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Quasi-hermitian lattices with imaginary zero-range interactions
Abstract
We study a general class of -symmetric tridiagonal Hamiltonians
with purely imaginary interaction terms in the quasi-hermitian representation
of quantum mechanics. Our general Hamiltonian encompasses many previously
studied lattice models as special cases. We provide numerical results regarding
domains of observability and exceptional points, and discuss the possibility of
explicit construction of general metric operators (which in turn determine all
the physical Hilbert spaces). The condition of computational simplicity for the
metrics motivates the introduction of certain one-parametric special cases,
which consequently admit closed-form extrapolation patterns of the
low-dimensional results.
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A Su-Schrieffer-Heeger model with added PT-symmetric boundary
term is studied in the framework of pseudo-hermitian quantum mechanics. For two special cases, a complete set of pseudometrics is constructed in closed form. When complemented with a condition of positivity, the pseudometrics determine all the physical inner
products of the considered model.
We study the effect of -symmetric complex potentials on the
transport properties of non-Hermitian systems, which consist of an infinite
linear chain and two side-coupled defect points with -symmetric
complex on-site potentials. By analytically solving the scattering problem of
two typical models, which display standard Fano resonances in the absence of
non-Hermitian terms, we find that the -symmetric imaginary
potentials can lead to some pronounced effects on transport properties of our
systems, including changes from the perfect reflection to perfect transmission,
and rich behaviors for the absence or existence of the prefect reflection at
one and two resonant frequencies. Our study can help us to understand the
interplay between the Fano resonance and symmetry in
non-Hermitian discrete systems, which may be realizable in optical waveguide
experiments.
The realization of a genuine phase transition in quantum mechanics requires
that at least one of the Kato's exceptional-point parameters becomes real. A
new family of finite-dimensional and time-parametrized quantum-lattice models
with such a property is proposed and studied. All of them exhibit, at a real
exceptional-point time t=0, the Jordan-block spectral degeneracy structure of
some of their observables sampled by Hamiltonian H(t) and site-position
Q(t). The passes through the critical instant t=0 are interpreted as
schematic simulations of non-equivalent versions of the Big-Bang-like quantum
catastrophes.
Three classes of finite-dimensional models of quantum systems exhibiting
spectral degeneracies called quantum catastrophes are described in detail.
Computer-assisted symbolic manipulation techniques are shown unexpectedly
efficient for the purpose.
We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator
type defined on a two-dimensional noncommutative space can be diagonalized
exactly by making use of pseudo-bosonic operators. The model admits an
antilinear symmetry and is of the type studied in the context of PT-symmetric
quantum mechanics. Its eigenvalues are computed to be real for the entire range
of the coupling constants and the biorthogonal sets of eigenstates for the
Hamiltonian and its adjoint are explicitly constructed. We show that despite
the fact that these sets are complete and biorthogonal, they involve an
unbounded metric operator and therefore do not constitute (Riesz) bases for the
Hilbert space \Lc^2(\Bbb R^2), but instead only D-quasi bases. As recently
proved by one of us (FB), this is sufficient to deduce several interesting
consequences.
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of PT symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories. {copyright} {ital 1998} {ital The American Physical Society}
We introduce a very simple, exactly solvable \mathcal{PT} -symmetric non-Hermitian model with a real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.
We extend the formulation of pseudo-Hermitian quantum mechanics to
eta-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator
eta. In particular, we give the details of the construction of the physical
Hilbert space, observables, and equivalent Hermitian Hamiltonian for the case
that H has a real and discrete spectrum and its eigenvectors belong to the
domain of eta and consequently its positive square root.
Several examples of Jacobi matrices with an explicitly solvable spectral
problem are worked out in detail. In all discussed cases the spectrum is
discrete and coincides with the set of zeros of a special function. Moreover,
the components of corresponding eigenvectors are expressible in terms of
special functions as well. Our approach is based on a recently developed
formalism providing us with explicit expressions for the characteristic
function and eigenvectors of Jacobi matrices. This is done under an assumption
of a simple convergence condition on matrix entries. Among the treated special
functions there are regular Coulomb wave functions, confluent hypergeometric
functions, q-Bessel functions and q-confluent hypergeometric functions. In
addition, in the case of q-Bessel functions, we derive several useful
identities.
