2017年度秋季総合分科会(於:山形大学)

SUMMARY: We study the quantum dynamics of a charged particle in the plane in the presence of a periodically pulsed magnetic field perpendicular to the plane. We show that by controlling the cycle when the magnetic filed is switched on and off appropriately, the result of the asymptotic completeness of wave operators can be obtained under the assumption that the potential \(V\) satisfies the decaying condition \(|V(x)| \leq C (1+|x|)^{-\rho } \), for some \(\rho >0\).

SUMMARY: Let \(X\) be a smooth manifold and \(Y\) a smooth submanifold of \(X\). Take \(G' \subset G\) to be a pair of Lie groups that act transitively on \(Y \subset X\), respectively. Suppose that \(\mathcal {V}\to X\) and \(\mathcal {W}\to Y\) are \(G\)- and \(G'\)-equivariant vector bundles over \(X\) and \(Y\) with fibers \(V\) and \(W\), respectively. Then we call a differential operator \(\mathcal {D} \colon C^\infty (X,\mathcal {V}) \to C^\infty (Y,\mathcal {W})\) between the spaces of smooth sections a differential symmetry breaking operator (differential SBO) if \(\mathcal {D}\) is \(G'\)-intertwining.

In the last year, for the setting \((G,G',V,W) = (O(n+1,1), O(n,1), \wedge ^i(\mathbb {C}^n), \wedge ^j(\mathbb {C}^{n-1}))\) with \(n\geq 3\), we completely classified the differential SBOs with their explicit formulas. In other words, for any \(0\leq i \leq n\) and \(0\leq j \leq n-1\), we classified all the differential SBOs \(\mathcal {D}^{i \to j}\colon \mathcal {E}^i(S^n) \to \mathcal {E}^j(S^{n-1})\) from the space of differential \(i\)-forms \(\mathcal {E}^i(S^n)\) over the standard Riemann sphere \(S^n\) to that of differential \(j\)-forms \(\mathcal {E}^j(S^n)\) over the totally geodesic hypersphere \(S^{n-1}\). In this talk we would like to discuss how we classify such operators. This is a joint work with T. Kobayashi and M. Pevzner.

SUMMARY: Deformation/rigidity theory is initiated by S. Popa in 2001 to study non-amenable von Neumann algebras. Amenable von Neumann algebras naturally appear in physics and they were extensively studied. The study of non-amenable algebras also attracted attention but very few had been known until 1990s. Deformation/rigidity theory is a great success in the study of non-amenable algebras. It brought much progress and in fact solved lots of open problems for non-amenable algebras.

Most of technologies used in this new theory require so-called a trace (a generalization of traces on matrices). Since this requirement is crucial, the above development is mostly restricted to algebras with traces. Von Neumann algebras without any traces are called type III. For example, von Neumann algebras in physics are always of type III. They also appear in many other context such as ergodic theory, quantum groups, free probability theory etc.

In this talk, I focus on type III algebras, particularly non-amenable type III algebras. To study them, it is important to find a way of applying technologies of deformation/rigidity theory to type III algebras. This problem has been mainly studied by C. Houdayer and myself. I will survey recent progress on this problem.

SUMMARY: The logarithmic representation of invertible evolution families is introduced in Ref.[1]. The nonlinearity related to the logarithmic representation is discussed in terms of its similarity with the Coles–Hopf transform.

Ref: [1] Yoritaka Iwata, “Infinitesimal generators of invertible evolution families”, Methods Funct. Anal. Topology 23 1 (2017) 26–36.

SUMMARY: We consider \(N\)-body Stark Hamiltonians. By Herbst et al, it is shown that both pure point spectrum and singular continuous spectrum of \(N\)-body Stark Hamiltonians are empty. Their method is based on Mourre’s positive commutator method. We prove this result by applying the theory of generalized canonical commutation relations.

SUMMARY: We study multistate Schrödinger operators related to molecular dynamics. We consider potentials which do not necessarily decay including those homogeneous of degree zero. We prove absence of the singular continuous spectrum and propagation estimates which mean the scattering at speed larger than a positive constant and decay of the states with potentials higher than considered energy at infinity. We also consider the multistate Schrödinger operators with many-body structures. We obtain the Mourre estimate and the minimal velocity estimate for the many-body operators. The lower bound of the velocity is determined by the distance between the energy and thresholds below the energy.

