Polynomial supersymmetry for matrix Hamiltonians

We study intertwining relations for matrix non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any intertwining operator of minimal order there is operator that intertwines the same Hamiltonians in the opposite direction and such that the products of these operators are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and reducibility of a matrix intertwining operator are considered and the criteria of minimizability and reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.     

Two-dimensional Schrödinger Hamiltonians with effective mass in SUSY approach

The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties—symmetry, irreducibility, and separation of variables—are investigated.     

Intertwining operators for solving differential equations,with applications to symmetric spaces

The use of intertwining operators to solve both ordinary and partial differential equations is developed. Classes of intertwining operators are constructed which transform between Laplacians which are self-adjoint with respect to different non-trivial measures. In the two-dimensional case, the intertwining operator transforms a non-separable partial differential operator to a separable one. As an application, the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed.     

The group theoretic structure of the dual current vertex

We construct the Corrigan-Fairlie dual current vertex using an intertwining operator. This first necessitates the development of an isomorphism between certain function and Fock space representations of the two-dimensional conformal group. The intertwining operator is a mapping between the different representations and may then be constructed using an analogous procedure. Due to the group properties built into this operator the well-known gauge conditions obeyed by the vertex soon follow.     

Intertwining operator method and supersymmetry for effective mass Schrödinger equations

By application of the intertwining operator method to Schrödinger equations with position-dependent (effective) mass, we construct Darboux transformations, establish the supersymmetry factorization technique and show equivalence of both formalisms. Our findings prove equivalence of the intertwining technique and the method of point transformations.     

Logarithmic Intertwining Operators and the Space of Conformal Blocks Over the Projective Line

We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This can be viewed as a generalization of Zhu??s result for ordinary intertwining operators among ordinary modules.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

An Exchange Identity for Non-linear Fields

We establish a useful identity for intertwining a creation or annihilation operator with the heat kernel of a self-interacting bosonic field theory.     

Reconstruction of quantum well potentials

The intertwining operator technique is applied to the generalized Schrödinger equation with a position-dependent effective mass. It is shown on concrete examples how to construct the quantum well potential with a desired spectrum for the Schrödinger equation with a nonhermitian kinetic energy operator.     

Classification of screening systems for lattice vertex operator algebras

Letters in Mathematical Physics - We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra...     

Tempered distributions in infinitely many dimensions

In the present paper we continue investigating spaces of tempered distributions in infinitely many dimensions. In particular, we prove that those linear homogeneous transformations of the canonical pair of field operators, which preserve the commutation relations, can be implemented by an essentially unique intertwining operator. The dependence of this operator on the transformation is studied.     

A New Approach to the {\ast} -Genvalue Equation

We show that the eigenvalues and eigenfunctions of the star-genvalue equation can be completely expressed in terms of the corresponding eigenvalue problem for the quantum Hamiltonian. Our methods make use of a Weyl-type representation of the star-product and of the properties of the cross-Wigner transform, which appears as an intertwining operator.     

On Lambda and Time Operators: the Inverse Intertwining Problem Revisited

An exact theory of irreversibility was proposed by Misra, Prigogine and Courbage, based on non-unitary similarity transformations Λ that intertwine reversible dynamics and irreversible ones. This would advocate the idea that irreversible behavior would originate at the microscopic level. Reversible evolution with an internal time operator have the intertwining property. Recently the inverse intertwining problem has been answered in the negative, that is, not every unitary evolution allowing such Λ-transformation has an internal time. This work contributes new results in this direction.     

Ordering Ambiguity Revisited via Position Dependent Mass Pseudo-Momentum Operators

Ordering ambiguity associated with the von Roos position dependent mass (PDM) Hamiltonian is considered. An affine locally scaled first order differential introduced, in Eq. (9), as a PDM-pseudo-momentum operator. Upon intertwining our Hamiltonian, which is the sum of the square of this operator and the potential function, with the von Roos d-dimensional PDM-Hamiltonian, we observed that the so-called von Roos ambiguity parameters are strictly determined, but not necessarily unique. Our new ambiguity parameters’ setting is subjected to Dutra’s and Almeida’s, Phys. Lett. A. 275 (2000) 25 reliability test and classified as good ordering. PACS numbers: 03.65.Ge, 03.65. Fd, 03.65.Ca     

Effective-Mass Schrödinger Equation and Generation of Solvable Potentials

A one-dimensional Schrödinger equation with position-dependent effective mass in the kinetic-energy operator is studied in the framework of an so(2,1) algebra. New mass-deformed versions of Scarf II, Morse, and generalized Pöschl-Teller potentials are obtained. Consistency with the intertwining condition is pointed out.     

Deductive inference of nuclear boson models (IBM, TQM, FQM) based on SU(6) dynamical symmetry

A single deductive inference of Schwinger realization (= interacting boson model—IBM), Holstein-Primakoff realization (= truncated quadrupole phonon model—TQM) and Dyson realization (= finite quadrupole phonon model—FQM) of dynamical SU(6) quadrupole collective algebra (QCA) is presented with a full scope of their isomorphism on the level of representations. Dyson realization of QCA is explicitly constructed by using holomorphically parametrized generalized coherent state and explicit form of root vectors. Utilizing appropriate orthogonalizing operators Holstein-Primakoff realization of QCA has been derived from the Dyson realization. The carrier spaces of Schwinger and Holstein-Primakoff realizations are investigated on the same footing and Marshalek's boson is rigorously derived. The intertwining operator which connects Schwinger and Holstein-Primakoff realizations is constructed and its domain and image are determined. It is shown that the intertwining operator has well-defined inverse in a definite factor space of the IBM basis space which is proved to be isomorphic to the physical subspace of the TQM basis space, meaning equivalence of IBM and TQM on level of representations.     

Full Field Algebras

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V ^L and V ^R , is naturally a full field algebra and we introduce a notion of full field algebra over . We study the structure of full field algebras over using modules and intertwining operators for V ^L and V ^R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over and an invariant bilinear form on this algebra.     

Intertwining operator realization of non-relativistic holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov–Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.     

Quantum mechanical treatment of a constrained particle on two dimensional sphere

In this work, we study the motion of a particle on two dimensional sphere. By writing the Schrodinger equation, we obtain the wave function and energy spectra for three dimensional harmonic oscillator potential plus trigonometric Rosen–Morse non-central potential. By letting three special cases for intertwining operator, we investigate the energy spectra and wave functions for Smorodinsky–Winternitz potential model.     

Darboux transformations for the generalized Schrödinger equation

The intertwining operator technique is applied to the Schrödinger equation with an additional functional dependence h(r) on the right-hand side of the equation. The suggested generalized transformations turn into the Darboux transformations for both fixed and variable values of energy and angular momentum. A relation between the Darboux transformation and supersymmetry is considered.