Motivation and main results…

Guass Hypergeometric equation (in this thesis, we call it the Hypergeometric equation) x(1 − x)d²f (x)

dx² + (c − (a + b + 1)x)df (x)

dx − abf (x) = 0,

is a kind of Euler’s Hypergeometric differential equations, and it is a second-order ordinary differential equation. Here we denote it as E(a, b, c) for both the operator and the equation. The Hypergeometric series

2F1(a, b; c; x) :=

∞

X

n=1

(a; n)(b; n) (c; n)(1; n)xⁿ

is one of the solutions of this equation. They appear in many fields of “Exact Science”

from the era of Euler to now.

In this thesis, we ask a question: “How does the operator structure of the Hyper-geometric equation look like? Can we do its spectral resolutions?” We give a partially positive answer:“We can do that on the closed interval [0, 1].” (For other domain, see the problem in section 1.1.1 .) In particular, the question and the answer can be described as in the following dialogues:

“Does any potential application for this spectral resolution exist?” My short answer to this is : “Why, sir, there is every probability that you will soon be able to tax it.”

(by Michael Faraday,).

The dream hidden in this thesis is an analogue of “from hamonic function to eigen-function” in the context of Riemannian geometry and Hodge theory. Especially, the dream can also be considered as the classical philosophy in functional analysis “ker and eigen” of nice operators deeply involved with geometry and arithmetic, such as Laplacian operator and, in our case, Hypergeometric equation.

We object is the zeta determinant of a one-parameter family of a regular singular Sturm-Liouville operator. The closed form of it can be calculated by using the formulaes

1

doi:10.6342/NTU201701882 given by Matthias Lesch and a Ramanujan’s formula on the Hypergeometric series that

related to the derivative of the Schwarz map. In case a = b = ¹₂, c = 1 , the Schwarz map is the inverse of Modular lambda function:

τ = ²F₁(¹₂,¹₂; 1; 1 − x)

where our regular singular Sturm-Liouville operator:

L = d²

dx² + a(x) x²(1 − x)²

is the Sturm-Liouville form of the Hypergeometric operator (and the associated equation is called the Sturm-Liouville form of the Hypergeometric equation):

H^(µ_G^0,µ1,µ∞)= d² dx² +1

4((1 − µ²₀) + (−1 + µ²₀− µ²₁+ µ²_∞)x + (1 − µ²_∞)x²

x²(1 − x)² ).

For classical Laplacian operator 4 of a compact Reimannian manifold M , we defines the zeta determinant by

det 4 := exp(−ζ₄⁰ (0)),

where the Laplacian 4 has the discrete spectrum 0 < λ₁ < λ₂· · · , and we define the spectral zeta function of 4 by

ζ4(s) :=

The absolute convergence of ζ4(s) on {Re(s) > σ} (for some σ) depends on the geom-etry of M (c.f.Weyl asympotic formula). Moreover, We can use Mellin transform by the apriori estimate of the trace of heat kernel of 4: P∞

n=1exp(−λt) and the identity

∞

When the coefficient of the operator is smooth on whole domain, the estimate of it heat kernel is relatively easy. However, when the coefficients of operator have strong singularities in the domain, such as −_dx^d22 − _4x¹2 on [0, α) for any α > 0, the foundation of the study of the heat kernel need to be reconstructed. One of the fundamental work on this is the “singular asymptotics lemma” of Bruning and Seeley, see [7](see also [1]).

On the other-hand, the difficulties of evaluation of the zeta determinant of an opera-tor may be regarded as the ignorance of the exact form of eigenvalues and eigenfunctions.

However, in the case of regular singular Sturm-Liouville operators L = d²

dx² + a(x) x²(1 − x)² 2

doi:10.6342/NTU201701882 on [0, 1](where a(x) is a smooth function defined on the cloesd interval [0, 1]), Lesch

gives a clear expression of the zeta determinant via only the a(0), a(1) and the pair of normalized solutions of L. For the whole details, we refer the readers to the papers of Bruning-Seeley’s[7] and Lesch’s [2].

The Lesch’s formula of the zeta determinant is Theorem 1.1.1. det_ζ(L) = exp(−ζ_L⁰(0)),

det_ζ(L) = πW(ψ, ϕ)

2^ν0+ν1Γ(ν₀+ 1)Γ(ν₁+ 1) (1.1) where ψ (ϕ,resp.) is the normalized solution of L at 0(at 1,resp.), ν₀ =q

1

4 + −¹₄(1 − µ²₀) (ν₁, µ₁, resp.), W(ψ, ϕ) = ψϕ_x− ϕψ_x=the Wronskian of ψ, ϕ.

Based on this, a particular case of the main result of this thesis is

det

H

= 1.

1.1.1 Motivation

I start this study since my advisor Chun-Chung Hsieh asked Masaaki Yoshida: “Why you study Schrodinger equation?” in the lectures of Yoshida on his lovely book <Hyper-geometric Functions, My Love> (see [3]) invited by Chang-Shou Lin in the autumn of 2014. If my understaning is correct, Yoshida has not considered the term “Schrodinger equation” in his lecture ever. In fact, Yoshida is considering the normal form(it this this we call it Sturm-Liouville from) of Hypergeometric equation at that time. For me, however, the question of Hsieh is not totally a nonsense. I dream on this diection, if the Hypergeometric equation have discrete spectrum

{0 = λ₀ < λ₁ ≤ · · · ≤ λ_k ≤ · · · } and assign the form the spectrum, we can see the eigenspaces

E_k= {g ∈ F (CP¹\ {0, 1, ∞}) : H_Gg = λ_kg}.

of H_G = H^(µ_G^0,µ1,µ∞)are dimension two. (Here F (CP¹\ {0, 1, ∞}) is a unknown function space that we want to find it but not success yet.) Therefore, we may get a series of linear monodromy representations if we can do analytic continuation for the eigenfuctions,

ρk : π1(CP¹\ {0, 1, ∞}) → GL(Ek) ∼= GL2(C).

Formally, we have

ρ : π₁(CP¹\ {0, 1, ∞}) → GL(M

k

E_k) ∼= GL∞(C),

3

doi:10.6342/NTU201701882 by using the naive direct sum of the monodromy matrices. One of the intention of the

author is to try to use this hypothetical construction to study the Belyi embedding Gal(Q

Q) ,→ Out(π₁(CP¹\ {0, 1, ∞})),

and the graded Grothendieck-Teichmuller Lie algebra. The beliefs of the author are based on the deep-interconnections between the Hypergeometric function and the con-formal field theory [4].And the recent work of Takashi Ichikawa: [6],[5], shows that there is a rigorous relation between Conformal theory and Grothendieck-Teichmuller lego game, this work is on the same line of the beliefs of the author.

However, this dream is still in the air. It has no any concrete result yet. As an opportunity and challenge, we use the spectral resolution of regular singular Sturm-Liouville operators in this thesis. For example, the spectral resolution on the domain of H_G is still not clear. Hence, we use a framework over the closed interval [0, 1]. There is a question: how would the eigenvalue or eigenfunction change when the domain are replaced as CP¹\ {0, 1, ∞}.

So here we have a challenge:

Remark 1.1.2. Can we find a functional space that allow us to do a spectral resolution of HG over CP¹\ {0, 1, ∞}? Which one is a “suitable” functional space?

1.2 Heat Kernel and Mellin Transform in the Case