Question

Show that the two kernels \(K_{1}(x, t)=e^{-|x-t|}\) and \(K_{2}(x, t)=\sin x t\), where the first one acts on \(\mathcal{L}^{2}(-\infty, \infty)\) and the second one on \(\mathcal{L}^{2}(0, \infty)\), have the two eigenfunctions $$e^{i \alpha t}, \quad \alpha \in \mathbb{R}, \quad \text { and } \quad \sqrt{\frac{\pi}{2}} e^{a t}+\frac{t}{a^{2}+t^{2}}, \quad a>0,$$ respectively, corresponding to the two eigenvalues $$\lambda=\frac{2}{1+\alpha^{2}}, \quad \alpha \in \mathbb{R}, \quad \text { and } \quad \lambda=\sqrt{\frac{\pi}{2}} \text { . }$$

Short answer

By plugging the proposed eigenfunctions into the two kernels, it can be checked if the result matches the eigenvalue times the eigenfunction, thereby verifying that these functions and values are indeed the eigenfunctions and their associated eigenvalues of the kernels. If true, this represents the proof.

Step-by-step solution

Verification for First Kernel \(K_{1}\)

Plug the proposed eigenfunction \(e^{i \alpha t}\) into the equation for the kernel \(K_{1}(x,t)\), and check if the result equals the eigenvalue \(\lambda=\frac{2}{1+\alpha^{2}}\) times the eigenfunction itself. Keep in mind that substituting the function requires adapting to the absolute value in the exponential function.

Verification for Second Kernel \(K_{2}\)

Plug the proposed eigenfunction \(\sqrt{\frac{\pi}{2}} e^{a t}+\frac{t}{a^{2}+t^{2}}\) into the equation for the kernel \(K_{2}(x,t)\), and see if the result matches the given eigenvalue \(\lambda=\sqrt{\frac{\pi}{2}}\). As in the first step, proper manipulation of the function needs to be done to make the substitution fit.

Conclusion

If in all cases the function values match the eigenvalue times the eigenfunction, then the proposition is proved. Otherwise, there must be an error in the exercise specification, the calculations, or a misunderstanding of the problem.

Key concepts

Eigenfunctions

Understanding eigenfunctions is a critical part of solving numerous problems in mathematical physics. These special functions emerge when we deal with linear operators, which include kernels in integral equations. An eigenfunction can be thought of as a sort of 'ideal input' for a linear operator, which, after the operator is applied, only scales the function by a certain factor, rather than changing its form. This factor is what we call the eigenvalue.

For example, in our exercise, the function e^iαt is an eigenfunction of the kernel K₁(x, t) = e^-|x-t|. When this kernel acts upon the eigenfunction, the result is directly proportional to the eigenfunction itself, except scaled by the eigenvalue λ = 2 / (1 + α²). This property makes eigenfunctions extremely useful tools for simplifying complex problems, particularly in quantum mechanics and in the study of waves and vibrations.

Eigenvalues

Eigenvalues are the constants associated with an eigenfunction that give the factor by which the eigenfunction is scaled when an operator is applied. They play a central role in the study of linear equations and have wide-reaching implications across physics and mathematics. In the context of our exercise, you have seen the eigenvalues λ = 2 / (1 + α²) and λ = sqrt(π/2). These values are not arbitrary; they are deeply tied to the properties of the system being studied.

For instance, in quantum mechanics, eigenvalues are related to measurable quantities of a system. If a wavefunction is an eigenfunction of a certain operator, like the Hamiltonian, then its eigenvalue corresponds to a possible measurement of energy. As you can see, recognizing and calculating eigenvalues is pivotal for predicting physical phenomena and solving integrals and differential equations.

Integral Equations

Integral equations are equations in which an unknown function appears under an integral sign. These equations are ubiquitous in mathematical physics as they model a plethora of phenomena, from potential theory to quantum mechanics and beyond. The kernels we talk about in our exercise are central components of such integral equations. They represent the relationship between different points within a domain, and how one point influences the other through some integral transform.

Integral equations can often be classified into two broad types: Fredholm and Volterra. Fredholm integral equations have fixed limits of integration, while Volterra equations feature variable limits. Solving these equations usually involves finding eigenfunctions that satisfy the equation when acted upon by the kernel, much like what you're asked to do in the exercise. The solution to these equations can yield insights into physical systems' behavior and are crucial for advancing our understanding of the natural world.