How to prove the orthogonality of eigenfunctions for a given Sturm-Liouville problem?

📚 What is Orthogonality of Eigenfunctions?

In the realm of Sturm-Liouville problems, orthogonality of eigenfunctions is a crucial property. Essentially, it means that different eigenfunctions are 'perpendicular' to each other in a specific mathematical sense. This property allows us to represent solutions to differential equations as a sum of these orthogonal eigenfunctions, much like how we can represent a vector as a sum of orthogonal basis vectors. This simplifies analysis and computation.

📜 History and Background

The study of Sturm-Liouville problems originates from the work of Jacques Charles François Sturm and Joseph Liouville in the mid-19th century. They investigated the properties of solutions to second-order linear differential equations, discovering that under certain boundary conditions, the eigenfunctions possess the property of orthogonality. Their work laid the foundation for many areas of mathematical physics and engineering, providing tools for analyzing heat flow, wave propagation, and quantum mechanics.

🔑 Key Principles

• 🔍 Sturm-Liouville Equation: The general form of the Sturm-Liouville equation is given by $ \frac{d}{dx} \left[p(x) \frac{dy}{dx}\right] + q(x)y + \lambda w(x)y = 0 $, where $p(x)$, $q(x)$, and $w(x)$ are known functions, $y(x)$ is the unknown function, and $ \lambda $ is a parameter.
• ⚖️ Weight Function: The function $w(x)$ is called the weight function. It plays a vital role in defining the orthogonality condition.
• 🚧 Boundary Conditions: The boundary conditions are typically of the form $a_1y(a) + a_2y'(a) = 0$ and $b_1y(b) + b_2y'(b) = 0$, where $a$ and $b$ define the interval of interest. These conditions are crucial for ensuring the self-adjointness of the Sturm-Liouville operator.
• ➗ Eigenvalues and Eigenfunctions: The values of $ \lambda $ for which non-trivial solutions exist are called eigenvalues, and the corresponding solutions $y(x)$ are called eigenfunctions.
• ➕ Orthogonality Condition: Two eigenfunctions, $y_m(x)$ and $y_n(x)$, corresponding to distinct eigenvalues $ \lambda_m $ and $ \lambda_n $, are orthogonal with respect to the weight function $w(x)$ if $ \int_a^b y_m(x) y_n(x) w(x) dx = 0 $ for $m \neq n$.

📝 Proof of Orthogonality

The proof generally involves the following steps:

1. ✍️ Start with the Sturm-Liouville equation for two different eigenfunctions, $y_m$ and $y_n$, corresponding to distinct eigenvalues $ \lambda_m $ and $ \lambda_n $: $$\frac{d}{dx} \left[p(x) \frac{dy_m}{dx}\right] + q(x)y_m + \lambda_m w(x)y_m = 0$$ $$\frac{d}{dx} \left[p(x) \frac{dy_n}{dx}\right] + q(x)y_n + \lambda_n w(x)y_n = 0$$
2. ✖️ Multiply the first equation by $y_n$ and the second by $y_m$: $$y_n \frac{d}{dx} \left[p(x) \frac{dy_m}{dx}\right] + q(x)y_m y_n + \lambda_m w(x)y_m y_n = 0$$ $$y_m \frac{d}{dx} \left[p(x) \frac{dy_n}{dx}\right] + q(x)y_n y_m + \lambda_n w(x)y_n y_m = 0$$
3. ➖ Subtract the second equation from the first: $$y_n \frac{d}{dx} \left[p(x) \frac{dy_m}{dx}\right] - y_m \frac{d}{dx} \left[p(x) \frac{dy_n}{dx}\right] + (\lambda_m - \lambda_n) w(x)y_m y_n = 0$$
4. ➗ Integrate both sides from $a$ to $b$: $$\int_a^b \left(y_n \frac{d}{dx} \left[p(x) \frac{dy_m}{dx}\right] - y_m \frac{d}{dx} \left[p(x) \frac{dy_n}{dx}\right]\right) dx + (\lambda_m - \lambda_n) \int_a^b w(x)y_m y_n dx = 0$$
5. 🔄 Integrate by parts to simplify the first integral: $$\int_a^b \left(y_n \frac{d}{dx} \left[p(x) \frac{dy_m}{dx}\right] - y_m \frac{d}{dx} \left[p(x) \frac{dy_n}{dx}\right]\right) dx = \left[p(x)\left(y_n \frac{dy_m}{dx} - y_m \frac{dy_n}{dx}\right)\right]_a^b$$
6. 📍 Apply the boundary conditions. For self-adjoint operators, the boundary term vanishes, so: $$\left[p(x)\left(y_n \frac{dy_m}{dx} - y_m \frac{dy_n}{dx}\right)\right]_a^b = 0$$
7. ✅ Since $ \lambda_m \neq \lambda_n $, we have: $$(\lambda_m - \lambda_n) \int_a^b w(x)y_m y_n dx = 0 \implies \int_a^b w(x)y_m(x) y_n(x) dx = 0$$

💡 Real-world Examples

• 🌡️ Heat Conduction: In solving the heat equation with specific boundary conditions, the eigenfunctions of the associated Sturm-Liouville problem represent the spatial modes of heat distribution, which are orthogonal.
• 🎸 Vibrating Strings: When analyzing the vibrations of a string fixed at both ends, the eigenfunctions represent the different modes of vibration, and their orthogonality is essential for determining the amplitude of each mode.
• ⚛️ Quantum Mechanics: In quantum mechanics, the solutions to the time-independent Schrödinger equation for certain potentials form a Sturm-Liouville problem. The orthogonality of eigenfunctions (wavefunctions) ensures the probabilistic interpretation of quantum states.

🔑 Conclusion

Proving the orthogonality of eigenfunctions in a Sturm-Liouville problem involves using the properties of the differential equation, the boundary conditions, and integration by parts. This property is not just a mathematical curiosity; it has profound implications in various fields, enabling the decomposition of complex problems into simpler, manageable components.