2019年度年会(於:東京工業大学)

SUMMARY: In this talk, we derive a Weyl-type lower bound for non-scattering energies (NSEs) of time-harmonic acoustic equations. A NSE is equivalent to the eigenvalue \(1\) of the S-matrix associated with the acoustic equation. Thus there exists an eigenspace of the S-matrix. If an energy is a NSE, there exists an incident wave such that its scattered wave vanishes. As far as the author knows, results on the existence of NSEs are scarce. Essentially, there are two cases. First one is the existence of infinitely many NSEs, when the inhomogeneity is compactly supported and spherically symmetric. On the other hand, if the inhomogeneity has suitable corners, it is known that the set of NSEs is empty.

In this talk, we consider a case where inhomogeneities exist in a bounded domain with smooth boundary. We assume that the index of refraction has a suitable discontinuity on the boundary. Then we show that there exists infinitely many NSEs with possible accumulation points are zero and infinity. Moreover, the number of NSEs satisfies a Weyl-type asymptotic lower bound at infinity. Our result is an application of the same type estimate for interior transmission eigenvalues. For the proof, we study the Dirichlet-to-Neumann (D-N) map. It is well-known that the S-matrix and the D-N map is equivalent. Then our problem can be reduced to the interior transmission eigenvalue problem.

SUMMARY: A unitary representation of a real reductive Lie group has unique irreducible decomposition. If the essential supremum of the multiplicities is finite, the representation is said to have uniformly bounded multiplicities. In this talk, we give several criteria for the uniform boundedness of multiplicities in the following three cases: restrictions of representations (branching laws); induced representations (harmonic analysis); and restrictions of parabolically induced representations. The criteria follow from a key result about an action of an algebra of invariant differential operators. We establish a relation between the essential supremum of multiplicities and an invariant of the algebra.

SUMMARY: I will overview how the quantum group techniques can be applied to the subfactors. First we observe that the classification of subfactors can be interpreted as a classification of actions of tensor categories, which can be seen as a slight generalization of actions of quantum groups and I will present some results on this direction. Then I explain approximation properties of tensor categories which is important in such classification.

SUMMARY: The notions of generalized Riesz systems and \(\cal D\)-quasi bases play an important rule for constructions physical operators. In this talk we introduce some results about the relationships between generalized Riesz systems and \(\cal D\)-quasi bases.

SUMMARY: We discuss the modified version of Euler Buckling problem as minimizing the problem by variational formulation. We discuss the smoothness of the problem and derive the equations of bifurcation set \(B\) and hysteresis set \(H\) up to order 3. From several numerical results, we draw approximations of \(B\) and \(H\) under suitable set-up and observe the change of \(B\) and \(H\). We observe that bifurcation set \(B\) and hysteresis set \(H\) have the same tangent at the origin and when length l is bigger then the narrow regions between bifurcation set and hysteresis set is also increasing.

SUMMARY: The dynamics of a quantum system is described by a certain type of differential equations on the set of states, and the solution is often assumed to be completely positive and trace-preserving. According to Choi, a completely positive operator has a particular representation so-called the Kraus representation. In this presentation, we will review some procedure to compute this type of representation for the solution of the differential equation. The point of this procedure is that it does not depend on the computations of eigenvalues and eigenvectors, which can be difficult if the dimension is high.

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. To this end, we have to differentiate the thermodynamic potential with respect to the temperature two times. Since there is the solution to the BCS-Bogoliubov gap equation in the form of the thermodynamic potential, we have to differentiate the solution with respect to the temperature two times. Therefore, we need to show that the solution to the BCS-Bogoliubov gap equation is differentiable with respect to the temperature two times as well as its existence and uniqueness. We carry out its proof on the basis of fixed point theorems.

SUMMARY: Kuramoto’s famous conjecture is that there is a positive constant \(K_c\) such that the system of oscillators becomes synchronized only when the coupling constant \(K>K_c\). H. Chiba proved this conjecture for the continuous version of Kuramoto model in 2015. His proof covers the cases that the initial distributions \(g(\omega )\) of frequencies of oscillators are Gaussian or Cauchy. His key tool is the precise analysis of the generalized eigenvalues of some unbounded linear operator \(T\) related to \(g(\omega )\) on some Hilbert space. This year, we get an asymptotic expansion of \((e^{tT}\varphi ,\psi )_g\) as \(t\to \infty \) directly for some much wider class of analytic distributions \(g(\omega )\). As a dirct application, we extend Chiba’s exponential decay result of synchronization parameter \(|\eta (t)|\) to such \(g(\omega )\).

