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I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator $$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \...

Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by $$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$ where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...

Let's consider the following differential equation on $\mathbb{R}$: $$-u''(x)+u(x)-V(x)u(x)=\lambda u(x),$$ where $\lambda<1$ and $V$ is a bounded. We consider only that solution $u(x) \in C^1$ ...

I was wondering if the following holds: If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{...