Motivated by the recent developments of pseudo-hermitian quantum mechanics,
we analyze the structure of unbounded metric operators in a Hilbert space. It
turns out that such operators generate a canonical lattice of Hilbert spaces,
that is, the simplest case of a partial inner product space (PIP space). Next,
we introduce several generalizations of the notion of similarity between
operators and explore to what extend they preserve spectral properties. Then we
apply some of the previous results to operators on a particular PIP space,
namely, a scale of Hilbert spaces generated by a metric operator. Finally, we
reformulate the notion of pseudo-hermitian operators in the preceding
formalism.
It is shown that if the C operator for a PT-symmetric Hamiltonian with simple
eigenvalues is not unique, then it is unbounded. Apart from the special cases
of finite-matrix Hamiltonians and Hamiltonians generated by differential
expressions with PT-symmetric point interactions, the usual situation is that
the C operator is unbounded. The fact that the C operator is unbounded is
significant because, while there is a formal equivalence between a PT-symmetric
Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the
two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is
so because the mapping from one Hilbert space to the other is unbounded. This
shows that PT-symmetric quantum theories are mathematically distinct from
conventional Hermitian quantum theories.
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!
Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.
The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
We study a non-Hermitian symmetric generalization of an N-particle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order N+1 which under perturbation unfold either into [(N+1)/2] eigenvalue pairs (and in case of N+1 odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to (N+1)th order are indicated.
A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics, the book features: Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and perturbation theory Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics and condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as , the true meaning and significance of the so-called charge operators and the -inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
We study the parity- and time-reversal PT symmetric non-Hermitian
Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials at two end sites. The SSH model is known as one of the simplest
two-band topological models which has topologically trivial and nontrivial
phases. We find that the non-Hermitian terms can lead to different effects on
the properties of the eigenvalues spectrum in topologically trivial and
nontrivial phases. In the topologically trivial phase, the system undergos an
abrupt transition from unbroken PT-symmetry region to spontaneously broken
-symmetry region at a certain , and a second
transition occurs at another transition point when further
increasing the strength of the imaginary potential . But in the
topologically nontrivial phase, the zero-mode edge states become unstable for
arbitrary nonzero and the -symmetry of the system is
spontaneously broken, which is characterized by the emergence of a pair of
conjugated imaginary modes.
PT symmetric Aubry-Andre model describes an array of N coupled optical
waveguides with position dependent gain and loss. We show that the reality of
the spectrum depends sensitively on the degree of disorder for small number of
lattice sites. We obtain the Hofstadter Butterfly spectrum and discuss the
existence of the phase transition from extended to localized states. We show
that rapidly changing periodical gain/loss materials almost conserves the total
intensity.
We review a surprising correspondence between certain two-dimensional
integrable models and the spectral theory of ordinary differential
equations. Particular emphasis is given to the relevance of this
correspondence to certain problems in PT-symmetric quantum mechanics.
We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.
PCT, a fundamental symmetry of quantum field theory, is derived from the assumption of Lorentz invariance and positivity of the spectrum. What happens if we assume only Lorentz invariance and PCT symmetry? Hamiltonians having this property need not be Hermitian, but, except when PCT is spontaneously broken, the energy levels of such Hamiltonians are all real and positive! In this talk I examine quantum mechanical and quantum field theoretic systems whose Hamiltonians are non-Hermitian but obey PCT symmetry. These systems have weird and remarkable properties, and the classical theories underlying these quantum systems also have strange behaviors. Simple examples of such Hamiltonians are H = p2 + ix3 and H = p2 - x4. Hamiltonians such as these may be regarded as complex deformations of conventional Hermitian Hamiltonians. Thus, in this talk I will be studying the analytic continuation of conventional classical mechanics and quantum mechanics into the complex plane.
This paper reviews a recently discovered link between integrable quantum
field theories and certain ordinary differential equations in the
complex domain. Along the way, aspects of {\cal P}{\cal T} -symmetric
quantum mechanics are discussed, and some elementary features of the
six-vertex model and the Bethe ansatz are explained.