SUMMARY: How to define self-adjoint time operators is an important open problem. In this talk we use a generalized commutation relation called the generalized weak Weyl relation (GWWR) to derive self-adjoint time operators for a free particle confined on a ring. Let \(\hat \pi _\theta \) be an angular momentum operator, \(\hat H=\hat \pi _\theta ^2/2I\) be a Hamiltonian operator and \(\hat f\) be a “periodic position operator”. We show that if \(\hat \pi _\theta \) and \(\hat f\) satisfy \([\hat \pi _\theta ,\hat f]=-i\hbar \hat {\mathscr C}\), then there exists a self-adjoint time operator which satisfies \([\hat H,\hat T]=i\hbar \hat {\mathscr C}\): These commutation relations are direct consequences of the GWWR and \(\hat T\) satisfies the GWWR. How to choose \(\hat f\) depends on the system of interest. We consider two specific examples, namely a time-of-arrival operator and the recently proposed quantum systems called “time crystal”. We surmise that time crystals are promising systems to define time operators.

SUMMARY: We consider a supersymmetric aspect of unitary operators with chiral symmetry and define an index for such a unitary operator when it has a Fredholm property. We give a criterion for the Fredholm property and an index formula in terms of a discriminant operator and birth eigenspaces.

SUMMARY: The existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition are proven. If the infrared singularity condition is imposed, then the absence of ground states is shown. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Byproducts of our analysis are a hypercontractivity bound for the semi-group and a new remark on Nelson’s operator theoretic renormalization procedure. We construct path measures associated with ground states of the renormalized Nelson operator whose analysis entails improved boson number decay estimates for ground state eigenvectors.

SUMMARY: We study the eigenvalues of the Zakharov–Shabat operator corresponding to the focusing nonlinear Schrödinger equation in the inverse scattering method. Although this operator is non-self-adjoint, all eigenvalues become purely imaginary when the potential is single-robe. We give an alternative approach to this fact by using the exact WKB method in the semiclassical limit. Moreover, we show that there are no purely imaginary eigenvalues when the potential is odd function and its square is double-robe.

SUMMARY: We discuss uniform bounds of the Birman–Schwinger operators in the discrete setting. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our results are weaker than in the continuous setting. As an application, we obtain unitary equivalence between the discrete Laplacian and the weakly coupled systems.

SUMMARY: Long-range scattering theory for discrete Schrödinger operators has been studied and it is known that Isozaki–Kitada modifiers, one of the modified wave operators, can be constructed for some kinds of lattices and that they give one-to-one correspondence between each scattering state of the unperturbed and perturbed operators. In this talk, We extend the above result to discrete Schrödinger operators on general lattices. In the proof, Isozaki–Kitada modifiers is given by the local construction of solution of the corresponding eikonal equations.

SUMMARY: Operator representation of Cole–Hopf transform is obtained based on the logarithmic representation of infinitesimal generators. For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of the spatial dimension, the Cole–Hopf transform is generalized to a transform between linear and nonlinear equations defined in Banach spaces. In conclusion a role of transform between the evolution operator and its infinitesimal generator is understood in the context of generating nonlinear semigroup.

[Ref] Y. Iwata, Methods Func. Anal. Topology, accepted; arXiv:1804.01338.

SUMMARY: We consider an arbitrary irreducible unitary representation \((\pi _{\lambda },V_{\lambda })\) of a compact semisimple Lie group, and apply the idea of Daubechies–Klauder (1985) and Yamashita (2011) on rigorous coherent-state path integrals to this representation. Our main theorem is two-fold: the first main theorem is in terms of Brownian motions and stochastic integrals, and proven using the Feynman–Kac–Itô formula on a vector bundle of a Riemannian manifold, due to Güneysu (2010). In the second main theorem, we consider a sequence \((\mu _{n})\) of finite measures on the space of smooth paths, and a ‘path integral’ is defined to be a limit of the integrals with respect to \((\mu _{n})\). The formulation and the proof of the second main theorem employ rough path theory originated by Lyons (1998).

SUMMARY: In the study of the Riemann zeta function, the functional equation plays a fundamental role, and it is well known that many kinds of zeta functions satisfy functional equations. M. Sato found that there is a big group action behind such functional equations, and reached the notion of prehomogeneous vector spaces. In this talk, we deal with zeta functions in several variables of solvable prehomogeneous vector spaces associated with homogeneous cones, and give an explicit formula of their functional equations.

