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Toshiaki Tanaka

Publications and source records attributed to Toshiaki Tanaka.

At least 19 recordsLinked to original sources
arXiv2014

Construction of Quasi-solvable Quantum Mechanical Matrix Models: Lie Superalgebra v.s. N-fold Supersymmetry

We construct quasi-solvable quantum mechanical matrix models by employing two different methods, the one is universal enveloping algebra of Lie superalgebra and the other is N-fold supersymmetry. For the former we examine the q(2) and osp(2/2) Lie-superalgebraic quasi-solvable matrix operators in the literature, and then compare them with the corresponding N-fold supersymmetric matrix systems. In the q(2) case, Lie-superalgebraic construction and the intertwining relation lead to the identical result. In the osp(2/2) case, however, some novel features emerge due to the difference in dimension of linear spaces which consist of the two-component invariant subspace. In both cases, the closure of N-fold superalgebra imposes stronger constraint on the admissible form of the systems and the concept of conjugation plays a key role in the formulation.

math-ph↗
arXiv2013

GL(3,C) Invariance of Type B 3-fold Supersymmetric Systems

Type B 3-fold supersymmetry is a necessary and sufficient condition for a quantum Hamiltonian to admit three linearly independent local solutions in closed form. We show that any such a system is invariant under GL(3,C) homogeneous linear transformations. In particular, we prove explicitly that the parameter space is transformed as an adjoint representation of it and that every coefficient of the characteristic polynomial appeared in 3-fold superalgebra is algebraic invariants. In the type A case, it includes as a subgroup the GL(2,C) linear fractional transformation studied in the literature. We argue that any N-fold supersymmetric system has a GL(N,C) invariance for an arbitrary integral N.

math-ph↗
arXiv2013

Type B 3-fold Supersymmetry and Non-polynomial Invariant Subspaces

We obtain the most general type B 3-fold supersymmetry by solving directly the intertwining relation. We then show that it is a necessary and sufficient condition for a second-order linear differential operator to have three linearly independent local analytic solutions. We find that there are eight linearly independent non-trivial linear differential operators of this kind. As a by-product, we find new quasi-solvable second-order operators preserving a monomial or polynomial subspace, one in type B, two in type C, and four in type X_2, all of which have been missed in the existing literature. In addition, we show that type A, type B, and type C 3-fold supersymmetries are connected continuously via one parameter. A few new quasi-solvable models are also presented.

math-ph↗
arXiv2012

N-fold Supersymmetric Quantum Mechanics with Reflections

We formulate N-fold supersymmetry in quantum mechanical systems with reflection operators. As in the cases of other systems, they possess the two significant characters of N-fold supersymmetry, namely, almost isospectrality and weak quasi-solvability. We construct explicitly the most general 1- and 2-fold supersymmetric quantum mechanical systems with reflections. In the case of N=2, we find that there are seven inequivalent such systems, three of which are characterized by three arbitrary functions having definite parity while the other four of which are by two. In addition, four of the seven inequivalent systems do not reduce to ordinary quantum systems without reflections. Furthermore, in certain particular cases, they are essentially equivalent to the most general two-by-two Hermitian matrix 2-fold supersymmetric quantum systems obtained previously by us.

math-ph↗
arXiv2012

N-fold Supersymmetry in Quantum Mechanical Matrix Models

We formulate N-fold supersymmetry in quantum mechanical matrix models. As an example, we construct general two-by-two Hermitian matrix 2-fold supersymmetric quantum mechanical systems. We find that there are two inequivalent such systems, both of which are characterized by two arbitrary scalar functions, and one of which does not reduce to the scalar system. The obtained systems are all weakly quasi-solvable.

math-ph↗
arXiv2012

Effect of Position-dependent Mass on Dynamical Breaking of Type B and Type X_2 N-fold Supersymmetry

We investigate effect of position-dependent mass profiles on dynamical breaking of N-fold supersymmetry in several type B and type X_2 models. We find that N-fold supersymmetry in rational potentials in the constant-mass background are steady against the variation of mass profiles. On the other hand, some physically relevant mass profiles can change the pattern of dynamical N-fold supersymmetry breaking in trigonometric, hyperbolic, and exponential potentials of both type B and type X_2. The latter results open the possibility of detecting experimentally phase transition of N-fold as well as ordinary supersymmetry at a realistic energy scale.

math-ph↗
arXiv2012

Two-step Shape Invariance in the Framework of N-fold Supersymmetry

We extensively investigate two-step shape invariance in the framework of N-fold supersymmetry. We first show that any two-step shape-invariant system possesses type A 2-fold supersymmetry with an intermediate Hamiltonian and thus has second-order parasupersymmetry as well. Employing the general form of type A 2-fold supersymmetry, we systematically construct two-step shape-invariant potentials. In addition to the well-known ordinary shape-invariant potentials, we obtain several new and novel two-step shape-invariant ones which are not ordinary shape invariant. Furthermore, some of the latter potentials are conditionally two-step shape invariant and thus are conditionally solvable.

math-ph↗
arXiv2011

On General Form of N-fold Supersymmetry

We analyze general structure of N-fold supersymmetry which provides a systematic framework to construct weakly quasi-solvable quantum mechanical systems. Main ingredients of our analysis are dimensional analysis and introduction of an equivalent class of linear differential operators associated with N-fold supersymmetry for each N. To illustrate how they work, we construct the most general form of N-fold supersymmetric systems for N=2, 3, and 4.

