Turning a scalar into a derivative operator (context: Schrödinger equation)

Question

At the sight of the Schrödinger equation, most of you will think "No, you are not in the right forum !". However, I'm not interested in the physical sense of this (fabulous) equation today. I would like you to explain me two things linked to pure maths.
Here is the Schrödinger equation at its simpliest form: $$H\psi(x) = E\psi(x) \ | \ E \in \mathbb{R}, \psi(x) \in C^2(I \subseteq \mathbb{R}, \mathbb{R})$$ To make it easier, let's suppose $V= 0$: $$H = T + V = \frac{mv^2}{2} = \frac{p^2}{2m} \in \mathbb{R}^+$$ First question: Can $H$ be considered as an operator despite it belongs to $\mathbb{R}$ ? If it is, we can consider that $E$ is the eigenvalue of $H$ and that $\psi(x)$ is its eigenfunction (else it has no sense to write $H\psi = E\psi$ since $H = E$).
Second question: In quantum physics, we turn $p$ into a derivative operator $-i\hbar \frac{\partial}{\partial x}$ and $E$ into $i\hbar \frac{\partial}{\partial t}$. What's the mathematical sense in it ? This really bothers me: you turn a real number into a derivate ! I'll show you a simple example of why it seems to have to sense to me: $$p = 1\overset{???}{\Longleftrightarrow} -i\hbar \frac{\partial}{\partial x} = 1$$ In my opinion, it has to sense to write a partial derivation whithout the function you want to derivate... I just don't understand what lays under this change of variables.
Thank you in advance for your answers !

Answer 1

This question is probably better suited for mathoverflow; but I will take a stab at it anyway. Keep in mind, I know nothing about the Schrodinger equation.

In mathematics, an operator is typically any mapping from functions to functions. Some operators are linear, some are not. Some use derivatives, others don't. Some have nice representations, others only exist in our imaginations and are basically not able to be defined explicitly.

Most operator development starts with simple operators that are then composed to create larger operators. For example, we choose operators $D$ and $x$, and then perform all multiplications, divisions, additions, infinite sums, etc. An entire theory can be built around just those two. However, they are nothing special, we could very well choose $\Delta$ defined as $\Delta (f) = f + 1$, and build a whole space of operators around $\Delta$.

In all cases, spaces of operators tend to enshrine the same forms as functions themselves. For example, we may wish to now study $\sum_{k=0}^\infty \Delta^k$ and ask questions like, is this well defined, what does it mean, does it make sense, what does it mean for operators to "converge", etc. Another example, $e^{D}$, what should this represent, how does it behave on a function, does it make sense that this is also know as the unit shift operator ($e^D f(x) = f(x+1)$), etc.

These questions are precisely the same questions we would have asked if we were studying polynomials themselves. For example, curiosities around $3x^3+x-1$ are very similar to curiosities about $x^3D^3+7x^5D-5x$, etc. The space of objects in $\mathbb{R}$ and the space of objects $f:\mathbb{R}\to\mathbb{R}$ are really not as different as they seem on a categorical level. Both will end up asking the same kinds of questions; "find all $v$ such that $f(v)=a*v$ for some $a$" vs "find all $f$ such that $T[f] = a*f$ for some $a$".

It is no surprise then, that constants would be swapped for derivatives, or derivatives replaced with other operators, etc. It is also of no surprise that there are theorems that transcend beyond these categories. For example, a multi-variate polynomial's properties $xy+1$ are intimately connected to the properties that the differential operator $xD+1$ exhibit on various functions. Why would we take a simple equation like $3x$ and explore something like $xy$ instead; because it extends the original problem in a new direction. Why would we not do that same thing with all the symbols we find in mathematics; replacing a constant with a variable is equivalent to going from studying algebra and now studying calculus.

To a mathematician, why would we replace and manipulate and change equations till they break? Why would we constantly try to generalize current equations into new categories? The real question is, why wouldn't we do that?

Sorry, kind-of going off on a tangent here. I think this mostly answers your open ended second question. To answer your first question; yes, if you define an operator, say $T$ to take any given function and multiple it by some constant $r$; that is, $T[f] = r\cdot f$, then absolutely, $T$ would be an operator and referring to it as $r$ would be consider standard convention. In context, we would certainly regard the constant $r$ as an operator that acts on functions.