I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator

$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$

where $\varphi: [0,1] \to \mathbb{R}$, $m$ is any nonnegative integer, and $V$ is say smooth and bounded and $-1 \leq V \leq 2$. This comes from a Schrödinger operator $L = \Delta + \overline{V}$ motivated by the Allen-Cahn equation (see here) on $D^2$ with some radial symmetry assumed.

I'm not sure if there are general sources about existence of eigenfunctions in this 1 dimensional case (or determining the index), especially since there is degeneracy as $r \to 1$. Any references would be appreciated!

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator

$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \varphi(1) = 0$$

where $\varphi: [0,1] \to \mathbb{R}$, $m$ is any nonnegative integer, and $V$ is say smooth and bounded and $-1 \leq V \leq 2$. This comes from a Schrödinger operator $L = \Delta + \overline{V}$ motivated by the Allen-Cahn equation (see here) on $D^2$ with some radial symmetry assumed.

I'm not sure if there are general sources about existence of eigenfunctions in this 1 dimensional case (or determining the index), especially since there is degeneracy as $r \to 0$. Any references would be appreciated!

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator

$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$

where $\varphi: [0,1] \to \mathbb{R}$, $m$ is any nonnegative integer, and $V$ is say smooth and bounded and $-1 \leq V \leq 2$. This comes from a schrodinger operator $L = \Delta + \overline{V}$ motivated by the Allen-Cahn equation (see here) on $D^2$ with some radial symmetry assumed.

I'm not sure if there are general sources about existence of eigenfunctions in this 1 dimensional case (or determining the index), especially since there is degeneracy as $r \to 1$. Any references would be appreciated!

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator

$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$

where $\varphi: [0,1] \to \mathbb{R}$, $m$ is any nonnegative integer, and $V$ is say smooth and bounded and $-1 \leq V \leq 2$. This comes from a Schrödinger operator $L = \Delta + \overline{V}$ motivated by the Allen-Cahn equation (see here) on $D^2$ with some radial symmetry assumed.

I'm not sure if there are general sources about existence of eigenfunctions in this 1 dimensional case (or determining the index), especially since there is degeneracy as $r \to 1$. Any references would be appreciated!

I'm interested in finding eigenvalues of the following operator

$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$

where $\varphi: [0,1] \to \mathbb{R}$, $m$ is any nonnegative integer, and $V$ is say smooth and bounded and $-1 \leq V \leq 2$. This comes from a schrodinger operator $L = \Delta + \overline{V}$ motivated by the Allen-Cahn equation (see here) on $D^2$ with some radial symmetry assumed.

I'm not sure if there are general sources about determining the index of schrodinger operators, especially since there is degeneracy as $r \to 1$. Any references would be appreciated!

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator

$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$

where $\varphi: [0,1] \to \mathbb{R}$, $m$ is any nonnegative integer, and $V$ is say smooth and bounded and $-1 \leq V \leq 2$. This comes from a schrodinger operator $L = \Delta + \overline{V}$ motivated by the Allen-Cahn equation (see here) on $D^2$ with some radial symmetry assumed.

I'm not sure if there are general sources about existence of eigenfunctions in this 1 dimensional case (or determining the index), especially since there is degeneracy as $r \to 1$. Any references would be appreciated!