2019年度秋季総合分科会(於:金沢大学)

SUMMARY: The Neumann–Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is \(C^{1, \alpha }\) smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.

Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to \(0\) for \(C^{1, \alpha }\) smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl’s law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl’s law, the Euler characteristic and the Willmore energy on the boundary surface. Furthermore, we present the asymptotic behavior of positive and negative NP eigenvalues separately under the condition of infinite smoothness of the boundary in three dimensions.

As an application, we analyze the localized surface plasmon resonance via the spectral theory of the NP operator. This is a particular class of metamaterials that allow the presence of negative material parameters such as negative permittivity and permeability in electromagnetism, and negative density and refractive index in acoustics, etc. Brief observations of NP operators reveal the mathematical meaning of these phenomena.

SUMMARY: With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds in the early 2000s. As an application of his propagation theorem of multiplicity-freeness property we can find that if a Lie group acts on a connected complex manifold strongly visibly then the space of holomorphic functions is multiplicity-free. I will show that the converse holds in an algebraic setting, namely, a complex spherical variety admits a strongly visible action of a compact real form.

This result and its proof have several applications. Huckleberry and Wurzbacher (1990) proved that for a connected compact Kähler manifold with a Kähler–Poisson action of a connected compact Lie group \(U\) the \(U\)-action is coisotropic if and only if it is an embedding of a complex spherical variety. Hence in this setting we can see that the coistropicity implies the visibility.

The proof of the visibility for spherical varieties has an application to harmonic analysis on Riemannian weakly symmetric spaces. By the same argument as the proof of the visibility in the affine homogeneous case we can show a \(KAK\)-decomposition for Gelfand pairs and from this we obtain an induction formula of spherical functions.

We also have an application to double coset decompositions. Again by the same argument we can show a Cartan decomposition for a real spherical reductive homogeneous space as conjectured by Kobayashi (1995). Further, we can describe generic double cosets with respect to pairs of absolutely spherical reductive subgroups under some conditions by using T. Matsuki’s results on double coset decompositions for symmetric pairs (1995).

SUMMARY: Throughout this talk, an operator \(A\) means a bounded linear operator acting on a complex Hilbert space \(H\). An operator \(A\) is positive, denoted by \(A \ge 0\), if \((Ax,x)\ge 0\) for all \(x \in H\). We denote \(A > 0\) if \(A\) is positive and invertible. The \(\alpha \)-geometric mean \(\#_\alpha \) for \(\alpha \in [0,1]\) is defined by \(A \#_\alpha B=A^{\frac 12}(A^{-\frac 12}BA^{-\frac 12})^\alpha A^{\frac 12}\) for \(A>0\) and \(B\ge 0\).

The core of log-majorization theorem due to Ando–Hiai is that \(A \#_\alpha B \le 1\) implies \(A^r \#_\alpha B^r \le 1\) for \(r \ge 1\). It holds for positive operators \(A, B\) on a Hilbert space, and is called the Ando–Hiai inequality (AH). A binary operation \(\natural _\alpha \) is defined by the same formula as the \(\alpha \)-geometric mean for \(\alpha \not \in [0,1]\). Very recently (AH) is extended by Seo as follows: For \(\alpha \in [-1,0]\), \(A \natural _\alpha B \le 1\) for \(A, B>0\) implies \(A^r \natural _\alpha B^r \le 1\) for \(r \in [0,1]\).

In this talk, we present two variable extension of it. As an application, we pose operator inequalities of type of Furuta inequality and grand Furuta inequality. Moreover, related to them, we propose norm inequalities of Bebiano–Lemos–Providência type.

SUMMARY: It is well-known in quantum theory that quantum states are perfectly distinguishable if and only if they are orthogonal. In this talk, we restrict available states to separable states and use a larger class of measurements. In this setting, we give a necessary and sufficient condition for two pure states \(\rho _1^A\otimes \rho _1^B\) and \(\rho _2^A\otimes \rho _2^B\) to be perfectly distinguishable. In particular, we find that there are two non-orthogonal states that are perfectly distinguishable in the above setting.

