## 1 Introduction  
Quarkonium spectroscopy indicates that between valence quarks inside hadrons, the potential on small scales has \(D=3\) Coulomb form and at hadronic scales has \(D=1\) Coulomb one (here we refer to Coulomb potential satisfying Gauss' law in \(D\) dimensions, cf. Section 2 or [1]). We may combine this two types of behavior and form an effective potential in which at small scales dominates the \(D=3\) component and at hadronic scale the \(D=1\) component: the Coulomb-plus-linear potential (the \"Cornell potential\" [2]),  
\[V(r)=-\frac{k}{r}+\frac{r}{a^{2}}=\mu\left(x-\frac{k}{x}\right),\;\mu=1/a=0.427 \,\mathrm{GeV},\;x=\mu r, \tag{1}\]  
where \(k=\frac{4}{3}a_{s}=0.52=x_{0}^{2}\), \(x_{0}=0.72\) and \(a=2.34\,\mathrm{GeV}^{-1}\) were chosen to fit the quarkonium spectra. An important step in the solution of a theoretical problem is to find a good initial approximation in the corresponding mathematical model. Then by small deformations and a few terms in perturbation expansion we describe a physical phenomenon. When a deformation parameter (e.g., coupling constant) value increases, in some region the initial approximation might change into a new form. In the case of Quantum chromodynamics (QCD), the coupling constant increases with increasing distance between quarks, and in the intermediate region (\(\sim\)0.5 fm) the three dimensional hadronic space becomes a fractal - a space with intermediate dimension. At the hadronic scale (\(\sim\)1 fm), we again have a nice classical picture (one dimensional space) and already one gluon exchange between valence quarks gives a confining potential. One of the main motivations for this work is to put the Cornell potential into a more geometrical setting. We aim to smoothly interpolate between the observed small and large distance behaviour of the quarkonium potential by considering dimension \(D(r)\) of space of hadronic matter to be dynamically changing with \(r\), i.e., the corresponding Coulomb potential is \(V_{D}(r)\sim r^{2-D(r)}\), where the effective dimension of space \(D(r)\) changes from 3 at small \(r\) to 1 at hadronic scales \(\sim\)1 fm. In this paper, we extend investigations started in [3] and construct such potentials and effective dimensions as functions of \(r\). Heavy quarkonium is a system which can probe all scales of QCD. Hence heavy quarkonium presents an ideal laboratory for testing the interplay between perturbative and nonperturbative QCD within a controlled environment. In the last decade, lots of states were found experimentally in the heavy quarkonium mass range that did not fit into the scheme predicted by the constituent quark model, for recent reviews see, for example, reference [4, 5, 6]. We hope our method of continual interpolation of the dimension of hadronic space will be useful in that beyond of the standart quarkonium model states. In Section 2 of this paper, we construct the point charge potential for different dimensions of space. The same problem is considered in Section 3 by using operator fractal calculus. In Section 4, we describe the quarkonium potential in terms of the potential of a point charge and identify the strong coupling fine structure constant dynamics. Then, in Section 5, we formulate renormdynamics in terms of hamiltonian dynamics and solve corresponding motion equations by numerical and graphical methods, we find corresponding asymptotics. Next, we deal with another example: spaces with compactified dimensions. In Section 6, \(1/r^{2}\) potential of a nonlinear extension of the quantum mechanics constructed. Such potentials are ingredients of space compactification problems considered in Section 7. Section 8 deals with Debye screening motivated potentials and its anti-screening counterparts. These potentials are supplemented by a confining factor corresponding to the proper account of the topological effects of the QCD vacuum. The form of the confining potential corresponds to the existence of halo states of nuclear matter. In Section 9, mass parameter effects are motivated and estimated.