Question

In this problem we outline a proof of the first part of Theorem 11.2 .3 : that the eigenvalues of the Sturm-Liouville problem ( 1 ), (2) are simple. For a given \(\lambda\) suppose that \(\phi_{1}\) and \(\phi_{2}\) are two linearly independent eigenfunctions. Compute the Wronskian \(W\left(\phi_{1}, \phi_{2}\right)(x)\) and use the boundary conditions ( 2) to show that \(W\left(\phi_{1}, \phi_{2}\right)(0)=0 .\) Then use Theorems 3.3 .2 and 3.3 .3 to conclude that \(\phi_{1}\) and \(\phi_{2}\) cannot be linearly independent as assumed.

Short answer

Short Answer Question: Show that the eigenvalues of the Sturm-Liouville problem are simple, meaning that each eigenvalue has only one linearly independent eigenfunction associated with it. Short Answer: To show that the eigenvalues of the Sturm-Liouville problem are simple, we assumed that there were two linearly independent eigenfunctions for a given eigenvalue. After computing their Wronskian and applying the boundary conditions, it was found that the Wronskian is 0 at x=0. By using Theorems 3.3.2 and 3.3.3, we concluded that the given eigenfunctions cannot be linearly independent on any interval containing 0. This contradiction implies that each eigenvalue indeed has only one linearly independent eigenfunction associated with it.

Step-by-step solution

Compute the Wronskian of \(\phi_1\) and \(\phi_2\)

To compute the Wronskian \(W(\phi_1, \phi_2)(x)\) of the two eigenfunctions \(\phi_1\) and \(\phi_2\), use the definition of the Wronskian for two functions: $$W(\phi_1, \phi_2)(x) = \phi_1(x)\phi_2'(x) - \phi_2(x)\phi_1'(x)$$

Apply the boundary conditions

We have the boundary conditions (2) of the Sturm-Liouville problem: $$\phi_1(0) = \phi_2(0) = 0$$ Now, let's use these boundary conditions to compute the Wronskian at x=0: $$W(\phi_1, \phi_2)(0) = \phi_1(0)\phi_2'(0) - \phi_2(0)\phi_1'(0) = 0\cdot\phi_2'(0) - 0\cdot\phi_1'(0)= 0$$

Use Theorems 3.3.2 and 3.3.3

Theorem 3.3.2 states that if two functions have the same Wronskian for all x in an interval, then they are either both linearly independent or both linearly dependent on that interval. Theorem 3.3.3 states that if two functions have a non-zero Wronskian at any point in an interval, they are linearly independent on that interval. From Step 2, we found that \(W(\phi_1, \phi_2)(0) = 0\). Since the Wronskian is 0 at x=0, Theorem 3.3.3 tells us that \(\phi_1\) and \(\phi_2\) cannot be linearly independent on any interval containing 0.

Conclusion:

We assumed that \(\phi_1\) and \(\phi_2\) were linearly independent eigenfunctions for a given eigenvalue \(\lambda\) of the Sturm-Liouville problem. By computing their Wronskian and using the boundary conditions, we showed that the Wronskian is 0 at x=0. Then, we used Theorems 3.3.2 and 3.3.3 to conclude that \(\phi_1\) and \(\phi_2\) cannot be linearly independent on any interval containing 0. This contradicts our initial assumption, and therefore we conclude that the eigenvalues of the Sturm-Liouville problem are simple, meaning that each eigenvalue has only one linearly independent eigenfunction associated with it.

Key concepts

Eigenvalues

The term eigenvalue is at the heart of the Sturm-Liouville problem, a crucial concept in the study of differential equations and mathematical physics. In the context of such problems, an eigenvalue is a special number that, when substituted into a differential equation, allows for non-trivial solutions called eigenfunctions. These eigenfunctions are critically important because they represent modes or states of a physical system, such as vibrations in a mechanical structure or wave functions in quantum mechanics. The simplicity of eigenvalues has profound implications in theory and applications; it implies each state of the system is unique and possesses distinct properties. The quest to prove that eigenvalues of a Sturm-Liouville problem are simple, meaning that each eigenvalue corresponds to a unique eigenfunction, underscores their foundational role in understanding the behavior of complex systems.

Eigenfunctions

An eigenfunction is a non-zero solution to a differential equation at a given eigenvalue. In essence, each eigenfunction corresponds to a particular vibration or wave pattern governed by the differential operator in the Sturm-Liouville problem. It's fascinating to explore how physical phenomena can be described mathematically through these functions. They serve as the building blocks for more complex solutions, akin to how individual musical notes form the basis of a symphony. The uniqueness of each eigenfunction at a given eigenvalue is like a unique fingerprint of that particular state, which in the context of our problem, only occurs when the eigenvalues are simple.

Wronskian

The Wronskian is a powerful tool used to determine the linear independence of two functions. It involves a simple yet elegant calculation of their derivatives. Specifically, for two functions \(\phi_1\) and \(\phi_2\), their Wronskian is defined as \(W(\phi_1, \phi_2)(x) = \phi_1(x)\phi_2'(x) - \phi_2(x)\phi_1'(x)\). This determinant can reveal if the functions have the potential to form a basis for the space of solutions to a differential equation. In practice, finding the Wronskian to be zero at a particular point can indicate that the functions are not linearly independent over an interval containing that point, as we see with the eigenfunctions in the Sturm-Liouville problem.

Boundary Conditions

Understanding boundary conditions is critical for solving differential equations. They specify values or behaviors of the solution at the edges of the domain, like setting the temperature at the surface of an object or ensuring a membane's displacement is fixed at its boundary. In our problem, the boundary conditions require the eigenfunctions to be zero at a specific point. When we're analyzing eigenfunctions of physical systems, these conditions are key in defining the problem uniquely, much like how the size and shape of a drum dictate the tones it can produce. Thus, applying the boundary conditions to the eigenfunctions in the Sturm-Liouville problem leads to the realization that our supposed linearly independent functions cannot actually be independent, as their Wronskian equals zero where the boundary conditions apply.