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80 papers · 1 filter
Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
Satoru Odake
Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum…
Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
Satoru Odake
We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)(-)Hahn, three kinds of -Krawtchouk and -Meixner). Based on them and…
Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics
Aina Mayumi, Gen Kimura, Hiromichi Ohno +1
By employing a weighted Frobenius norm with a positive matrix , we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of t…
Type II Multi-indexed Little -Jacobi and Little -Laguerre Polynomials
Satoru Odake
For the isospectral Darboux transformations of the discrete quantum mechanics with real shifts, there are two methods: type I and type II constructions. Based on the type I constru…
New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
Satoru Odake
The Hamiltonians of finite type discrete quantum mechanics with real shifts are real symmetric matrices of order . We discuss the Darboux transformations with higher degree ($…
Delocalized spectra of Landau operators on helical surfaces
Yosuke Kubota, Matthias Ludewig, Guo Chuan Thiang
On a flat surface, the Landau operator, or quantum Hall Hamiltonian, has spectrum a discrete set of infinitely degenerate Landau levels. We consider surfaces with asymptotically co…
Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
Satoru Odake
The discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts are investigated. We show that the discre…
The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
Yasumichi Matsuzawa, Akito Suzuki, Yohei Tanaka +2
It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined…
Constraints for the spectra of generators of quantum dynamical semigroups
Dariusz Chruscinski, Ryohei Fujii, Gen Kimura +1
Motivated by a spectral analysis of the generator of completely positive trace-preserving semigroup, we analyze a real functional $$ A,B \in M_n(\mathbb{C}) \to r(A,B) = \frac{1}{2…
The bulk-dislocation correspondence for weak topological insulators on screw-dislocated lattices
Yosuke Kubota
A weak topological insulator in dimension is known to have a topologically protected gapless mode along the screw dislocation. In this paper we formulate and prove this fact wi…
The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution
Keisuke Asahara, Daiju Funakawa, Motoki Seki +1
Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a pair of a unitary time-evolution and…
A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators
Yohei Tanaka
In this paper we shall focus on one-dimensional strictly local operators, the notion of which naturally arises in the context of discrete-time quantum walks on the one-dimensional…
Free Oscillator Realization of the Laguerre Polynomial
Satoru Odake
We revisit the radial oscillator from the free oscillator realization point of view. By using a free oscillator, namely the creation/annihilation operators of the harmonic oscillat…
Parameterization of quantum walks on cycles
Shuji Kuriki, Md Sams Afif Nirjhor, Hiromichi Ohno
This study investigate the unitary equivalence classes of quantum walks on cycles. We show that unitary equivalence classes of quantum walks on a cycle with vertices are parame…
Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
Satoru Odake
Corresponding to a certain Wronskian identity, we present two types of new Casoratian identities. We apply these identities to the Darboux transformations of quantum mechanical sys…
Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types
Satoru Odake
In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, Askey-Wilson, Racah and -Racah types. In t…
Explicit Diagonalization of Pair Interaction Models
Yasumichi Matsuzawa, Itaru Sasaki, Kyosuke Usami
We provide a general method for constructing bosonic Bogoliubov transformations that diagonalize a general class of quadratic Hamiltonians. These Hamiltonians describe the pair int…
Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
Satoru Odake
We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of the coordinate is the whole real line. These systems a…
The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins
Akito Suzuki, Yohei Tanaka
Chirally symmetric discrete-time quantum walks possess supersymmetry, and their Witten indices can be naturally defined. The Witten index gives a lower bound for the number of topo…
Continuous limits of linear and nonlinear quantum walks
Masaya Maeda, Akito Suzuki
In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove tha…
Time operators for continuous-time and discrete-time quantum walks
Daiju Funakawa, Yasumichi Matsuzawa, Akito Suzuki +2
We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discr…
On the Feynman path integral for the magnetic Schroedinger equation with a polynomially growing electromagnetic potential
Wataru Ichinose
The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For exa…
Dual Polynomials of the Multi-Indexed (-)Racah Orthogonal Polynomials
Satoru Odake
We consider dual polynomials of the multi-indexed (-)Racah orthogonal polynomials. The -indexed (-)Racah polynomials satisfy the second order difference equations and vari…
Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations
Toru Fuda, Daiju Funakawa, Akito Suzuki
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in \cite{HiSeSu}. In this paper, as an application of the spectral mappin…