Applications of the Differential Eignevalue Problem

This is why it is important to study and understand the general concept of the Differential Eigenvalue Problem. In summary, the Differential Eigenvalue Problem is considered interesting and is currently an active area of research because it is a fundamental concept in many fields of study, and particularly relevant in understanding quantum mechanics. Its complexity also poses challenges for solving large systems, making it an intriguing topic for further exploration and research.
  • #1
dev00790
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Homework Statement



following from this post; https://www.physicsforums.com/showthread.php?t=312075", I would like to know why people generally think the Differential Eigenvalue Problem is interesting? eg why is there a fair amount of current research into this?

Homework Equations



\lambda y = Ly, where L is a linear operator, \lambda is an eigenvalue. y is an eigenfunction corresponding to the eigenvalue.

The Attempt at a Solution



Searched Google, wikipedia, encylc. Britannica.

I know that large systems are hard to solve for the differential eigenvalue problem, so i am presuming this is partly why the research effort. But I'm stabbing in the dark. - I need something in detail please. Or a site / book where i can find the information?

Thanks :)
 
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  • #2
I can give you one example... One of the most important equations in Quantum Physics is the Schrodinger Equation, which is basically a differential eigenvalue problem.
 

1. What is the differential eigenvalue problem?

The differential eigenvalue problem is a mathematical concept that involves finding the eigenvalues and corresponding eigenvectors of a differential equation. It is commonly used in physics, engineering, and other scientific fields to model and analyze systems with continuous variables.

2. How are differential eigenvalue problems used in real-world applications?

Differential eigenvalue problems have a wide range of applications, including in quantum mechanics, fluid dynamics, and structural analysis. They are used to understand the behavior and properties of complex systems and to make predictions about their future behavior.

3. What is the difference between a differential eigenvalue problem and a regular eigenvalue problem?

The main difference is that a regular eigenvalue problem deals with discrete variables, while a differential eigenvalue problem deals with continuous variables. This means that the solutions to a differential eigenvalue problem are functions, rather than individual numbers.

4. How are eigenvalues and eigenvectors calculated for a differential eigenvalue problem?

Eigenvalues and eigenvectors for a differential eigenvalue problem are typically calculated using numerical methods, such as the finite element method or the finite difference method. These methods involve discretizing the differential equation and solving the resulting matrix equations.

5. What are some limitations of using differential eigenvalue problems in applications?

One limitation is that the solutions to a differential eigenvalue problem may not always have a physical interpretation. Additionally, the calculations can become computationally intensive for complex systems. Also, the accuracy of the solutions depends on the chosen numerical method and the level of discretization used.

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