Quark Model, Large Order Behavior and Nonperturbative Wave Functions in QCD

We discuss a few, apparently different (but actually, tightly related) problems: 1. The relation between QCD and valence quark model, 2. The evaluation of the nonlocal condensate $ \la \bar{q}(x)q(0)\ra $, its relation to heavy-light $\bar{q}Q$ quark system and to constituent quark mass, 3. The asymptotic behavior of the nonperturbative pion wave function $\psi(\k, x)$ at $x\rightarrow 0,~1, \k \rightarrow \infty$ and 4. The large order behavior of perturbative series. The analysis is based on such general methods as dispersion relations, duality and PCAC. We use the steepest descent method (also known as semiclassical, or instanton calculus), introduced by Lipatov, in order to calculate the $n-$th moment of the $\psi(\k, x)$ with result $\la\vec{k}_{\perp}^{2n}\ra \sim n!$. This information is converted into the fixing of the asymptotic behavior of $wf$ at large $\k$. This behavior it turns out to be Gaussian commonly used in the phenomenological analyses.The same method determines the asymptotic behavior of the mixed local vacuum condensates $\la\bar{q}G_{\mu\nu}^nq\ra\sim n!$ at large $n$ as well as the nonlocal vacuum condensate $ \la \bar{q}(x)q(0)\ra $ which is naturally arises in the description of the heavy-light $\bar{q}Q$ quark system. The relation between nonlocal condensate and constituent quark mass is also discussed.