arXiv subjects

Tatsuya Tate

Publications and source records attributed to Tatsuya Tate.

15 recordsLinked to original sources

arXiv2026

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution

The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach enables us to compare the horizontal Laplacian with the usual Laplacian on a Riemannian covering over the base manifold, and, when the holonomy group is infinite and amenable, we prove a coincidence of the essential spectrum, which strengthen, in our special setup, a result due to Kordyukov in the context of geometric analysis on foliated manifolds.

math.DG↗

arXiv2021

An eigenfunction expansion formula for one-dimensional two-state quantum walks

The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional 2-state quantum walks, which is an analog of that for Sturm-Liouville operators due to Weyl, Stone, Titchmarsh and Kodaira. In the context of the theory of CMV matrix it had been already established by Gesztesy-Zinchenko. Our approach is restricted to the class of quantum walks mentioned above whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model.

math-ph↗

arXiv2019

Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for $3$- and $4$-valent graphs

We are concerned with spectral problems of the Goldberg-Coxeter construction for $3$- and $4$-valent finite graphs. The Goldberg-Coxeter constructions $\mathrm{GC}_{k,l}(X)$ of a finite $3$- or $4$-valent graph $X$ are considered as "subdivisions" of $X$, whose number of vertices are increasing at order $O(k^2+l^2)$, nevertheless which have bounded girth. It is shown that the first (resp. the last) $o(k^2)$ eigenvalues of the combinatorial Laplacian on $\mathrm{GC}_{k,0}(X)$ tend to $0$ (resp. tend to $6$ or $8$ in the $3$- or $4$-valent case, respectively) as $k$ goes to infinity. A concrete estimate for the first several eigenvalues of $\mathrm{GC}_{k,l}(X)$ by those of $X$ is also obtained for general $k$ and $l$. It is also shown that the specific values always appear as eigenvalues of $\mathrm{GC}_{2k,0}(X)$ with large multiplicities almost independently to the structure of the initial $X$. In contrast, some dependency of the graph structure of $X$ on the multiplicity of the specific values is also studied.

math.CO↗

arXiv2017

Up and down grover walks on simplicial complexes

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatorial structures of simplicial complexes. But the shift operators are modified so that it can contain information of orientations of each simplex in the simplicial complex. It is well-known that the spectral structures of this kind of unitary operators are completely determined by its discriminant operators. It has strong relationship with combinatorial Laplacian on simplicial complexes and geometry, even topology, of simplicial complexes. In particular, theorems on a relation between spectrum of up and down discriminants and orientability, on a relation between symmetry of spectrum of discriminants and combinatorial structure of simplicial complex are given. Some examples, both of finite and infinite simplicial complexes, are also given. Finally, some aspects of finding probability and stationary measures are discussed.

math.SP↗

arXiv2017

Eigenvalues of quantum walks of Grover and Fourier types

A necessary and sufficient conditions for certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of $\pm 1$ as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has $1$ as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general criterion for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics such as Grover walks on the triangular lattice, products and deformations of Grover walks are also discussed.

math.FA↗

arXiv2017

Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a complex vector space, which is a generalization of the discrete-time quantum walks with constant coin matrices, are discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of the periodic unitary transition operators is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and the long time average of the transition probabilities coincides with the point-wise norm of the projection of the initial state to the direct sum of eigenspaces.

math.FA↗

arXiv2013

The Hamiltonians generating one-dimensional discrete-time quantum walks

An explicit formula of the Hamiltonians generating one-dimensional discrete-time quantum walks is given. The formula is deduced by using the algebraic structure introduced previously. The square of the Hamiltonian turns out to be an operator without, essentially, the `coin register', and hence it can be compared with the one-dimensional continuous-time quantum walk. It is shown that, under a limit with respect to a parameter, which expresses the magnitude of the diagonal components of the unitary matrix defining the discrete-time quantum walks, the one-dimensional continuous-time quantum walk is obtained from operators defined through the Hamiltonians of the one-dimensional discrete-time quantum walks. Thus, this result can be regarded, in one-dimension, as a partial answer to a problem proposed by Ambainis.

math.FA↗

arXiv2013

An algebraic structure for one-dimensional quantum walks and a new proof of the weak limit theorem

An algebraic structure for one-dimensional quantum walks is introduced. This structure characterizes, in some sense, one-dimensional quantum walks. A natural computation using this algebraic structure leads us to obtain an effective formula for the characteristic function of the transition probability. Then, the weak limit theorem for the transition probability of quantum walks is deduced by using simple properties of the Chebyshev polynomials.

math.FA↗

arXiv2011

Asymptotic behavior of quantum walks on the line

This paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line. The formulae depend heavily on the `normalized' position of the walk. When the position is in the support of the weak-limit distribution obtained by Konno, one observes, in addition to the limit distribution itself, an oscillating phenomenon in the leading term of the asymptotic formula. When the position lies outside of the support, one can establish an asymptotic formula of large deviation type. The rate function, which expresses the exponential decay rate, is explicitly given. Around the boundary of the support of the limit distribution (called the `wall'), the asymptotic formula is described in terms of the Airy function.

math.PR↗

arXiv2010

Asymptotic Euler-Maclaurin formula over lattice polytopes

An asymptotic expansion formula of Riemann sums over lattice polytopes is given. The formula is an asymptotic form of the local Euler-Maclaurin formula due to Berline-Vergne. The proof given here for Delzant lattice polytopes is independent of the local Euler-Maclaurin formula. But we use it for general lattice polytopes. As corollaries, an explicit formula for each term in the expansion over Delzant polytopes in two dimension and an explicit formula for the third term of the expansion for Delzant polytopes in arbitrary dimension are given. Moreover, some uniqueness results are given.

math.CO↗

arXiv2009

A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

An asymptotic formula for the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold is given under a integrality assumption on the spectrum of the Hamiltonian. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer.

math.FA↗

arXiv2008

Bernstein measures on convex polytopes

We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard $m$-simplex obtained by Abel-Ivan and Hörmander. These are different from the Bergman-Bernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch's Bergman-Bernstein approximations.

math.FA↗

arXiv2008

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of certain operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

math.RT↗

arXiv2003

Distribution laws for integrable eigenfunctions

We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.

math.CV↗

arXiv2003

Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers

We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers $V_λ^{\otimes N}$ of an irreducible representation $V_λ$ of a compact connected Lie group $G$. The weights are allowed to depend on $N$, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope.

math.RT↗