SUMMARY: A variant of the Gagliardo–Nirenberg inequality in Hat–Sobolev spaces is proved, which improves certain classes of classical Sobolev embeddings. Some continuation criterion for the incompressible Navier–Stokes system is established as an application. A direct proof of the fractional Gagliardo–Nirenberg inequality in end-point Besov and Fourier–Herz spaces is established.

SUMMARY: Abstract evolution equations are discussed in Besov spaces. By means of the logarithmic representation of infinitesimal generators [1], the solovability is extended to non-parabolic evolution equations.

[1] Y. Iwata, Methods Funct. Anal. Topology (2017) 1, 26–36.

SUMMARY: We show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. Here we have no magnetic field. Moreover we obtain the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature.

SUMMARY: We consider the scattering theory for one-dimensional two-state quantum walks. The S-matrix appears in the Fourier transform of the scattering operator associated with the position-dependent QWs. Usually, the scattering operator is defined by the wave operator in a time-dependent manner. In this talk, we consider the spectral theory for QW in the time-independent argument. Moreover, we show that the S-matrix appears in the singularity expansion of the generalized eigenfunction in \( \ell ^{\infty } ({\bf Z} ; {\bf C}^2 )\).

SUMMARY: It is recently shown by A. Suzuki (Shinshu University) that chirally symmetric discrete-time quantum walks possess supersymmetry, and that their associated Witten indices can be naturally defined. Such quantum walks are referred to as supersymmetric quantum walks (SUSYQWs). In this talk, we are going to consider a well-known one-dimensional two-phase model (split-step quantum walk) as a prototype example of a SUSYQW. A complete classification of the Witten index associated with this model will be given.

*This is joint work with A. Suzuki.

SUMMARY: We consider the spectrum of the two dimensional Schrödinger operator with homogeneous magnetic field. The non-perturbed operator has eigenvalues with infinite multiplicity called Landau levels. The perturbation, which decays at infinity, may create eigenvalues with finite multiplicity around each Landau level. In this talk, we give the Bohr–Sommerfeld type quantization condition for the two dimensional magnetic Schrödinger operator as the strength of the magnetic field tends to infinity.

SUMMARY: We give a self-adjointness criterion of the Schrödinger operator with infinitely many point interactions, which is applicable in the case the support of the point interactions is the Poisson configuration. We also calculate the spectrum of the Schrödinger operator with point interactions of Poisson–Anderson type.

SUMMARY: In this session, we talk about existence of the outgoing/incoming resolvents of repulsive Schrödinger operators which may not be essentially self-adjoint on the Schwartz space. As a consequence, we construct \(L^2\)-eigenfunctions associated with complex eigenvalues by a standard technique of scattering theory. In particular, we give another proof of the classical result via microlocal analysis: The repulsive Schrödinger operators with large repulsive exponent are not essentially self-adjoint on the Schwartz space.

SUMMARY: In the last 10 years, explicit formulas for wave operators have been obtained for several continuous quantum scattering systems, namely Schrödinger operators on a Euclidean space. Such formulas enable us to give a topological interpretation to Levinson’s theorem, which relates the scattering part to the number of bound states of the underlying system. In this talk we report new formulas for the wave operators associated with a discrete Schrödinger operators on the half-line.

SUMMARY: We consider the quantum dynamics of a charged particle on the plane \({\bf R}^2\) in the presence of a time-periodic magnetic field \({\bf B}(t) = (0,0,B(t))\) with \(B(t+T) =B(t)\) which is always perpendicular to this plane. Then the charged particle has the following three states accordingly to the mass of the particle, charge of the particle and \(B(t)\); (I). For any \(t\), the particle is in some compact region (bound state). (II). The particle goes to a distance with velocity \(O(t)\). (III) The particle goes to a distance with velocity \(O(e^{|t|})\). In this talk, we focus on the case (III) and see that the Hamiltonian of case (III) is closely related to so called homogeneous repulsive Hamiltonian. By using this similarity, we prove the Mourre estimate for the case (III).

