We study selected non-Hermitian quantum models, indecomposable representations of the de Sitter group, and coherent state quantization in infinite well Systems. The first part is focused on the investigation of spectral and bases properties of PT-symmetric operators. At first, we consider Schrodinger operators subjected to non-self-adjoint boundary conditions. The existence and the structure of similarity transformations to normal or self-adjoint operators is examined and general results are illustrated on explicitly solvable models. A physical motivation for the study of Schrodinger operators with non-self-adjoint Robin-type boundary conditions is established in a reflectionless scatterin. Further, irregular boundary conditions are considered and operators with very non-normal spectral ; properties are found. Finally, an operator with PT-symmetric version of Coulomb potential is investigate an effective behavior of the kinetic term is discussed and the reflection and transmission coefficients for this System are calculated. The second part deals with elements of the discrete scalar series of representations of the de Sitter group Physical motivations and origins of the so-called zero mode problem are explained. The generalization of the Gupta-Bleuler-like quantization of the massless minimally coupled field, based on the construction an indecomposable representation, is worked out and put into the mathematical framework of the cohomology of representations and Gupta-Bleuler triplets. The third part is oriented on the construction of coherent states for the Poschl-Teller. The obtained stat exhibit remarkable semi-classical properties.
We study selected non-Hermitian quantum models, indecomposable representations of the de Sitter group, and coherent state quantization in infinite well Systems. The first part is focused on the investigation of spectral and bases properties of PT-symmetric operators. At first, we consider Schrodinger operators subjected to non-self-adjoint boundary conditions. The existence and the structure of similarity transformations to normal or self-adjoint operators is examined and general results are illustrated on explicitly solvable models. A physical motivation for the study of Schrodinger operators with non-self-adjoint Robin-type boundary conditions is established in a reflectionless scatterin. Further, irregular boundary conditions are considered and operators with very non-normal spectral ; properties are found. Finally, an operator with PT-symmetric version of Coulomb potential is investigate an effective behavior of the kinetic term is discussed and the reflection and transmission coefficients for this System are calculated. The second part deals with elements of the discrete scalar series of representations of the de Sitter group Physical motivations and origins of the so-called zero mode problem are explained. The generalization of the Gupta-Bleuler-like quantization of the massless minimally coupled field, based on the construction an indecomposable representation, is worked out and put into the mathematical framework of the cohomology of representations and Gupta-Bleuler triplets. The third part is oriented on the construction of coherent states for the Poschl-Teller. The obtained stat exhibit remarkable semi-classical properties.
The main motivation behind the call for this special issue was to gather recent results, developments and open problems in quantum physics with non-Hermitian operators. There have been previous special issues in this journal [1, 2] and elsewhere on this subject. The intention of this issue is to reflect the current state of this rapidly-developing field. It has therefore been open to all contributions containing new results on non-Hermitian theories that are explicitly -symmetric and/or pseudo-Hermitian or quasi-Hermitian. In the last decade these types of systems have proved to be viable self-consistent physical theories with well defined unitary time-evolution and real spectra. As the large number of responses demonstrates, this is a rapidly evolving field of research. A consensus has been reached regarding most of the fundamental problems, and the general ideas and techniques are now readily being employed in many areas of physics. Nonetheless, this issue still contains some treatments of a more general nature regarding the spectral analysis of these models, in particular, the physics of the exceptional points, the breaking of the -symmetry, an interpretation of negative energies and the consistent implementation of the WKB analysis. This issue also contains a treatment of a scattering theory associated with these types of systems, weak measurements, coherent states, decoherence, unbounded metric operators and the inclusion of domain issues to obtain well defined self-adjoint theories. Contributions in the form of applications of the general ideas include: studies of classical shock-waves and tunnelling, supersymmetric models, spin chain models, models with ring structure, random matrix models, the Pauli equation, the nonlinear Schrödinger equation, quasi-exactly solvable models, integrable models such as the Calogero model, Bose–Einstein condensates, thermodynamics, nonlinear oligomers, quantum catastrophes, the Landau–Zener problem and pseudo-Fermions. Applications close to experimental realization are proposed in optics, including short light pulse models, waveguides and laser systems, and also in electronics. We hope that this issue will become a valuable reference and inspiration for the broader scientific community working in mathematical and theoretical physics. References [1] Fring A, Jones H F and Znojil M (ed) 2008 J. Phys. A: Math. Theor. 41 240301 [2] Geyer H, Heiss D and Znojil M (ed) 2006 J. Phys. A: Math. Gen. 39 9963
This paper is devoted to the construction of what we will call exactly solvable models, i.e., of quantum mechanical systems described by an Hamiltonian H whose eigenvalues and eigenvectors can be explicitly constructed out of some minimal ingredients. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
In this thesis we aim to show that the Schrödinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian.After a brief introduction into non-Hermiticity, we will focus on solving the Schrödinger equation with a special class of non-Hermitian Hamiltonians, namely PTsymmetric Hamiltonians.PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing specifically on obtaining the real and discrete energy spectrum.Various methods for solving this problematic Schrödinger equation will be considered.After starting with perturbation theory, we will move on to numerical methods.Three different categories of methods will be discussed.First there is the shooting method based on a Runge-Kutta solver.Next, we investigate various implementations of the spectral method.Finally, we will look at the Riccati-Padé method, which is a numerical implemented analytical method.PT -symmetric potentials need to be solved along a contour in the complex plane.We will propose modifications to the numerical methods to handle this.After solving the widely documented PT -symmetric Hamiltonian H = p 2 -(ix) N with these methods, we give a discussion and comparison of the obtained results.Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their influence on the results.
In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac's kets and bras, using the momentum operator as an example. The mathematical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian operators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.