SUMMARY: We consider an interior transmission eigenvalue (ITE) problem on some compact \(C^{\infty } \)-Riemannian manifolds with smooth boundary. In particular, we do not assume that two domains are diffeomorphic, but we impose some conditions of Riemannian metrics and indices of refraction on the boundary. Then we prove the discreteness of the set of ITEs, the existence of infinitely many ITEs, and its Weyl type lower bound.

SUMMARY: In this talk, we prove the Strichartz estimates for the Schrödinger equations with a harmonic potential with a time-decaying coefficient by introducing the time weighted Lebesgue space.

SUMMARY: In this talk, we study the spectral structure of periodic Schrödinger operators on zigzag nanotubes with multiple chemical bonds. Utilizing the Floquet–Bloch theory for the corresponding quantum graph, we see that the spectrum has the band-gap structure. Namely, the spectrum consists of flat band and the absolutely continuous spectrum. In the talk, we see the difference between the case of the single bond and the multiple bonds.

SUMMARY: We consider two-point boundary value problems for bending of a beam supported by uniformly distributed springs with spring constant \(q>0\) on a fixed floor under tension \(p>0\). The tension is relatively strong, that is \((p/2)^2>q\). We have treated periodic, Dirichlet, Dirichlet–Neumann and Neumann boundary conditions and found their Green functions. As an application, we have found the best constants of the corresponding Sobolev inequality, which are equal to the maximum of diagonal values of Green functions.

SUMMARY: We first study the spectral properties of two relativistic Hamiltonians; one is the square root of a Pauli operator with an electric potential growing polynomially at infinity, and the other differs from it only in the sign of the potential. We next give a resonance free region for the latter. Moreover, we show that resonances (eigenvalues) of each of them converge to resonances (eigenvalues) of the corresponding Pauli operators with the same potential in the nonrelativistic limit.

SUMMARY: We investigate resonance free regions of Dirac operators with a bounded magnetic potential and an electric potential diverging at infinity with the help of the dilation analytic method and the FW transform. In this work two square roots of \(c\)-dependent Pauli operators play an important role.

SUMMARY: We introduce the condition related to the peripheral spectrum and the multiplication for maps between uniform algebras to be linear and isometric. We can describe maps related to algebra isomorphisms by some property of peripheral spectra and multiplication. We also generalize weakly peripherally-multiplicative maps and peripherally monomial-preserving maps and give some examples.

SUMMARY: Let \(C^1([0,1])\) be a linear space of all continuously differentiable complex valued functions on the closed unit interval \([0,1]\). \(C^1([0,1])\) is a Banach space with respect to the following norms: \(\displaystyle \| f \|_C = \sup _{t \in [0,1]} (|f(t)| + |f'(t)|)\), \(\| f \|_\Sigma = \| f \|_\infty + \| f' \|_\infty \) and \(\| f \|_\sigma = |f(0)| + \| f' \|_\infty \), where \(\| \cdot \|_\infty \) denotes the supremum norm on \([0,1]\). We give the characterization of surjective isometries on \(C^1([0,1])\) with respect to the above norms.

SUMMARY: For the M\(\rm \ddot o\)bius gyrovector spaces introduced by A. A. Ungar, we reveal the structure of finitely generated gyrovector subspaces, present a notion of orthogonal gyrodecomposition with respect to any gyrovector subspace which is closed under the Poincare metric. Moreover, we show a concrete procedure to obtain orthogonal gyroexpansion in a M\(\rm \ddot o\)bius gyrovector space, like as the classical orthogonal expansion in a Hilbert space.

SUMMARY: We study unital homomorphisms on admissible quadruples. In particular, we exhibit results that every unital homomorphism between admissible quadruples with certain conditions on maximal ideal spaces is of type BJ.

SUMMARY: Immanants are generalizations of the determinant and the permanent, and are labeled by Young diagrams. The limit of immanants of a correlation matrix is an interesting problem in terms of inequality problems, whose origins are Schur’s inequality and Lieb’s permanental dominance conjecture. In this talk, we give some results of the limit of immanants depending on the arms and legs of the Young diagrams, applying the Littlewood–Richardson rule, which is one of the most important property to describe the representations of the symmetric group. Also, we observe the behavior of the Littlewood–Richardson rule that becomes simple under some conditions.

