Prove that the eigenvalues of the Strum-Liouville problem are real.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Proving the Realness of Eigenvalues in the Sturm-Liouville Problem
**Objective:**
To demonstrate that the eigenvalues of the Sturm-Liouville problem are real.
**The Problem Statement:**
Prove that the eigenvalues of the Sturm-Liouville problem are real.
**Explanation:**
The Sturm-Liouville problem is a type of differential equation that can be written in the form:
\[ (p(x) y')' + q(x) y + \lambda r(x) y = 0 \]
where \( p(x) \), \( q(x) \), and \( r(x) \) are given functions, and \( \lambda \) is a parameter. The boundary conditions generally associated with this problem ensure specific behaviors of the functions at the boundaries of the domain.
**Key Points to Consider:**
1. **Self-Adjointness:**
The operator in the Sturm-Liouville equation is self-adjoint. This is a crucial property because self-adjoint operators have real eigenvalues. For an operator \( L \) to be self-adjoint, it must hold that \( \langle Lf, g \rangle = \langle f, Lg \rangle \) for all functions \( f \) and \( g \) in the function space.
2. **Inner Product and Orthogonality:**
The eigenfunctions corresponding to different eigenvalues of a self-adjoint operator are orthogonal with respect to the inner product defined by \( \langle f, g \rangle = \int_a^b f(x) g(x) \, w(x) \, dx \), where \( w(x) \) is a weight function. This inner product structure is pertinent to proving the reality of eigenvalues.
3. **Boundary Conditions:**
Suitable boundary conditions, such as Dirichlet or Neumann conditions, must be specified to ensure that the operator remains self-adjoint. These conditions ensure that the function space over which we are working remains appropriate for the definitions of self-adjointness and orthogonality.
**Proof Outline:**
1. Show that the operator \( L \) is self-adjoint by verifying the condition \( \langle Lf, g \rangle = \langle f, Lg \rangle \).
2. Use the properties of self-adjoint operators to argue that their eigenvalues must be real. This can often be done by considering](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ddf1e52-17d0-47a9-956c-00bd0ce93716%2F53adb7a9-fd26-48f3-9f5d-46c3b968c337%2Fg0r2srt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Proving the Realness of Eigenvalues in the Sturm-Liouville Problem
**Objective:**
To demonstrate that the eigenvalues of the Sturm-Liouville problem are real.
**The Problem Statement:**
Prove that the eigenvalues of the Sturm-Liouville problem are real.
**Explanation:**
The Sturm-Liouville problem is a type of differential equation that can be written in the form:
\[ (p(x) y')' + q(x) y + \lambda r(x) y = 0 \]
where \( p(x) \), \( q(x) \), and \( r(x) \) are given functions, and \( \lambda \) is a parameter. The boundary conditions generally associated with this problem ensure specific behaviors of the functions at the boundaries of the domain.
**Key Points to Consider:**
1. **Self-Adjointness:**
The operator in the Sturm-Liouville equation is self-adjoint. This is a crucial property because self-adjoint operators have real eigenvalues. For an operator \( L \) to be self-adjoint, it must hold that \( \langle Lf, g \rangle = \langle f, Lg \rangle \) for all functions \( f \) and \( g \) in the function space.
2. **Inner Product and Orthogonality:**
The eigenfunctions corresponding to different eigenvalues of a self-adjoint operator are orthogonal with respect to the inner product defined by \( \langle f, g \rangle = \int_a^b f(x) g(x) \, w(x) \, dx \), where \( w(x) \) is a weight function. This inner product structure is pertinent to proving the reality of eigenvalues.
3. **Boundary Conditions:**
Suitable boundary conditions, such as Dirichlet or Neumann conditions, must be specified to ensure that the operator remains self-adjoint. These conditions ensure that the function space over which we are working remains appropriate for the definitions of self-adjointness and orthogonality.
**Proof Outline:**
1. Show that the operator \( L \) is self-adjoint by verifying the condition \( \langle Lf, g \rangle = \langle f, Lg \rangle \).
2. Use the properties of self-adjoint operators to argue that their eigenvalues must be real. This can often be done by considering
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