Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ the Laplacian associated to $g$, where $\nabla^g$ is the Levi-Civita connection and $d^{\ast_g}$ is the formal adjoint of the exterior derivative with respect to the $L^2$ inner product determined by $g$. I am interested in the following spectral problem on $(\mathbb{D},g)$:

$$\Delta_g f = \mu^2 f + \kappa$$

where $\mu\in \mathbb{R}$, $f\in C^{\infty}(\mathbb{D})$, and $\kappa\in C^{\infty}(\mathbb{D})$ is a given fixed function. Note that in this problem the term $\mu^2 f$ has the "wrong sign" and therefore if $\kappa = 0$ (if I am not mistaken) then there are no $L^2$ integrable solutions. On the other hand, if $\kappa \neq 0$, since $\mu^2$ does not belong to the essential spectrum of $\Delta_g$, the resolvent is well-defined and we can always find a solution that is in addition unique. Is this correct? Can this resolvent be computed for functions that are not $L^2$ integrable?

I am interested in this spectral problem beyond $L^2$ integrable functions. I have found a lot of information for the same spectral problem but with the "right sign". I wonder, what is known about the case with the "wrong sign"? Any pointer to the literature will be highly appreciated.

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ the Laplacian associated to $g$, where $\nabla^g$ is the Levi-Civita connection and $d^{\ast_g}$ is the formal adjoint of the exterior derivative with respect to the $L^2$ inner product determined by $g$. I am interested in the following spectral problem on $(\mathbb{D},g)$:

$$\Delta_g f = \mu^2 f + \kappa$$

where $\mu\in \mathbb{R}$, $f\in C^{\infty}(\mathbb{D})$, and $\kappa\in C^{\infty}(\mathbb{D})$ is a given fixed function. Note that in this problem the term $\mu^2 f$ has the "wrong sign" and therefore if $\kappa = 0$ (if I am not mistaken) then there are no $L^2$ integrable solutions. On the other hand, if $\kappa \neq 0$, since $\mu^2$ does not belong to the essential spectrum of $\Delta_g$, the resolvent is well-defined and we can always find a solution that is in addition unique. Is this correct? Can this resolvent be computed for functions that are not $L^2$ integrable?

I am interested in this spectral problem beyond $L^2$ integrable functions. I have found a lot of information for the same spectral problem but with the "right sign" and $L^2$ solutions. I wonder, what is known about the case with the "wrong sign"? Any pointer to the literature will be highly appreciated.

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ the Laplacian associated to $g$, where $\nabla^g$ is the Levi-Civita connection and $d^{\ast_g}$ is the formal adjoint of the exterior derivative with respect to the $L^2$ inner product determined by $g$. I am interested in the following spectral problem on $(\mathbb{D},g)$:

$\Delta_g f = \mu^2 f + \kappa$

where $\mu\in \mathbb{R}$, $f\in C^{\infty}(\mathbb{D})$, and $\kappa\in C^{\infty}(\mathbb{D})$ is a given fixed function. Note that in this problem the term $\mu^2 f$ has the "wrong sign" and therefore if $\kappa = 0$ (if I am not mistaken) then there are no $L^2$ integrable solutions. On the other hand, if $\kappa \neq 0$, since $\mu^2$ does not belong to the essential spectrum of $\Delta_g$, the resolvent is well-defined and we can always find a solution that is in addition unique. Is this correct? Can this resolvent be computed for functions that are not $L^2$ integrable?

I am interested in this spectral problem beyond $L^2$ integrable functions. I have found a lot of information for the same spectral problem but with the "right sign". I wonder, what is known about the case with the "wrong sign"? Any pointer to the literature will be highly appreciated.

Thanks.

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ the Laplacian associated to $g$, where $\nabla^g$ is the Levi-Civita connection and $d^{\ast_g}$ is the formal adjoint of the exterior derivative with respect to the $L^2$ inner product determined by $g$. I am interested in the following spectral problem on $(\mathbb{D},g)$:

$$\Delta_g f = \mu^2 f + \kappa$$

where $\mu\in \mathbb{R}$, $f\in C^{\infty}(\mathbb{D})$, and $\kappa\in C^{\infty}(\mathbb{D})$ is a given fixed function. Note that in this problem the term $\mu^2 f$ has the "wrong sign" and therefore if $\kappa = 0$ (if I am not mistaken) then there are no $L^2$ integrable solutions. On the other hand, if $\kappa \neq 0$, since $\mu^2$ does not belong to the essential spectrum of $\Delta_g$, the resolvent is well-defined and we can always find a solution that is in addition unique. Is this correct? Can this resolvent be computed for functions that are not $L^2$ integrable?

I am interested in this spectral problem beyond $L^2$ integrable functions. I have found a lot of information for the same spectral problem but with the "right sign". I wonder, what is known about the case with the "wrong sign"? Any pointer to the literature will be highly appreciated.