SUMMARY: In this talk, the authors prove the existence of spectral degeneracies in the asymmetric quantum Rabi model (AQRM) when the symmetry-breaking parameter \(\epsilon \) is a half-integer. The degeneracy had been previously established for the case \(\epsilon = \frac {1}{2}\) by Wakayama (2017) and verified experimentally the general case by Li and Batchelor in 2015. The main result is established by the study of certain (so-called constraint) polynomials appearing from the finite-dimensional irreducible representations of \(\mathfrak {sl}_2(\mathbb {R})\) in the representation theoretical picture of the AQRM. Two independent proofs are given of the main result, each one giving a better understanding of the structure of the spectrum of the AQRM.

SUMMARY: We study elementary spherical functions on a non-compact real simple Lie group of finite center associated with a small \(K\)-type in the sense of Wallach. We prove that in most cases, the radial parts of elementary spherical functions for small \(K\)-types are written by hypergeometric functions of Heckman and Opdam. As an application, we give explicit formulae for Harish-Chandra’s \(\boldsymbol {c}\)-functions for small \(K\)-types and obtain the inversion formulae for the spherical transforms for small \(K\)-types.

SUMMARY: In this talk the speaker presents the result on the explicit construction of embedding maps between two holomorphic discrete series representations. Today we mainly deal with the embedding of the holomorphic discrete series representation of \(U(s',s'')\) into that of \(Sp(s,\mathbb {R})\), where \(s=s'+s''\).

SUMMARY: In this talk, by virtue of the matrix geomtric mean and the polar decomposition, we present new Wielandt type inequalities for matrices of any size. To this end, based on results due to J. I. Fujii, we reform a matrix Cauchy–Schwarz inequality, which differs from ones due to Marshall and Olkin.

SUMMARY: Tsallis relative operator entropy was firstly formulated by Fujii and Kamei as an operator version of Uhlmann’s relative entropy. Afterwards, Yanagi, Kuriyama and Furuichi reformulated Tsallis relative operator entropy as an operator version of Tsallis relative entropy. In this talk, we define Tsallis relative operator entropy with negative parameters of (non-invertible) positive operators on a Hilbert space and show some properties.

SUMMARY: Let \(A\) and \(B\) be bounded positive invertible operators on a Hilbert space and let \(\Psi _{A,B}(t)\equiv A\ \natural _t \ B\) be an operator valued smooth function, where \(A\ \natural _t \ B\equiv A^{\frac {1}{2}} ( A^{-\frac {1}{2}}BA^{-\frac {1}{2}})^t A^{\frac {1}{2}}\) \((t\in {\mathbb R})\) is a path passing through \(A\) and \(B\). We consider the following functions \(\Psi ^{[n]}_{A,B}: {\mathbb R}^2\to B(\mathcal {H})\): \(\Psi ^{[1]}_{A,B}(x,y)\equiv \frac {\Psi _{A,B}(x) - \Psi _{A,B}(y)}{x-y}\) and \(\Psi ^{[n]}_{A,B}(x,y)\equiv \frac {\Psi ^{[n-1]}_{A,B}(x,y)-\Psi ^{[n-1]}_{A,B}(y,y)}{x-y}\) (\(n\geq 2\)). Since Petz–Bregman divergence \(D_{FK}(A|B)\) can be represented by \(\Psi ^{[1]}_{A,B}(1,0)-\Psi ^{[1]}_{A,B}(0,0)\), we can give the \(n\)-th Petz–Bregman divergence \(D^{[n]}_{FK}(A|B)\equiv \Psi ^{[n]}_{A,B}(1,0)-\Psi ^{[n]}_{A,B}(0,0)\). Moreover, we treat the \(n\)-th divergences related with some operator valued divergences defined by the difference between the relative operator entropies.

SUMMARY: As generalizations of the arithmetic and geometric means for positive real numbers \(a\) and \(b\), the power difference mean \(J_{q}(a,b)=\frac {q}{q+1}\frac {a^{q+1}-b^{q+1}}{a^{q}-b^{q}}\), the Lehmer mean \(L_{q}(a,b)=\frac {a^{q+1}+b^{q+1}}{a^{q}+b^{q}}\), and the Heron mean \(K_{q}(a,b)=(1-q)\sqrt {ab}+q\frac {a+b}{2}\) are well known. Recently, we have shown estimations of the power difference mean by the Heron mean.