SUMMARY: We determine the deficiency indices and the spectrum of a time operator of unitary operator. We show that, for a discrete-time quantum walk, the time operator can be self-adjoint if the time evolution operator has a non-zero winding number.

SUMMARY: In this talk we show pointwise bounds of eigenvectors in quantum field theory. Upper and lower bounds of eigenvectors are given by using Feynman–Kac formula.

SUMMARY: The quantum Rabi model (QRM), and its generalization, asymmetric quantum Rabi model (AQRM), are the simplest models used in quantum optics to describe the interaction of light and matter. Both models were shown to be integrable in 2011 by showing the existence of a \(G\)-function whose zeros correspond to a part of the spectrum of QRM. We show that the remaining eigenvalues, called exceptional correspond to removable singularities of the \(G\)-function for certain values of the parameters. In the general case, we define a complete \(G\)-function that captures the complete spectrum of QRM. Moreover, we show that this completed \(G\)-function is, up to an entire non-vanshing function, equal to the spectral determinant of the QRM, defined in terms of the zeta regularized product of its spectral zeta function.

SUMMARY: The quantum Rabi model (QRM) is one of simplest and most fundamental systems describing quantum light-matter interaction. In this talk we give a closed form of the heat kernel of the Hamiltonian of the QRM using the Trotter–Kato product formula. To the best knowledge of the authors, this is the first explicit derivation of the heat kernel for any non-trivial interacting quantum system. From the explicit expression of the heat kernel we also obtain a formula for the partition function of the QRM. As an application, we investigate basic properties of the spectral zeta function for the QRM via the Mellin transform of the partition function of the QRM.

SUMMARY: We talk about the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillators (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, mainly on an observation on a relation between the generating functions of the Apery-like numbers arising from the special values of the spectral zeta function and the logarithmic Mahler measures for certain Laurent polynomials and the automorphic integrals used to describe the generating functions.

SUMMARY: In this talk, we give a brief summary that any complex Heisenberg homogeneous space has a strongly visible action of some closed subgroup of the Heisenberg Lie group.

SUMMARY: We show that the values of the Schur polynomials in all \(n\)th primitive roots of unity are \(1\), \(0\), or \(-1\), if \(n\) has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the cyclotomic polynomial.

SUMMARY: The immanant of a matrix is a generalization of both the determinant and the permanent in terms of the representations of the symmetric group. Since the discovery of the correspondence between the product of Schur functions and the minor expansion of immanants, it has played the important role in the representation theory and the invariant theory, etc.

In this talk, we consider some immanant identities corresponding to plethysm, another type of the product of Schur functions, which arises in the representations of the general linear group. Following Littlewood’s approach, we review invariant matrices and the contribution of immanants to the plethysm. We give the immanant identities corresponding to the most simplest formula of the plethysm, and discuss more general cases.

SUMMARY: We realize the Hermitian symmetric space \(U(p,q)/U(p)\times U(q)\) as a bounded symmetric domain \(D_{p,q}\subset M(p,q;\mathbb {C})\), and consider the weighted Bergman space \(\mathcal {H}_\lambda (D_{p,q})\subset \mathcal {O}(D_{p,q})\). In this talk we present a result on the computation of the inner product of a polynomial on the subspace \(M(p',q';\mathbb {C})\oplus M(p'',q'';\mathbb {C})\) and an exponential function on \(M(p,q;\mathbb {C})\). Also, as an application, we present a result on explicit construction of intertwining operators from representations of \(U(p,p)\) to those of the subgroup \(U(p',p'')\times U(p'',p')\).