SUMMARY: Let \(D_n\) be the \(n\)-dimensional Hermitian symmetric space of rank 2, realized as a bounded symmetric domain in \(\mathbb {C}^n\), and we consider a polynomial on a subspace \(\mathbb {C}^{n''}\). Then the speaker presents the result that the weighted Bergman inner product on \(D_n\) of an exponential function on \(\mathbb {C}^n\) and a polynomial on \(\mathbb {C}^{n''}\) is represented by using a hypergeometric polynomial. As an application, the speaker presents a result on construction of differential symmetry breaking operators from representations of \(SO_0(2,n)\) to those of \(SO_0(2,n')\times SO(n'')\).

SUMMARY: We give an explicit formula for the Harish-Chandra \(c\)-function for a small \(K\)-type on a noncompact real split Lie group of type \(G_2\). As an application we give an explicit formula for spherical inversion for a small \(K\)-type.

SUMMARY: We give a higher order analogue of the Pfaffian version of the Cayley–Hamilton theorem. Using this theorem, we describe the vector space \((\mathcal {P}(\Lambda _2(V)) \otimes \Lambda _2(V)^{\otimes r})^{Sp(V)}\). This result is similar to the description of the algebra \((\mathcal {P}(V \otimes V^*) \otimes (V \otimes V^*)^{\otimes r})^{GL(V)}\) using the Cayley–Hamilton theorem of higher order (the result reported at the last MSJ meeting).

SUMMARY: Let \(G\) be a real reductive Lie group and \(\mathcal {D}\) an intertwining differential operator for \(G\). Since \(\mathcal {D}\) respects the action of \(G\), the space \(\mathcal {S}ol(\mathcal {D})\) of solutions to \(\mathcal {D}\) is naturally a representation space of \(G\). In fact it is known that “smallest” infinite-dimensional representations, so-called minimal representations, are realized on such spaces. In this talk, with the motivation above, we discuss the space \(\mathcal {S}ol(\mathcal {D})_K\) of \(K\)-finite solutions to \(\mathcal {D}\). More prcisely, we first give a Peter–Weyl type decomposition theorem for \(\mathcal {S}ol(\mathcal {D})_K\). We then apply the decomposition theorem for the case \(G=\widetilde {SL}(3,\mathbb {R})\) to realize on \(\mathcal {S}ol(\mathcal {D})\) all irreducible unitary representations that are attached to the minimal nilpotent orbit.

SUMMARY: We are concerned with holomorphically induced representations \(\rho \) of exponential solvable Lie groups \(G\). Decomposing \(\rho \) into a direct integral of irreducible representations of \(G\), we discuss some reciprocity in distribution sense and give concrete examples.

SUMMARY: Let \(G\) be the split orthogonal group of degree \(2n\) over an arbitrary infinite field \(\mathbb {F}\) of characteristic not \(2\). In this talk, we introduce multiple flag varieties \(G/P_1\times \cdots \times G/P_k\) of finite type. Here a multiple flag variety is said to be of finite type if it has a finite number of \(G\)-orbits with respect to the diagonal action of \(G\).

SUMMARY: Let \(G\) be the split orthogonal group of degree \(2n\) over an arbitrary infinite field \(\mathbb {F}\) of characteristic not \(2\). In this talk, we classify multiple flag varieties \(G/P_1\times \cdots \times G/P_k\) of finite type. Here a multiple flag variety is said to be of finite type if it has a finite number of \(G\)-orbits with respect to the diagonal action of \(G\).

SUMMARY: We treat with a bifunction V(x,y) defined for any x,y in a smooth Banach space E, which gives a generalized projection in E, and we give a definition of a V-strongly non-expansive mapping T on E that is characterized by this bifunction V. The mapping T has a property that the class of this mapping T includes the class of firmly non-expansive mappings, and it is non-expansive in a Hilbert space. However, we could show that there exists a Banach space where the mapping T is not non-expansive mapping. This mapping T is elastic according to types of Banach spaces. In this talk, we shall introduce results with respect to fixed point theorems of this mapping T, which are some convergence theorems and existence theorems for fixed points of this mapping T.

SUMMARY: We introduce a framework of norms on \(C^1([0,1])\), and give the characterization of surjective isometries on it.

SUMMARY: We introduce a notion of generalized bi-circular idempotents on normed spaces. Then we characterize generalized bi-circular idempotents on \(C^1([0,1])\) with respect to several norms.

SUMMARY: We talk about surjective isometries on a Banach space of analytic functions on the open unit disc.

SUMMARY: We study 2-local surjective isometeries on certain spaces of complex-valued continuous functions. We do not assume linearity for the isometries. We mainly consider the Banach algebra of continuously differentiable functions on the closed unit interval with the sum norm.

SUMMARY: We present Cauchy–Bunyakovsky–Schwarz type inequalities related to the Möbius operations, which are extensions of what was obtained early in 2018.

