special type of function that arises in the context of differential equations and linear operators. It is commonly associated with the solutions to the Sturm-Liouville problem, a type of boundary value problem that appears in various branches of physics and engineering. When a linear differential operator acts on an eigenfunction, the result is a scaled version of the original function, with the scaling factor being the eigenvalue.

In simple terms, given a Sturm-Liouville problem, we seek functions \(\phi(x)\) such that:

• The differential operator \(-\frac{d}{dx}\left[p(x)\frac{d}{dx}\phi(x)\right] + q(x)\phi(x)\) equals \(\lambda w(x)\phi(x)\).
• \(\phi(x)\) satisfies predefined boundary conditions.

Here, \(p(x)\), \(q(x)\), and \(w(x)\) are given functions that define the specific problem, and \(\lambda\) is the eigenvalue associated with the eigenfunction \(\phi(x)\). These eigenfunctions form an orthogonal set, meaning they can be used as base functions in expansions that approximate other functions.

Eigenvalue

Eigenvalues are the constants that come into play when we explore eigenfunctions. In the context of the Sturm-Liouville problem, the eigenvalue \(\lambda\) is a crucial part of the equation. It represents the scaling factor when the linear differential operator is applied to the eigenfunction.

To break it down further, if \(\phi(x)\) is an eigenfunction corresponding to a certain eigenvalue \(\lambda\), then applying the Sturm-Liouville differential operator to \(\phi(x)\) results in a product of \(\lambda\) and the weight function \(w(x)\) times the eigenfunction itself:\[-\frac{d}{dx} \left[p(x) \frac{d}{dx} \phi(x)\right] + q(x) \phi(x) = \lambda w(x) \phi(x)\]The significance of eigenvalues comes from their role in determining the behavior and nature of the solutions to the differential equations. Different eigenvalues correspond to different modes of behavior or patterns, expressed by the associated eigenfunctions. Finding the eigenvalues is essential in many physical contexts, such as in vibration analysis, quantum mechanics, and stability assessment.

Linearly dependent

Linear dependence is a fundamental concept concerning relationships between functions or vectors. Specifically, a set of functions (or vectors) is considered linearly dependent if one of the functions can be expressed as a linear combination of the others. In the problem given, it mentions that eigenfunctions \(U(x)\) and \(V(x)\) are linearly dependent.

For \(U(x)\) and \(V(x)\) being linearly dependent, there exists some constant \(C\) such that:\[V(x) = CU(x)\]When functions are linearly dependent, it implies that they do not provide additional, independent information about the problem's solution. This is crucial in the context of the Sturm-Liouville problem because it means any complex part of the eigenfunction \(\phi(x) = U(x) + iV(x)\) can be reduced to its real part, making the problem easier to manage.

Understanding the linear dependence among functions can simplify problems considerably. It helps reveal that under certain conditions, complex eigenfunctions can effectively be interpreted in terms of purely real functions, contributing to clearer and more intuitive solutions.