It is well-known that in 1 dimension, the parabolic cylinder function occurs in the solution of Weber's equation. See wiki.
I wonder that there maybe exists a counterpart of it.
My first guess is that it is the solution of the following PDE:
$$\triangle u(x) = (|x|^2+c)u(x),$$
but I failed to find any function associated with that equation.
So is there any function like that? Any help is appreciated.
It is doubtful that a generalized parabolic cylinder function was standardized in more that one dimension. Nevertheless, one can consider the product of parabolic functions with different variables (which is less general).
For example, in two dimensions :
$$\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2}=(x^2+y^2+c)u(x,y)$$ Looking for particular solutions on the form $\quad u(x,y)=X(x)Y(y)$ : $$X''Y+XY''=(x^2+y^2+c)XY$$ $$\frac{X''}{X}+\frac{Y''}{Y}=x^2+y^2+c$$ $$\begin{cases} \frac{X''}{X}-x^2=\alpha \\ \frac{Y''}{Y}-y^2=\beta \end{cases}\qquad \text{with}\quad \alpha+\beta=c$$ or, equivalently : $$\begin{cases} \frac{X''}{X}-x^2=\frac{c}{2}+\lambda \\ \frac{Y''}{Y}-y^2=\frac{c}{2}-\lambda \end{cases}\qquad \text{with any constant }\lambda $$ $$\begin{cases} X(x)=c_1 D_{\nu_1}(\sqrt{2}\:x)+c_2 D_{\nu_2}(\sqrt{2}\:x) \\ Y(y)=c_3 D_{\nu_1}(\sqrt{2}\:y)+c_2 D_{\nu_2}(\sqrt{2}_:y) \end{cases} \qquad \begin{cases} \nu_1=\frac{c+2\lambda}{4}-\frac12 \\ \nu_2=-\frac{c+2\lambda}{4}-\frac12 \end{cases}$$ The solutions of the PDE can be expressed as the sum of terms each one on the form : $$u_{_\lambda}(x,y)=C_\lambda\:D_{\pm\frac{c+2\lambda}{4}-\frac12 }(\sqrt{2}\:x)\:D_{\pm\frac{c+2\lambda}{4}-\frac12 }(\sqrt{2}\:y)$$ $\lambda$ and $C_\lambda$ are arbitrary constants. $D_\nu(z)$ is the parabolic cylinder function.
On discret form : $$u(x,y)=\sum_{\text{any }\lambda}C_\lambda\:D_{\pm\frac{c+2\lambda}{4}-\frac12 }(\sqrt{2}\:x)\:D_{\pm\frac{c+2\lambda}{4}-\frac12 }(\sqrt{2}\:y)$$ More generally : $$u(x,y)=\int f(\lambda)\:D_{\pm\frac{c+2\lambda}{4}-\frac12 }(\sqrt{2}\:x)\:D_{\pm\frac{c+2\lambda}{4}-\frac12 }(\sqrt{2}\:y)\:d\lambda$$ The $f(\lambda)$ are arbitrary functions, each one corresponding to one combination of the $\pm$ in the indexes of $D_{\pm\frac{c+2\lambda}{4}-\frac12} $.
The functions $f(\lambda)$ have to be determined according to the boundary conditions.
Of course, this can be similarly done in more than two dimensions.