SUMMARY: Let \(G\) be a real reductive group and \(X\) a homogeneous \(G\)-manifold. The Plancherel measure for \(X\) describes how \(L^2(X)\) decomposes into irreducible unitary representations of \(G\). We show that the support of Plancherel measure looks like asymptotically the moment map image of the cotangent bundle of \(X\) via correspondence between the unitary dual of \(G\) and the coadjoint orbits. In particular, we obtain a sufficient condition for the existence of discrete series. This is a joint work with Benjamin Harris.

SUMMARY: Let \(G\) be a noncompact connected simple Lie group of finite center and \(K\) a maximal compact subgroup. For a certain class of \(K\)-type, associated elementary spherical functions can be expressed by Opdam’s nonsymmetric hypergeometric function. As an application, we give an explicit inversion formula for the spherical transform.

SUMMARY: Let \( G = GL_n \) be a general linear group. We generalize the Steinberg theory, which gives a geometric interpretation of Robinson–Schensted correspondence for permutations, to the case of partial permutations.

SUMMARY: Let \( G = GL_{2n} \) be a general linear group and \( K = GL_n \times GL_n \) a symmetric subgroup. Let \( P \) be a parabolic subgroup of \( G \) stabilizing \( n \) dimensional subspace of \( \mathbb {C}^{2n} \) whose Levi part is isomorphic to \( K \). We consider a double flag variety \( X = K/B_k \times G/P \), where \( B_K \) denotes a Borel subgroup of \( K \).

We study the conormal variety of the diagonal action of \( K \) in \( X \) and its moment map. It leads us to the study of combinatorial correspondence involving partial permutations and signed Young diagrams, which we call exotic Robinson–Schensted correspondence.

SUMMARY: We introduce a notion of \(\lambda \)-graph bisystem, that consists of a pair \(({\frak L}^-, {\frak L}^+)\) of two labeled Bratteli diagrams \({\frak L}^-, {\frak L}^+\), respectively, and satisfy certain compatibility condition of their labeling on edges. It yields a pair of \(C^*\)-algebra written \({{\mathcal {O}}_{{\frak L}^-}^+}, {{\mathcal {O}}_{{\frak L}^+}^-}\). If a \(\lambda \)-graph bisystem comes from a \(\lambda \)-graph system of a finite directed graph, then \({\mathcal {O}}_{{\frak L}^-}^+\) is isomorphic to \({\mathcal {O}}_A,\) whereas \({\mathcal {O}}_{{\frak L}^+}^-\) is isomorphic to \(C(\Lambda _A)\times _{\sigma _A^*}\mathbb {Z}\) of the two-sided topological Markov shift \((\Lambda _A, \sigma _A)\).

SUMMARY: The Cuntz–Toeplitz algebra is a C*-algebra generated by isometries with mutually orthogonal ranges. We consider the automorphism group of the Cuntz–Toeplitz algebra and compute its homotopy groups. In this talk, we would like to introduce the above result and explain its relation between M. Dadarlat’s work about Cuntz algebras.

SUMMARY: Following the results known in the case of a finite abelian group action on C*-algebras we prove the following two theorems;

(1) an inclusion \(P\subset A\) of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property).

(2) an inclusion \(P\subset A\) of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property).

SUMMARY: We study structures of Polish groups which arise as closed subgroups of the unitary group on an infinite-dimensional Hilbert space.

SUMMARY: I will explain that, if the closed unit ball of a normed space \(X\) has sufficiently many extreme points, then every mapping \(\Phi \) from \(X\) into itself with the following property is affine: For any pair of points in \(X\), there exists a (not necessarily linear) surjective isometry on \(X\) that coincides with \(\Phi \) at the two points. We also consider properties of such a mapping in the setting of C\(^*\)-algebras.