In this talk, similarly to these results, we get estimations of the Lehmer mean by the Heron mean. In other words, we obtain the greatest value \(\alpha =\alpha (q)\) and the least value \(\beta =\beta (q)\) such that the double inequality \(K_{\alpha }(a,b)<L_{q}(a,b)<K_{\beta }(a,b)\) holds for any \(q \in \mathbb {R}\). We can also obtain operator inequalities for bounded linear operators on a Hilbert space.

SUMMARY: In this talk, we shall consider an extension of the Petz–Hasegawa function. In fact, we shall give upper and lower bounds, and operator monotonicity of this function with elementary proofs.

SUMMARY: Let \(a_1\ge a_2 \ge \ldots \ge a_K>0\) and \(b_1 \ge b_2 \ge \ldots \ge b_K>0\). We consider the function \[ g(x) = \prod _{i=1}^K \frac {\sinh a_ix}{\sinh b_ix}. \] It is kwnown that the positive definiteness of such functions are related to some operator norm inequalities. In the case of \(K=2\), \(g\) is positive definite if and only if \(a_1\le b_1\) and \(a_1+a_2\le b_1+b_2\). Unfortunately we do not know such an equivalent condition when \(K\ge 3\). We consider the condition which related to the positive definiteness of \(g\).

SUMMARY: We will consider the asymptotic toeplitzness associated with weighted composition operators on the Hardy–Hilbert space \(H^2\).

SUMMARY: P. R. Halmos has presented his famous results of a classification of 2 subspaces under unitary equivalence. In this talk we discuss two subspaces under more weak equivalence. We point out that operator ranges are crucially important to study a classification of 2 subspaces under more weak equivalence. Under this equivalence, we give continuously many non-isomorphic examples of systems of 2 subspaces. We also give a relation between operator ranges, Hilbert representations of \(A_{2}\) Dynkin quiver and particular systems of 3 subspaces.

SUMMARY: In this talk, we present analysis of the core of the C*-algebras associated with self-similar maps associated with Sierpinski carpet. Although, the branch set if a infinite set and the structure of branching is complex for Sierpinski carpet, we can classify finite traces and ideals of the core, and can describe the matrix representation of the core.

SUMMARY: We consider the K-theory of the group and subgroup \(C\)*-algebras of the special or general linear groups over the ring of integers and of their canonical subgroups. We further consider the K-theory of the associated, crossed product \(C\)*-algebras.

SUMMARY: A Smale space is a hyperbolic dynamical system with local product structure. D. Ruelle constructed C*-algebras from Smale spaces. The algebras are regarded as higher dimensional analogues of Cuntz–Krieger algebras. We introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their étale groupoids and their asymptotic Ruelle algebras with their dual actions, respectively.

SUMMARY: We show that pure infiniteness of reduced crossed product is inherited to minimal extensions.

SUMMARY: In this tallk, we classify trace scaling automorphisms of \(\mathcal {W}\otimes \mathbb {K}\) up to outer conjugacy, where \(\mathcal {W}\) is a certain simple separable nuclear stably projectionless C\(^*\)-algebra having trivial \(K\)-groups.

SUMMARY: In the recent breakthrough by Tikuisis, White, and Winter, it is shown that the universal coefficient theorem and nuclearity imply quasidiagonality for separable \(\mathrm {C}^*\)-algebras. Precisely, they showed a faithful tracial state is quasi-diagonal under the natural assumption required in the classification theory. In this talk, we explain a technical ingredient in their proof and several alternative approaches to obtaining quasidiagonality.

SUMMARY: When a von Neumann algebra acts on a Hilbert space, the relative tensor product of the Hilbert spaces is defined. The notion of the relative tensor product was introduced by Alain Connes. There are two ways which we define it by changing left or right Hilbert space into the operator space. We call them the left and right relative tensor products respectively. We will show that the two categories consisting of all bimodules (i.e. Hilbert spaces on which von Neumann algebras act from left and right) with left and right relative tensor products are equivalent. This is a joint work with Shigeru Yamagami.