SUMMARY: We consider a monotone map on von Neumann algebras with trivial center (factors). At first we characterize continuous functional calculus on factor. Using this fact, we can show that monotone maps on factor with some property are given by a operator monotone function.

SUMMARY: In this talk, we present some reverses of the Golden–Thompson type inequalities with Specht’s ratio and Olson order. The same inequalities are also provided with other constants. The obtained inequalities improve some known results.

SUMMARY: In this talk, we show norm inequalities related to the quasi geometric mean of negative power, the chaotic geometric mean and \(A^{1-\beta }B^{\beta }\) for positive definite matrices \(A,B\).

SUMMARY: Let \(A\) and \(B\) be bounded positive invertible operators on a Hilbert space, \(n\in \mathbb {N}\), \(\alpha \), \(\mu \in [0,1]\) and \(r\in [-1,1]\). We regard power mean \(A\ \sharp _{\mu ,r}\ B\equiv A^{\frac {1}{2}}\{(1-\mu )I+\mu (A^{\frac {-1}{2}}BA^{\frac {-1}{2}})^r\}^{\frac {1}{r}}A^{\frac {1}{2}}\) as a path connecting \(A\) and \(B\). As relative operator entropies on the path \(A\ \sharp _{\mu , r}\ B\),

\(\ \ \ \ \ \ \ \ \ \ S_{\alpha , r}(A|B)\equiv A^{\frac {1}{2}}\left ( \left \{ 1-\alpha +\alpha ( A^{\frac {-1}{2}}BA^{\frac {-1}{2}} )^r \right \}^{\frac {1}{r}-1}\cdot \frac {(A^{\frac {-1}{2}}BA^{\frac {-1}{2}})^r-I }{r} \right )A^{\frac {1}{2}} \)

and \(T_{\alpha , r}(A|B) \equiv \frac {A\ \sharp _{\alpha ,r}\ B - A}{\alpha }\) \((\alpha \neq 0)\), \(T_{0, r}(A|B) \equiv \lim _{\alpha \to +0} T_{\alpha , r}(A|B)\) are known. In this talk, we define the \(n\)-th relative operator entropies \(S^{[n]}_{\alpha , r}(A|B)\) and \(T^{[n]}_{\alpha , r}(A|B)\) based on the Taylor’s expansion of the path \(A\ \natural _{\mu , r}\ B\) and show some properties of them.

SUMMARY: Arveson have introduced the notion of product systems and classified E\(_0\)-semigroups on type I factors. We shall classify E\(_0\)-semigroups on a general von Neumann algebra by product systems of W\(^*\)-bimodules. The classification is reflected by Bhat–Skeide’s one.

SUMMARY: We introduce the notion of asymptotic flip conjugacy, which implies asymptotic continuous orbit equivalence,and show that flip conjugate Smale spaces are asymptotically flip conjugate. Several equivalent conditions of asymptotic flip conjugacy of Smale spaces in terms of their groupoids and their Ruelle algebras with dual actions are presented. We finally characterize the flip conjugacy classes of irreducible two-sided topological Markov shifts in terms of the associated Ruelle algebras with its \(C^*\)-subalgebras.

SUMMARY: Let \(P \subset A\) be an inclusion of \(\sigma \)-unital C*-algebras with a finite index in the sense of Pimsner–Popa. Then we introduce the Rokhlin property for a conditional expectation \(E\) from \(A\) onto \(P\) and show that if \(A\) is simple and satisfies any of the property like pure infiniteness, stable rank one, real rank zero, the nuclear dimension \(n\), \(\mathcal {D}\)-absorption for a strongly self-absorbing C*-algebra \(\mathcal {D}\), simplicity, \(AF\), \(AI\), \(AT\)-properties, the strict comparison property for Cuntz semigroup, and \(E\) has the Rokhlin property, then so does \(P\).

SUMMARY: The nuclear dimension is a relatively new concept introduced by E. Kirchberg, W. Winter, J. Zacharias. Currently, it is regarded as a central research subject of classification theory of nuclear C*-algebras and attracts the attention of many researchers. In my research so far, we have used the von Neumann algebraic approach to calculate the finiteness of nuclear dimensions and calculated it. In this presentation, we give a calculation method of nuclear dimensions by C*-algebraic theory and clarify the nature of nuclear dimensions and other associated ranks. Besides, we try to refine the calculation of nuclear dimensions for crossed products obtained from the dynamical system of C*-algebras.

SUMMARY: In this talk, we present a method to represent the dimension group of the core of the C*-algebra associated with the self similar map giving the Sierpinski gasket, which is a typical example of self similar figure, using model traces on the core. For the case of the Sierpinski gasket, we need values of generators of K-group of the core at the three series of model traces.