Showing papers on "Ladder operator published in 2015"
TL;DR: In this paper, a numerical radius inequality for general n × n operator matrices is given, which improves a well-known inequality of J.C. Hou and H.K. Du.
74 citations
TL;DR: In this paper, a scalar constraint operator for loop quantum gravity is defined on the recently introduced space of partially diffeomorphism invariant states, and this space is preserved by the action of the operator.
Abstract: We present a concrete and explicit construction of a new scalar constraint operator for loop quantum gravity. The operator is defined on the recently introduced space of partially diffeomorphism invariant states, and this space is preserved by the action of the operator. To define the Euclidean part of the scalar constraint operator, we propose a specific regularization based on the idea of so-called "special" loops. The Lorentzian part of the quantum scalar constraint is merely the curvature operator that has been introduced in an earlier work. Due to the properties of the special loops assignment, the adjoint operator of the non-symmetric constraint operator is densely defined on the partially diffeomorphism invariant Hilbert space. This fact opens up the possibility of defining a symmetric scalar constraint operator as a suitable combination of the original operator and its adjoint. We also show that the algebra of the scalar constraint operators is anomaly free, and describe the structure of the kernel of these operators on a general level.
58 citations
TL;DR: In this paper, the authors present the construction of a physical Hamiltonian operator in the deparametrized model of loop quantum gravity coupled to a free scalar field, based on the use of the recently introduced curvature operator, and on the idea of so-called "special loops".
Abstract: We present the construction of a physical Hamiltonian operator in the deparametrized model of loop quantum gravity coupled to a free scalar field. This construction is based on the use of the recently introduced curvature operator, and on the idea of so-called "special loops". We discuss in detail the regularization procedure and the assignment of the loops, along with the properties of the resulting operator. We compute the action of the squared Hamiltonian operator on spin network states, and close with some comments and outlooks.
50 citations
TL;DR: In this article, a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory is presented, where representation theoretic aspects and connections to vertex operator algebras are emphasized.
Abstract: This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or quantum field theory is assumed.
48 citations
TL;DR: In this article, an algebraic structure, the division algebras, can be seen to underlie a generation of quarks and leptons, and a simple hermitian form, built from these ladder operators, results uniquely in the nine generators of SU c ( 3 ) and U em ( 1 ).
41 citations
TL;DR: In this paper, the average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive-semidefinite part of the non-hermitians.
Abstract: In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, the reality of an eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators that may admit real eigenvalues under some symmetry conditions. In general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive-semidefinite part of the non-Hermitian operator. We also prove an uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under a quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula, and measuring the product of noncommuting projectors.
41 citations
TL;DR: In this paper, a new symmetric Hamiltonian constraint operator is proposed for loop quantum gravity, which is well defined in the Hilbert space of diffeomorphism invariant states up to non-planar vertices with valence higher than three.
32 citations
TL;DR: In this paper, two versions of the Dirichlet-to-Neumann operator on exterior domains were defined and convergence properties when the domain is truncated were studied under the truncation condition.
Abstract: We define two versions of the Dirichlet-to-Neumann operator on exterior domains and study convergence properties when the domain is truncated.
27 citations
TL;DR: In this paper, the connection between a 1D quantum Hamiltonian involving the fourth Painleve transcendent P${\rm IV}, obtained in the context of second-order supersymmetric quantum mechanics and third-order ladder operators, with a hierarchy of families of quantum systems called $k$-step rational extensions of the harmonic oscillator and related with multi-indexed Hermite exceptionnal orthogonal polynomials of type III, is established at the level of the equivalence of the Hamiltonians using rational solutions of the fourth painleve equation in
Abstract: The purpose of this communication is to point out the connection between a 1D quantum Hamiltonian involving the fourth Painleve transcendent P$_{\rm IV}$, obtained in the context of second-order supersymmetric quantum mechanics and third-order ladder operators, with a hierarchy of families of quantum systems called $k$-step rational extensions of the harmonic oscillator and related with multi-indexed $X_{m_{1},m_{2},...,m_{k}}$ Hermite exceptionnal orthogonal polynomials of type III. The connection between these exactly solvable models is established at the level of the equivalence of the Hamiltonians using rational solutions of the fourth Painleve equation in terms of generalized Hermite and Okamoto polynomials. We also relate the different ladder operators obtained by various combinations of supersymmetric constructions involving Darboux-Crum and Krein-Adler supercharges, their zero modes and the corresponding energies. These results will demonstrate and clarify the relation observed for a particular case in previous papers.
