Variational justification of the dimensional-scaling method in chemical physics: the H-atom

The dimensional scaling (D-scaling) method first originated from quantum chromodynamics by using the spatial dimension D as an order parameter. It later has found many useful applications in chemical physics and other fields. It enables, e.g., the calculation of the energies of the Schr?dinger equation with Coulomb potentials without having to solve the partial differential equation (PDE). This is done by imbedding the PDE in a D-dimensional space and by letting D tend to infinity. One can avoid the partial derivatives and then solve instead a reduced-order finite dimensional minimization problem. Nevertheless, mathematical proofs for the D-scaling method remain to be rigorously established. In this paper, we will establish this by examining the D-scaling procedures from the variational point of view. We show how the ground state energy of the hydrogen atom model can be calculated by justifying the singular perturbation procedures. In the process, we see in a more clear and mathematical way confirming (Herschbach J Chem Phys 85:838, 1986 Sect. II.A) how the D-dimensional electron wave function “condenses into a particle,” the Dirac delta function, located at the unit Bohr radius.     

Simple zero property of some holomorphic functions on the moduli space of tori

We prove that some holomorphic functions on the moduli space of tori have only simple zeros.Instead of computing the derivative with respect to the moduli parameter τ, we introduce a conceptual proof by applying Painlevé Ⅵ equation. As an application of this simple zero property, we obtain the smoothness of the degeneracy curves of trivial critical points for some multiple Green function.     

Resolution of Chern–Simons–Higgs Vortex Equations

It is well known that the presence of multiple constraints of non-Abelian relativisitic Chern–Simons–Higgs vortex equations makes it difficult to develop an existence theory when the underlying Cartan matrix K of the equations is that of a general simple Lie algebra and the strongest result in the literature so far is when the Cartan subalgebra is of dimension 2. In this paper we overcome this difficulty by implicitly resolving the multiple constraints using a degree-theorem argument, utilizing a key positivity property of the inverse of the Cartan matrix deduced in an earlier work of Lusztig and Tits, which enables a process that converts the equality constraints to inequality constraints in the variational formalism. Thus this work establishes a general existence theorem that settles a long-standing open problem in the field regarding the general solvability of the equations.     

Sharp estimates for solutions of mean field equations with collapsing singularity

The pioneering work of Brezis-Merle [7 Brezis, H., Merle, F. (1991). Uniform estimates and blow-up behavior for solutions of ?Δu = V(x)e^u in two dimensions. Commun. Partial Differential Equation 16:1223–1254.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Li-Shafrir [27 Li, Y.Y., Shafrir, I. (1994). Blow-up analysis for solutions of ?Δu = V(x)e^u in dimension two. Indiana Univ. Math. J. 43:1255–1270.[Crossref], [Web of Science ®] , [Google Scholar]], Li [26 Li, Y.Y. (1999). Harnack inequality: the method of moving planes. Commun. Math. Phys. 200:421–444.[Crossref], [Web of Science ®] , [Google Scholar]], and Bartolucci-Tarantello [3 Bartolucci, D., Tarantello, G. (2002). Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229:3–47.[Crossref], [Web of Science ®] , [Google Scholar]] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30 Lin, C.S., Tarantello, G. (2016). When “blow-up” does not imply “concentration”: A detour from Brezis-Merle’s result. C. R. Math. Acad. Sci. Paris 354:493–498.[Crossref], [Web of Science ®] , [Google Scholar]] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

Liouville Systems of Mean Field Equations

In this article, we discuss the recent work of Lin and Zhang on the Liouville system of mean field equations: $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} ({\frac{{h_j}e^{u_{j}}}{\int_{M}{h_{j}e^{u_{j}}}}-{\frac{1}{|M|}}})=0\,\, \quad{\rm on}\, M,$$ where M is a compact Riemann surface and |M| is the area, or $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} \frac{{h_j}e^{u_{j}}}{\int_{\Omega}{h_{j}e^{u_{j}}}}=0\,\, \quad{\rm in}\, \Omega,$$ $${u_j}=0,\,\, \quad{\rm on}\, \partial\Omega, $$ where ?? is a bounded domain in ${\mathbb{R}^2}$ . Among other things, we completely determine the set of non-critical parameters and derive a degree counting formula for these systems.