Himeji Conference on Partial Differential Equations
Titles and abstracts 2021
FURUTANI, Kenro (Advanced Mathematical Institute, Osaka City University)
Title: Calabi-Yau structure and a Bargmann type transformation on theCayley projective plane
Abstract: The purpose of our talk is(1) to show the existence of a Calabi-Yau structure on the punctured cotangent bundle
T0∗ (P 2 O) of the Cayley projective plane P 2 O,(2) to construct a Bargmann type transformation between a space of holomorphic functions on T0∗ (P 2 O) and the L2 -space on P 2 O.
A Kähler structure on T0∗ (P 2 O) was shown by identifying it with a quadrics in thecomplex space C27 \{0} and the natural symplectic form of the cotangent bundle T0∗ (P 2 O)
is expressed as a Kähler form.
Our method to construct the transformation is the pairing of polarizations, one is thenatural Lagrangian foliation given by the projection map q : T0∗ (P 2 O) −→ P 2 O and thepositive complex polarization defined by the Kähler structure.
The transformation gives a quantization of the geodesic flow in terms of one parametergroup of elliptic Fourier integral operators whose canonical relations are defined by thegraphs of the geodesic flow action at each time. It turn out that for the Cayley projective
plane the results are a little bit different from other cases of the original Bargmanntransformation for Euclidean space, spheres and other projective spaces.
This is based on a joint work with Kurando Baba (TUS): arXiv 2101.07505: CalabiYau structure and Bargmann type transformation on the Cayley projective plane.

HIGUCHI, Kenta (Ritsumeikan University)
Title: Semiclassical resonances generated by crossings of classical trajectories
Abstract: We consider a one-dimensional semiclassical coupled Schrödinger operator
P1 hW
P (h) =in L2 (R) ⊕ L2 (R).
(1)
hW ∗ P2
Here h > 0 denotes the semiclassical parameter,
22 d+ Vj (x),Pj = −hdx2x ∈ R, (j = 1, 2)

are Schrödinger operators with Vj ∈ C ∞ (R; R), W = W (x, hDx ) a first-order semiclassicaldifferential operator and W ∗ its formal adjoint. We study the resonances of P near anenergy E0 ∈ R at which both the Hamiltonian systems corresponding to P1 and P2 arenon-trapping. It is well known that, near E0 , each Pj has no resonance with imaginary
part of order h log(1/h) (see [Ma]). We will see however that the coupled operator P (h)Figure 1: A case with a closed curve

Figure 2: A case without any closed curves
has resonances at this order if the Hamiltonian trajectories corresponding to P1 and P2
cross and make a closed trajectory. We give precise asymptotics of such resonances interms of the geometry of these trajectories and the crossings.
[Hi] K. Higuchi, Resonance free domain for a system of Schrödinger operators with energylevel crossings, Rev. Math. Phys., Online Ready.
[Ma] A. Martinez, Resonance free domains for non globally analytic potentials, Annales
Henri Poincaré 3 (2002), 739–756.

HIROSHIMA, Fumio (Kyushu University)
Title: Localization of the ground state of the Nelson model
Abstract: The ground state of the renormalized and non-renormalized Nelson model inquantum field theory is studied. Localizations of the ground state are an important problem. The ground state can be seen as a function of three argumetns: the position variable
x, the number of bosons n and the field variable φ. We show the spatial exponential decay
in x in terms of the Agmon metric and Gaussian domination in φ. The super exponential
decay of the number of bosons is also shown. To show the localizations we apply theGibbs measure associated with the ground state.

ISHIWATA, Satoshi (Yamagata University)
Title: Geometric analysis on manifolds with ends
Abstract: In this talk, we consider the behavior of the heat kernel on manifold with ends
(ex. connected sum of tube and plane) and some related results. This talk is based on ajoint work with A. Grigor’yan (Bielefeld) and L. Saloff-Coste (Cornell).

KAMEOKA, Kentaro (The University of Tokyo)
Title: Resonances and complex absorbing potential method for theWigner-von Neumann type Hamiltonian
Abstract: We study the Wigner-von Neumann type Hamiltonian, which has an oscillatory
and slowly decaying potential. We define the complex resonances by proving the existence of meromorphic continuations of matrix elements of resolvent. We also characterize
resonances by complex absorbing potential method. The proofs are based on our newcomplex distortion of Hamiltonian. This is joint work with Shu Nakamura.