math-ph↗
arXiv2010

N-fold Supersymmetry and Quasi-solvability Associated with X_2-Laguerre Polynomials

We construct a new family of quasi-solvable and N-fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomial subspace of codimension 2. We show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the X_2-Laguerre polynomials of the second kind. In addition, we find that the two kinds of the X_2-Laguerre polynomials are ingeniously connected with each other by the N-fold supercharge.

math-ph↗
arXiv2010

Existence of Different Intermediate Hamiltonians in Type A N-fold Supersymmetry II. The N=3 Case

We continue the previous study on the existence of different intermediate Hamiltonians in type A N-fold supersymmetric systems and carry out an exhaustive investigation on the N=3 case. In contrast with the N=2 case, we find various patterns in the existence of intermediate Hamiltonians due to the presence of two different intermediate positions in a factorized type A 3-fold supercharge. In addition, all the N=3 models are strictly restricted to at most elliptic type, which enables us to make the complete classification of the systems which admit intermediate Hamiltonians. Finally, we show realizations of third-order parasupersymmetry and variant generalized 3-fold superalgebras by such systems.

hep-th↗
arXiv2009

Existence of Different Intermediate Hamiltonians in Type A N-fold Supersymmetry

Type A N-fold supercharge admits a one-parameter family of factorizations into product of N first-order linear differential operators due to an underlying GL(2,C) symmetry. As a consequence, a type A N-fold supersymmetric system can have different intermediate Hamiltonians corresponding to different factorizations. We derive the necessary and sufficient conditions for the latter system to possess intermediate Hamiltonians for the N=2 case. We then show that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and a generalized 2-fold superalgebra. As an illustration, we construct a set of generalized Poeschl--Teller potentials of this kind.

hep-th↗
arXiv2007

N-fold Parasupersymmetry

We find a new type of non-linear supersymmetries, called N-fold parasupersymmetry, which is a generalization of both N-fold supersymmetry and parasupersymmetry. We provide a general formulation of this new symmetry and then construct a second-order N-fold parasupersymmetric quantum system where all the components of N-fold parasupercharges are given by type A N-fold supercharges. We show that this system exactly reduces to the Rubakov-Spiridonov model when N=1 and admits a generalized type C 2N-fold superalgebra. We conjecture the existence of other `N-fold generalizations' such as N-fold fractional supersymmetry, N-fold orthosupersymmetry, and so on.

hep-th↗
arXiv2007

Parasupersymmetry and N-fold Supersymmetry in Quantum Many-Body Systems II. Third Order

Based on the general formalism of parafermionic algebra and parasupersymmetry proposed previously by us, we explicitly construct third-order parafermionic algebra and multiplication law, and then realize third-order parasupersymmetric quantum systems. We find some novel features in the third-order, namely, the emergence of a fermionic degree of freedom and of a generalized parastatistics. We show that for one-body cases the generalized Rubakov-Spiridonov model can be constructed also in our framework and find that it admits a generalized 3-fold superalgebra. We also find that a three-body system can have third-order parasupersymmetry where three independent supersymmetries are folded. In both cases, we also investigate the new concept of quasi-parasupersymmetry introduced by us and find that those of order (3,3) are indeed realized under less restrictive conditions than (ordinary) parasupersymmetric cases.

hep-th↗
arXiv2006

Parasupersymmetry and N-fold Supersymmetry in Quantum Many-Body Systems I. General Formalism and Second Order

We propose an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems without recourse to any specific matrix representation of parafermionic operators and any kind of deformed algebra. Within our formulation, we show generically that every parasupersymmetric quantum system of order p consists of N-fold supersymmetric pairs with N<p or N=p and thus has weak quasi-solvability and isospectral property. We also propose a new type of non-linear supersymmetries, called quasi-parasupersymmetry, which is less restrictive than parasupersymmetry and is different from N-fold supersymmetry even in one-body systems though the conserved charges are represented by higher-order linear differential operators. To illustrate how our formulation works, we construct second-order parafermionic algebra and three simple examples of parasupersymmetric quantum systems of order 2, one is essentially equivalent to the one-body Rubakov-Spiridonov type and the others are two-body systems in which two supersymmetries are folded. In particular, we show that the first model admits a generalized 2-fold superalgebra.

hep-th↗
arXiv2006

General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space

In our previous work, we proposed a mathematical framework for PT-symmetric quantum theory, and in particular constructed a Krein space in which PT-symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral property and PT symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical-quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding PT-symmetric quantum system, which indicates that PT-symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in the framework.

hep-th↗
arXiv2006

On Existence of a Biorthonormal Basis Composed of Eigenvectors of Non-Hermitian Operators

We present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators. As an illustration, we examine these conditions in the case of normal operators. We also provide a generalization of the conditions which is applicable to non-diagonalizable operators by considering not only eigenvectors but also all root vectors.

quant-ph↗
arXiv2006

PT-Symmetric Quantum Theory Defined in a Krein Space

We provide a mathematical framework for PT-symmetric quantum theory, which is applicable irrespective of whether a system is defined on R or a complex contour, whether PT symmetry is unbroken, and so on. The linear space in which PT-symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every PT-symmetric operator is P-Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the PT-symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K(H).

hep-th↗