20 citations
01 Jun 2015
TL;DR: In this article, a class of operators acting on continuous linear operators, called the quantum Ornstein-Uhlenbeck (O-U) semigroup, was defined, and the solution of the Cauchy problem associated with the quantum number operator was shown to be expressed in terms of such operators.
Abstract: Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth condition with two variables, we define a class of operators which gives in particular three semigroups acting on continuous linear operators, called the quantum Ornstein–Uhlenbeck (O–U) semigroup, the left quantum O–U semigroup and the right quantum O–U semigroup. Then, we prove that the solution of the Cauchy problem associated with the quantum number operator, the left quantum number operator and the right quantum number operator, respectively, can be expressed in terms of such semigroups. Moreover, probabilistic representations of these solutions are given. Eventually, using a new notion of positive white noise operators, we show that the aforementioned semigroups are Markovian.
18 citations
TL;DR: In this paper, an algebraic treatment of shape-invariant quantum-mechanical position-dependent effective mass systems is discussed using shape invariance, a general recipe for construction of ladder operators and associated algebraic structure of the pertaining system, is obtained.
Abstract: An algebraic treatment of shape-invariant quantum-mechanical position-dependent effective mass systems is discussed. Using shape invariance, a general recipe for construction of ladder operators and associated algebraic structure of the pertaining system, is obtained. These operators are used to find exact solutions of general one-dimensional systems with spatially varying mass. We apply our formalism to specific translationally shape-invariant potentials having position-dependent effective mass.
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TL;DR: In this article, a polynomial Heisenberg algebras (PHA) was proposed for the SUSY transformation of the harmonic and radial oscillators, and a new Wronskian formula was obtained for the confluent supersymmetric quantum mechanics (SUSY QM) for the inverted oscillator.
Abstract: We study first the supersymmetric quantum mechanics (SUSY QM), specially the cases of the harmonic and radial oscillators. Then, we obtain a new Wronskian formula for the confluent SUSY transformation and apply the SUSY QM to the inverted oscillator.
After that, we present the polynomial Heisenberg algebras (PHA). We study the general systems described by PHA: for zeroth- and first-order we obtain the harmonic and radial oscillators, respectively; for second- and third-order PHA, the potential is determined in terms of solutions to Painleve IV and V equations ($P_{IV}$ and $P_{V}$), respectively.
Later on, we review the six Painleve equations and we study the cases of $P_{IV}$ and $P_V$. We prove a reduction theorem for $2k$th-order PHA to be reduced to second-order algebras. We also prove an analogous theorem for the $(2k+1)$th-order PHA to be reduced to third-order ones. Through these theorems we find solutions to $P_{IV}$ and $P_V$ given in terms of confluent hypergeometric functions. For some special cases, those can be classified in several solution hierarchies. In this way, we find real solutions with real parameters and complex solutions with real and complex parameters for both equations.
Finally, we study the coherent states (CS) for the SUSY partners of the harmonic oscillator that are connected with $P_{IV}$, which we will call Painleve IV coherent states. Since these systems have third-order ladder operators $l_k^\pm$, we seek first the CS as eigenstates of the annihilation operator $l_k^-$. We also define operators analogous to the displacement operator and we get CS departing from the extremal states in each subspace in which the Hilbert space is decomposed. We conclude our treatment applying a linearization process to the ladder operators in order to define a new displacement operator to obtain CS involving the entire Hilbert space.
TL;DR: In this paper, the authors have discussed the reduction of noise due to peak amplitude and power ratio in multi carrier modulation scheme like OFDM process using frame theory, where the frame operator of the frame which is positive, self adjoint, invertible and commutes with synthesis operator.
