# Tagged Questions

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### Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form ...
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### Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi]$ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
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### Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
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### Spectrum of the Orr Sommerfeld equation

The Orr Sommerfeld equation is as follows $$\psi''-k^2 \psi - \frac{U''}{U-c}\psi=0$$ where $\psi(y)$ is a complex valued function on $[0,2\pi]$ satisfying Dirichlet boundary conditions ...
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Consider the eigenvalue equation for the Hill operator $$L(y)= -y''+ v(x) y = \lambda y, \quad x\in \mathbb{R},$$ where $v(x)$ is any potential and $\lambda$ is the spectral parameter. If $v(x) ... 0answers 57 views ### What are the connections between spectral expansion and differential operator? For instance, for a nice function$f$on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ... 1answer 154 views ### Compact resolvent Given that the operator $$Hf(x) = -xf''(x) + (x - 1)f'(x)$$ on the Hilbert space$L^2([0,\infty),e^{-x}dx)$possesses, for each$n \in \mathbb{N}$, an eigenvalue$\lambda_n = n$with eigenvector ... 1answer 119 views ### Finding the spectrum of the Schrodinger operator Let$H(f) = -f'' + V(x) f$be the Schrodinger operator on$\mathbb R$. I am trying to calculate the spectrum (eigenvalues) of the operator$H$in$L^2(\mathbb R)$for various choices of$V$. In ... 0answers 54 views ### Compute Rayleigh quotient for ODE I am trying to find Rayleigh quotient for this equation:$u''(r) + [\frac{1-4n^2}{4r^2} + \lambda - 2n\beta -\beta^2r^2]u(r) = 0$, where$0 \le r \le 1$. Is there any way to compute eigenvalue ... 2answers 161 views ### Symbol Of Differential Operators When we are given a differential operator of the form$Lf : = \sum_\alpha a_{\alpha}(x) D^ \alpha f(x) $, we can define the symbol associated with it to be the function:$a(x,y) := \sum_\alpha ...
I am trying to find the eigenvalues and eigenvectors of the Laplacian with mixed boundary conditions on $[0,L]$: More precisely: $$X''(x) = \lambda X(x)$$ with $X'(0)=0$ and $X(L)=a$. I know how to ...