KAMIMOTO, Joe (Kyushu University)
Title: Asymptotic analysis of oscillatory integrals with degenerate phases
Abstract:
form:This talk is concerned with the behavior of scalar oscillatory integrals of theZI(t) =eitf (x) ϕ(x)dx
Rnfor large values of the real papameter t. The above integrals appear in various mathematical fields and there have been many interesting studies about their behavior.
In the case when the Hessian of the phase is nondegenerate, the phase can be simply
expressed by coordinate changes (via the Morse lemma), which can compute an exact
asymptotic expansion of oscillatory integrals. On the other hand, when the phase is degenerate, it is quite difficult to understand their bahavior since regular coordinate changes
are not always sufficiently useful in this case. In the history of studies in the degeneratecase, methods based on “resolution of singularities” provide extremely powerful tools forunderstanding their behavior. The seminal result in this direction is due to A. Varchenko.
In this talk, after recalling his work, we explain many kinds of results which generalize
and improve it. These contain recent important works due to the real analysis school ofE.M. Stein and our results with T. Nose.

LEVITINA, Galina (Australian National University)
Title: Cwikel-type estimates on open domains
Abstract: Estimates on the rate of decay of the singular values of the operators of thetype f(x)g(-iv) take their origin in the study of the number of bound states of Schrodinger
operators. Estimates in the weak-Schatten ideals Lp,∞ , were first conjectured by Simonand later proved by Cwikel for p > 2. In this talk Cwikel-type estimates in weak-Schatten
ideal L1,∞ in bounded open sets of the Euclidean space are discussed. Different boundary
conditions for a self-adjoint Laplacian are considered.

NAKANO, Fumihiko (Tohoku University)
Title: Scaling limit of eigenvalues and eigenfunctions of 1-dimensional random
Schrödinger operators
Abstract: 1-dimensional Schrödinger operator with random decaying potential has various spectral and level statistics properties depending on the decay rate of the potential
at infinity. In this talk we consider the random measure associated to the eigenfunctions
and its scaling limit.

NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Title: Edge states of Schrödinger equations on graphene with zigzag
boundaries
Abstract: Recently, topological insulators have been garnering attention in condensed
matter physics. They behave as an insulator in its interior, but their surfaces containconducting states (edge states). These properties are explained by comparing the spectumof a Hamiltonian in the whole space with it in the half space with a boundary.
In this talk, we discuss spectral structures of Schrödinger operators on graphene withzigzag boundaries from the point of view of quantum graphs. Let Γ = (E, V ) be thehexagonal lattice (graphene), Γ] = (E ] , V ] ) the half of Γ with zigzag boundaries. Here, E(E ] , resp.) and V (V ] , resp.) are the set of edges and vertices of Γ (Γ] , resp.), respectively.
Assume that the length of each edge is equal to 1. We define the Hamiltonian H in L2 (Γ)
and H ] in L2 (Γ] ) as follows. For each e ∈ E ] ,(H ] y)(x) = −y 00 (x) + q(x)y(x),where x ∈ (0, 1) is the local coordinate on e and q ∈ L2 (0, 1) is a potential. Let y ∈Dom(H ] ) satisfy (i) the Kirchhoff–Neumann vertex condition at any v ∈ V ] except zigzag
boundaries and (ii) the Dirichlet boundary condition on zigzag boundaries. On the otherhand, the operator H acts as(Hy)(x) = −y 00 (x) + q(x)y(x)on each x ∈ E. In addition, y ∈ Dom(H) satisfies the Kirchhoff–Neumann vertex condition at any v ∈ V . If q is even, then the spectrum of H is discussed by P. Kuchment
and O. Post in 2007. Thus, we study σ(H ] ) for a general q ∈ L2 (0, 1). Then, we compareσ(H ] ) with σ(H) for an even q ∈ L2 (0, 1).
Using
integral decompositions, we have a unitary equivalenceR ⊕ the theory of direct],
whereH(µ)
is a fiber operator for H ] for a quasi-momentum µ ∈H ] ' S 1 H ] (µ) dµ2π
S 1 := [−π, π). Studying σ(H ] (µ)) for each µ ∈ S 1 , we derive the results on σ(H ] ).
To state main results, we introduce notations. Let σD be the set of eigenvalues of theproblem −y 00 + qy = λy and y(0) = y(1). Moreover, we put σp] = σp (H ] ) \ σD and call aneigenfunction corresponding to λ ∈ σp] an edge state.