Abstract: In this study, we have to discuss the reduction of noise due to peak amplitude and power ratio in multi carrier modulation scheme like Orthogonal Frequency Division Multiplexing (OFDM) process using Frame theory. The frame operator of the frame which is positive, self adjoint, invertible and it commutes with synthesis operator. If X is dual frame in H with frame operator S and analysis operator T, then T is quasi normal operator. If Y is dual frame for X, T is quasi unitary operator. If T and Q are pseudo inverse and sum of its inverse with its ad joint multiplication is frame operator, then X is Bessel’s sequence and dual frame.
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TL;DR: This master thesis discusses how the theory of operator algebras, also called operator theory, can be applied in quantum computer science.
Abstract: In this master thesis, I discuss how the theory of operator algebras, also called operator theory, can be applied in quantum computer science.
TL;DR: In this paper, the second order Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation and exact bound state solutions are obtained using convolution theorem.
Abstract: The second order \(N\)-dimensional Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution theorem. The Ladder operators are also constructed for the Mie-type potentials in \(N\)-dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.
TL;DR: In this article, a non-Hermitian fermionic model with the γ5 extension of mass was considered and the relation between this theory and a geometric theory with the maximum mass in the quantum mechanical approximation was established.
Abstract: We consider a non-Hermitian fermionic model with the γ5-extension of mass: m → m1 + γ5m2. We establish the relation between this theory and a geometric theory with the maximum mass in the quantum mechanical approximation. We propose a more detailed condition of PT-symmetry preservation in the theory. It implies segregating the initial domain of PT-symmetry preservation into subdomains corresponding to descriptions of standard and exotic particles. We calculate the operator C in the new scalar product in such a theory with a non-Hermitian Hamiltonian and describe some consequences of introducing this operator. We find the eigenvalues and eigenvectors of this Hamiltonian.
TL;DR: In this paper, the authors consider the functional difference operator and derive an explicit formula for the resolvent of the self-adjoint operator on the Hilbert space and prove the eigenfunction expansion theorem, which is a?-deformation of the well-known Kontorovich-Lebedev transform in the theory of special functions.
Abstract: We consider the functional-difference operator , where and are the Weyl self-adjoint operators satisfying the relation , , . The operator has applications in the conformal field theory and representation theory of quantum groups. Using the modular quantum dilogarithm (a?-deformation of the Euler dilogarithm), we define the scattering solution and Jost solutions, derive an explicit formula for the resolvent of the self-adjoint operator on the Hilbert space , and prove the eigenfunction expansion theorem. This theorem is a?-deformation of the well-known Kontorovich-Lebedev transform in the theory of special functions. We also present a?formulation of the scattering theory for?.
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TL;DR: In this paper, the authors studied spectral properties of the Schroedinger operator with an imaginary sign potential on the real line and showed that the pseudospectra of this operator are highly non-trivial, because of a blow-up of the resolvent at infinity.
Abstract: We study spectral properties of the Schroedinger operator with an imaginary sign potential on the real line. By constructing the resolvent kernel, we show that the pseudospectra of this operator are highly non-trivial, because of a blow-up of the resolvent at infinity. Furthermore, we derive estimates on the location of eigenvalues of the operator perturbed by complex potentials. The overall analysis demonstrates striking differences with respect to the weak-coupling behaviour of the Laplacian.
Journal Article•
TL;DR: A survey of several topics that have led to fruitful interactions between operator theory and harmonic analysis can be found in this article, including operator and spectral synthesis, Schur and Herz-Schur multipliers, and reflexivity.
Abstract: The artice is a survey of several topics that have led to fruitful interactions between Operator Theory and Harmonic Analysis, including operator and spectral synthesis, Schur and Herz-Schur multipliers, and reflexivity. Some open questions and directions are included in a separate section.
TL;DR: In this paper, the authors make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) as eigenstates of a deformed annihilation operator and ii) by application of adeformed displacement operator to the vacuum state.
Abstract: In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrodinger equation. We show that both methods yield coherent states with identical algebraic structure.
13 Apr 2015
TL;DR: In this article, the Darboux-Crum and Krein-Adler transformations are used to construct ladder operators for two-dimensional superintegrable systems constructed from rationally-extended potentials.
