I'm trying to find the eigenvalues and eigenfunctions for the integral operator $Ku=\displaystyle \int_{-1}^1 (1-|x-y|) \,u(y) \, dy$Since I want to find $\mu,u$ such that $Ku=\mu u$, we get the equation$\mu u(x) = \displaystyle \int_{-1}^x \,u(y) \, dy - x \displaystyle \int_{-1}^x \,u(y) \, dy + \displaystyle \int_{-1}^x y \,u(y) \, dy - \displaystyle \int_{1}^x \,u(y) \, dy + \displaystyle \int_{1}^x y \,u(y) \, dy - x \displaystyle \int_{1}^x \,u(y) \, dy$Taking derivatives on both sides I get:$\mu u'(x)= - \displaystyle \int_{-1}^x \,u(...Read more

It is well known that eigenfunctions of a regular Sturm-Liouville problem form a complete orthogonal basis, but what about eigenfunctions of a singular Sturm-Liouville problem? Under what conditions are they complete? I'm primarily interested in the semi-infinite case $L_2(0,\infty)$ with well-behaved (continuously differentiable) coefficient and weight functions. I would be very grateful for proof sketches and references....Read more

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-Beltrami operator and the natural vibration analysis of objects. I wonder, is my intuition true? What is the physical meaning of Laplace-Beltrami eigenfunctions?For now, I only know that the eigenfunctions of the Laplace-Beltrami operator are real and orthogonal, thus they could be used as the basis of functions on the manifold where the functions are...Read more

Having solved the eigenvalue problem $$y''+ λ y=0, 0 \leq x \leq L$$$$y(0)=y(L)=0$$which solution is:$$\text{The eigenvalues are: } λ_n=(\frac{n \pi}{L})^2$$$$\text{ and the eigenfuctions are: }y_n=\sin (\frac{n \pi x}{L})$$I am asked to expand the function $f(x)=2$ to the eigenfunctions of the problem.At an other exercise in my notes there is the following:$$\text{Since the problem is Sturm-Liouville, each function } f(x), 0 \leq x \leq L \text{ with } f(0)=f(L)=0, \text{ can be written as a sum of the eigenfunctions, so}$$$$f(x)=\sum_{n=1}^{\...Read more

Given a sequence of functions orthogonal over some interval, which satisfy Dirichlet boundary conditions at that Interval, can we construct a Sturm Liouville problem that gives these as its eigenfunctions?For example, if we take the sequence of scaled Bessel's functions $J_n (\zeta_i x)$ for all positive integral values of $i$ where $\zeta_i$s are roots of the Bessel's function. Note that here, $n$ is fixed and non - negative. We already know that these form an orthogonal basis over the weight function $x$ and range $[0,1]$ such that $$\int_...Read more

Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction Consider the linear differential operator: $$ L = \frac{1}{4}(1+x^2)\frac{d^2}{dx^2}+\frac{1}{2}x(1+x^2)\frac{d}{dx}+a $$ acting on functions defined in $-1 \le x \le 1$ and vanishing at the endpoints of the interval. (a) Is $L$ Hermitian? (b) Determine the weight function necessary to make $L$ Hermitian. (c) Show explicitly that $$ \int_{-1}^{1}V^*(x)W(x)Lu(x)dx = \int_{-1}^{1}(LV)^*W(x)u(x)dx$$ and thereby determine the condition on 'a'. ...Read more

Given this Sturm-Liouville problem:$$X'' + \lambda X = 0$$There are general solutions (Eigenfunctions) for three cases on $\lambda$:$$\lambda > 0$$ Has the characteristic equation: $r^2+\lambda = 0$ with roots $r_1 = i \sqrt{\lambda}, r_2 = -i \sqrt{\lambda}$, inserting them into the derivative eigenfunction $e^{rx}$ results into: $$X(x) = c_1 e^{i \sqrt{\lambda}x} = c_1cos\sqrt{\lambda}x + c_2 sin \sqrt{\lambda}x$$For the case:$$\lambda < 0 $$$$X(x) = c_1 e^{\sqrt{\lambda}x} + c_2 e^{-\sqrt{\lambda}x}$$Finally, for case $\lambda = 0$ the...Read more

I have the eigenvalue problem, $\frac{d}{dx}\big((1-x^2)\frac{du}{dx}\big)+\lambda u=0$,on $[-1,1]$ subject to single boundary condiction $u(-1) = u(1)$.Assume that there is an eigenfunction of the form $u(x)=a_0+a_1x+a_2x^2$. Find the possible eigenvalues for such an eigenfunction.To solve this problem, I plug the $u(x)$ and the 2nd derivative $u(x)$ into the $\frac{d}{dx}\big((1-x^2)\frac{du}{dx}\big)+\lambda u=0$, solve for $x$.But I couldn't get any eigenvalues. Any help will be appreciated....Read more