A short resume is given about the nature of exceptional points (EPs) followed
by discussions about their ubiquitous occurrence in a great variety of physical
problems. EPs feature in classical as well as in quantum mechanical problems.
They are associated with symmetry breaking for -symmetric
Hamiltonians, where a great number of experiments have been performed in
particular in optics, and to an increasing extent in atomic and molecular
physics. EPs are involved in quantum phase transition and quantum chaos, they
produce dramatic effects in multichannel scattering, specific time dependence
and more. In nuclear physics they are associated with instabilities and
continuum problems. Being spectral singularities they also affect approximation
schemes.
The Lee model is an elementary quantum field theory in which mass, wave-function, and charge renormalization can be performed exactly. In early studies of this model in the 1950's it was found that there is a critical value of g
2, the square of the renormalized coupling constant, above which g
02, the square of the unrenormalized coupling constant, is negative. For g
2 larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. In this non-Hermitian regime a new state appears whose norm is negative. This state is called a ghost. It has always been thought that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a time-independent operator called C. In terms of C one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown that time evolution in such a theory is unitary. In this talk the C operator for the Lee model in the ghost regime is constructed in the V/Nθ sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of g
2.
We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. This involves the construction of a metric (if it exists) for the given set of non-Hermitian observables. We discuss uniqueness of this metric. We also show that it is not always necessary to construct the metric for the whole set of observables under consideration, but that it is sufficient for some calculational purposes to construct it for a subset only, even though this metric is, in general, not unique. The restricted metric turns out to be particularly useful in the implementation of a variational principle, which we also formulate.
Complete set of inner products for a discrete PT-symmetric square-well Hamiltonian
M. Znojil. "Complete set of inner products for a discrete PT-symmetric square-well Hamiltonian". J. Math. Phys. 50 (2009), p. 122105. arXiv:0911.0336 [math-ph].
Quantum big bang without fine-tuning in a toy-model
M. Znojil. "Quantum big bang without fine-tuning in a toy-model". J. Phys. Conf. S. 343
(2012), p. 012136. arXiv:1105.1282 [gr-qc].
Quantum inner-product metrics via the recurrent solution of the Dieudonné equation
M. Znojil. "Quantum inner-product metrics via the recurrent solution of the Dieudonné
equation". J. Phys. A 45 (2012), p. 085302. arXiv:1201.2263 [math-ph].
Cryptohermitian Hamiltonians on Graphs
M. Znojil. "Cryptohermitian Hamiltonians on Graphs". Int. J. Theor. Phys. 50 (2011), pp. 1052-1059. arXiv:1008.2082 [quant-ph].
An exactly solvable quantum-lattice model with a tunable degree of nonlocality
M. Znojil. "An exactly solvable quantum-lattice model with a tunable degree of nonlocality".
J. Phys. A 44 (2011), p. 075302. arXiv:1101.1183 [math-ph].
Solvable non-hermitian discrete square well with closed-form physical inner product
M. Znojil. "Solvable non-hermitian discrete square well with closed-form physical inner product".
J. Phys. A 47 (2014), p. 435302. arXiv:1409.3788v1 [quant-ph].
Gegenbauer-solvable quantum chain model
M. Znojil. "Gegenbauer-solvable quantum chain model". Phys. Rev. A 82 (2010), p. 052113.
arXiv:1011.4803 [quant-ph].
N-site-lattice analogues of V (x) = ix 3
M. Znojil. "N-site-lattice analogues of V (x) = ix 3 ". Ann. Phys, 327 (2012), pp. 893-913.
arXiv:1111.0484 [math-ph].
Non-hermitian Heisenberg representation
M. Znojil. "Non-hermitian Heisenberg representation". Phys. Letters A 379 (2015), pp. 2013-2017. arXiv:1505.01036 [quant-ph].
Three-Hilbert-space formulation of quantum mechanics
M. Znojil. "Three-Hilbert-space formulation of quantum mechanics". SIGMA 5 (2009), p. 1.
arXiv:0901.0700 [quant-ph].
Discrete quantum square well of the first kind
M. Znojil. "Discrete quantum square well of the first kind". Phys. Letters A 375 (2011),
pp. 2503-2509. arXiv:1105.1863 [quant-ph].