Abstract: Exceptional orthogonal polynomials constitute the main part of the bound-state wavefunctions of some solvable quantum potentials, which are rational extensions of well-known shape-invariant ones. The former potentials are most easily built from the latter by using higher-order supersymmetric quantum mechanics (SUSYQM) or Darboux method. They may in general belong to three different types (or a mixture of them): types I and II, which are strictly isospectral, and type III, for which k extra bound states are created below the starting potential spectrum. A well-known SUSYQM method enables one to construct ladder operators for the extended potentials by combining the supercharges with the ladder operators of the starting potential. The resulting ladder operators close a polynomial Heisenberg algebra (PHA) with the corresponding Hamiltonian. In the special case of type III extended potentials, for this PHA the k extra bound states form k singlets isolated from the higher excited states. Some alternative constructions of ladder operators are reviewed. Among them, there is one that combines the state-adding and state-deleting approaches to type III extended potentials (or so-called Darboux-Crum and Krein-Adler transformations) and mixes the k extra bound states with the higher excited states. This novel approach can be used for building integrals of motion for two-dimensional superintegrable systems constructed from rationally-extended potentials.
TL;DR: This paper gives rules for combining bold operator tensors such that, for a circuit, they give a probability distribution over the possible outcomes and forms a natural starting point for an operational approach to quantum field theory.
Abstract: In this paper, we present a formulation of quantum theory in terms of bold operator tensors. A circuit is built up of operations where an operation corresponds to a use of an apparatus. We associate collections of operator tensors (which together comprise a bold operator) with these apparatus uses. We give rules for combining bold operator tensors such that, for a circuit, they give a probability distribution over the possible outcomes. If we impose certain physicality constraints on the bold operator tensors, then we get exactly the quantum formalism. We provide both symbolic and diagrammatic ways to represent these calculations. This approach is manifestly covariant in that it does not require us to foliate the circuit into time steps and then evolve a state. Thus, the approach forms a natural starting point for an operational approach to quantum field theory.
TL;DR: In this article, a model for the analysis of the rovibrational structure of a molecule based on anharmonic ladder operators associated with the vibrational degrees of freedom is presented.
Abstract: A novel model for the analysis of the rovibrational structure of a molecule based on anharmonic ladder operators associated with the vibrational degrees of freedom is presented. This is devised as an alternative method for the global spectral analysis of rovibrational data considering vibrational anharmonicities from the outset. The present method is thought up with an effective rovibrational Hamiltonian written in terms of angular momentum components and anharmonic Morse ladder operators, associated with rotational and vibrational degrees of freedom, respectively. The resulting Hamiltonian is diagonalized in a symmetry-adapted basis set expressed as a product of rotational states and individual 1D-Morse wave functions for each local vibrational degree of freedom. This approach has been successfully applied to the study of the vibrational structure (up to polyad 14) and the rovibrational structure (up to polyad 2 and Jmax = 20) of hydrogen sulfide. It was shown that this new global analysis formalism is a...
TL;DR: In this article, a quantized fluctuational electrodynamics (QFED) formalism was developed to describe the quantum aspects of local thermal balance formation and to formulate the electromagnetic field ladder operators so that they no longer exhibit the anomalies reported for resonant structures.
Abstract: We have recently developed a quantized fluctuational electrodynamics (QFED) formalism to describe the quantum aspects of local thermal balance formation and to formulate the electromagnetic field ladder operators so that they no longer exhibit the anomalies reported for resonant structures. Here we show how the QFED can be used to resolve between the left and right propagating fields to bridge the QFED and the quantum optical input-output relations commonly used to describe selected quantum aspects of resonators. The generalized model introduces a density of states concept describing interference effects, which is instrumental in allowing an unambiguous separation of the fields and related quantum operators into left and right propagating parts. In addition to providing insight on the quantum treatment of interference, our results also provide the conclusive resolution of the long-standing enigma of the anomalous commutation relations of partially confined propagating fields.
06 Jul 2015
TL;DR: In this article, it was shown that a given operator algebra is scattered if and only if its associated partial order is, equivalently, continuous (a domain), algebraic, atomistic, quasi-continuous, or quasialgebraic.