SUMMARY: Let \(A,B\subset M\) be inclusions of \(\sigma \)-finite von Neumann algebras such that \(A\) and \(B\) are images of faithful normal conditional expectations. In this article, we investigate Popa’s intertwining condition \(A\preceq _MB\) using their modular actions. In the main theorem, we prove that if \(A\preceq _MB\) holds, then an intertwining element for \(A\preceq _MB\) also intertwines some modular flows of \(A\) and \(B\). As a result, we deduce a new characterization of \(A\preceq _MB\) in terms of their continuous cores. Using this new characterization, we prove the first W\(^*\)-superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components.

SUMMARY: Let \(A\) be a simple separable nuclear C\(^*\)-algebra with a unique tracial state and no unbounded traces, and let \(\alpha \) be a strongly outer action of a finite group \(G\) on \(A\). We show that \(\alpha \otimes \mathrm {id}\) on \(A\otimes \mathcal {W}\) has the Rohlin property.

SUMMARY: We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III\(_1\) with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type III\(_1\).

SUMMARY: In this talk, we present a method to represent the dimension group of the core of the C*-algebra associated with self-similar maps using model traces. In particular, for the case of Sierpinski Gasket, the \(\mathrm K_0\) group of the core is isomorphic to \(\mathbb Z^{\infty }\), and the canonical endomorphism on the \(\mathrm K_0\) group is isomorphic to a unilateral shift of multiplicity 3.

SUMMARY: Anari, Oveis Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. In this talk, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial of a complete graph can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space. Furthermore, we prove that an algebra associated to a graphic matroid satisfies the strong Lefschetz property at degree one.

SUMMARY: The weighted power and Heron means are well known as generalizations of the weighted arithmetic, geometric and harmonic ones, and also the logarithmic and Heinz means are known as kinds of non-weighted means. Recently, Pal, Singh, Moslehian and Aujla introduced the weighted logarithmic mean of two positive numbers or operators.

In this talk, we propose the notion of a transpose symmetric path of weighted \(\mathfrak {M}\)-means for a symmetric operator mean \(\mathfrak {M}\), and we introduce a new family of operator means including the weighted logarithmic mean by Pal et al. This family also includes the weighted Heron mean, and newly produces the weighted Heinz mean.

SUMMARY: In this talk, we present some inequalities involving operator decreasing functions and operator means. These inequalities provide some reverses of operator Aczél inequality dealing with the weighted geometric mean.

SUMMARY: In this talk, we discuss a difference counterpart to the information monotonicity and variants of Ando–Hiai type inequality for operator power means due to Lawson–Lim–Pálfia.

SUMMARY: Some means can be expressed as algebraic midpoints. For example, the geometric mean can be expressed as the gyromidpoint of a gyrgroup. In this talk, we study gyrogroups for means.

SUMMARY: Let \(A\) and \(B\) be strictly positive operators on a Hilbert space, \(n\in \mathbb {N}\) and \(x\in \mathbb {R}\). A path \(A\ \natural _x\ B \equiv A^{\frac {1}{2}}(A^{\frac {-1}{2}}BA^{\frac {-1}{2}})^x A^{\frac {1}{2}}\) passing through \(A\) and \(B\). We have defined the \(n\)-th relative operator entropy \(S^{[n]}(A|B)\equiv \frac {1}{n!}A^{\frac {1}{2}} (\log A^{\frac {-1}{2}}BA^{\frac {-1}{2}})^n A^{\frac {1}{2}}\) and the \(n\)-th Tsallis relative operator entropy \(T^{[1]}_x(A|B) \equiv \frac {A\ \natural _x\ B-A}{x}\) and \(T^{[n]}_x(A|B) \equiv \frac {T^{[n-1]}_x(A|B)-S^{[n-1]}(A|B)}{x}\) for \(n\geq 2\). We have also introduced the \(n\)-th Petz–Bregman divergence \(D_{FK}^{[n]}(A|B) \equiv T^{[n]}_{1}(A|B) - S^{[n]}(A|B)\). In this talk, we regard the differences between the \(n\)-th relative operator entropies as \(n\)-th operator divergences and show relations between these \(n\)-th operator divergences and the \(n\)-th Petz–Bregman divergence.