Theorem 1 (to appear in ”Results in Mathematics”) Fix q ∈ L2 (0, 1), which is notnecessarily even.
−+−+(i) (Basic spectral structure) There exists some sequence λ+0 < λ1 ≤ λ1 < λ2 ≤ λ2 <+· · · < λ−j ≤ λj < · · · → +∞ such that]σ(H ) = (∞[Bj ) ∪ σD ∪ σp] ,j=1−
where Bj = [λ+j−1 , λj ] for each j ∈ N.
(ii) (Existence of edge states) We have σp] = {λ ∈ R| θ(1, λ) + 2ϕ0 (1, λ) = 0} as theinfinite set, where θ(x, λ) and ϕ(x, λ) are the solutions to −y 00 + qy = λy as well as theinitial conditions (θ(0, λ), θ0 (0, λ)) = (1, 0) and (ϕ(0, λ), ϕ0 (0, λ)) = (0, 1), respectively. In
particular, we have σp] 6= ∅.
+−+(iii) (Location of the eigenvalues) Putting Gj = (λ−j , λj ), ∂Gj = {λj , λj } and Gj =+[λ−j , λj ] for each j ∈ N, we haveσD ⊂∞[G2nandσp] ⊂n=1∞[G2n−1 .

n=1SRemark 1 In 2007, P. Kuchment and O. Post proved that σ(H) = ( ∞j=1 Bj ) ∪ σD ,2]provided that q ∈ L (0, 1) is even. In this sense, σp stands the set of the eigenvalues forthe edge states.

SHIMAKURA, Norio
Title: Resolvent kernel of an elliptic operator
Abstract: – See Appendix –TAKEI, Yumiko (Kwansei Gakuin University)
Title: WKB analysis via topological recursion for hypergeometricdifferential equations
Abstract: The exact WKB analysis is a method to analyze ordinary differential equations
with a small parameter ~. The main ingredient of the exact WKB analysis is a formal
solution for ~, called a WKB solution. When we study differential equations by using theexact WKB analysis, the so-called Voros coefficients play an important role. The Voros
coefficient is defined as a contour integral of the logarithmic derivative of WKB solutions.
On the other hand, the topological recursion is introduced by B. Eynard and N. Orantin[EO] to study the correlation functions in the random matrix theory and it gives a generalization of the loop equations for the matrix model.
Recently, a surprising connection between the WKB analysis and the topological recursion has been discovered, that is, it is found that WKB solutions can be constructed
via the topological recursion [BE].

In this talk, we prove that the Voros coefficients for hypergeometric differential equations are described by the generating functions of free energies defined in terms of thetopological recursion. Furthermore, as its applications we show the following objects canbe explicitly computed for hypergeometric equations: (i) three-term difference equations
that the generating function of free energies satisfies, (ii) explicit form of the free energies,and (iii) explicit form of Voros coefficients [IKT,T].
[BE] Bouchard, V. and Eyanard, B., Reconstructing WKB from topological recursion,
Journal de l’Ecole polytechnique – Mathematiques, 4 (2017), 845–908.
[EO] Eynard, B. and Orantin, N., Invariants of algebraic curves and topological expansion,Communications in Number Theory and Physics, 1 (2007), 347–452.
[IKT] Iwaki, K., Koike, T., and Takei, Y.-M., Voros coefficients for the hypergeometricdifferential equations and Eynard-Orantin’s topological recursion, Part I, arXiv:1805.10945& Part II, Journal of Integrable Systems 3 (2019), 1–46.
[T] Takei, Y.-M., Voros Coefficients and the Topological Recursion for a Class of the Hypergeometric Differential Equations associated with the Degeneration of the 2- dimensional
Garnier System, arXiv: 2005.08957.
Organized by : FUJIIE, S., HOSHIRO, T., ISOZAKI, H., IWASAKI, C.,
NAGAYASU, S., NOMURA, Y., RICHARD, S., UMEDA, T.,
and WATANABE, T.