Abstract: Operator algebras provide uniform semantics for deterministic, reversible, probabilistic, and quantum computing, where intermediate results of partial computations are given by commutative sub algebras We study this setting using domain theory, and show that a given operator algebra is scattered if and only if its associated partial order is, equivalently: continuous (a domain), algebraic, atomistic, quasi-continuous, or quasialgebraic In that case, conversely, we prove that the Lawson topology, modelling information approximation, allows one to associate an operator algebra to the domain
TL;DR: In this paper, a multilevel computation scheme for time-harmonic fields in three dimensions was formulated with a new Gaussian translation operator that decays exponentially outside a circular cone centered on the line connecting the source and observation groups.
Abstract: A multilevel computation scheme for time-harmonic fields in three dimensions will be formulated with a new Gaussian translation operator that decays exponentially outside a circular cone centered on the line connecting the source and observation groups. This Gaussian translation operator is directional and diagonal with its sharpness determined by a beam parameter. When the beam parameter is set to zero, the Gaussian translation operator reduces to the standard fast multipole method translation operator. The directionality of the Gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy. The sampling rate can be determined straightforwardly to achieve any desired accuracy. The use of the computation scheme will be illustrated through a near-field scanning problem where the far-field pattern of a source is determined from near-field measurements with a known probe. Here the Gaussian translation operator improves the condition number of the matrix equation that determines the far-field pattern. The Gaussian translation operator can also be used when the probe pattern is known only in one hemisphere, as is common in practice. Also, the Gaussian translation operator will be used to solve the scattering problem of the perfectly conducting sphere.
TL;DR: In this article, the generalized k-difference operator was defined and the discrete version of Leibnitz theorem was presented according to the generalized K-Difference operator, and the sum of product of arithmetic and geometric progressions in the field of Numerical analysis was obtained.
Abstract: In this paper, we define the generalized k-difference operator and present the discrete version of Leibnitz theorem according to the generalized k-difference operator. Also, by defining its inverse, we obtain the sum of product of arithmetic and geometric progressions in the field of Numerical Analysis. Mathematics Subject Classification: 39A70, 47B39, 97N40
TL;DR: In this article, the authors derive a one-step extension of the Swanson oscillator that describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected to an extended harmonic oscillator model.
13 Apr 2015
TL;DR: In this paper, it was shown that ladder operators obtained from alternative construction based on combinations of supercharges in the Krein-Adler and Darboux Crum (or state deleting and creating) approaches can be used to generate a set of integrals of motion and a corresponding polynomial algebra that provides an algebraic derivation of the full spectrum and total number of degeneracies.
Abstract: Four new families of two-dimensional quantum superintegrable systems are constructed from k-step extension of the harmonic oscillator and the radial oscillator. Their wavefunctions are related with Hermite and Laguerre exceptional orthogonal polynomials (EOP) of type III. We show that ladder operators obtained from alternative construction based on combinations of supercharges in the Krein-Adler and Darboux Crum (or state deleting and creating) approaches can be used to generate a set of integrals of motion and a corresponding polynomial algebra that provides an algebraic derivation of the full spectrum and total number of degeneracies. Such derivation is based on finite dimensional unitary representations (unirreps) and doesn't work for integrals build from standard ladder operators in supersymmetric quantum mechanics (SUSYQM) as they contain singlets isolated from excited states. In this paper, we also rely on a novel approach to obtain the finite dimensional unirreps based on the action of the integrals of motion on the wavefunctions given in terms of these EOP. We compare the results with those obtained from the Daskaloyannis approach and the realizations in terms of deformed oscillator algebras for one of the new families in the case of 1-step extension. This communication is a review of recent works.
TL;DR: In this paper, the Pegg-Barnett phase operator can be obtained by the application of the discrete Fourier transform to the number operator defined in a finite-dimensional Hilbert space.
Abstract: In quantum mechanics the position and momentum operators are related to each other via the Fourier transform. In the same way, here we show that the so-called Pegg-Barnett phase operator can be obtained by the application of the discrete Fourier transform to the number operator defined in a finite-dimensional Hilbert space. Furthermore, we show that the structure of the London-Susskind-Glogower phase operator, whose natural logarithm give rise the Pegg-Barnett phase operator, is contained into the Hamiltonian of circular waveguide arrays. Our results may find applications in the development of new finite-dimensional photonic systems with interesting phase-dependent properties.