1,020 research outputs found
Complex WKB Analysis of a PT Symmetric Eigenvalue Problem
The spectra of a particular class of PT symmetric eigenvalue problems has
previously been studied, and found to have an extremely rich structure. In this
paper we present an explanation for these spectral properties in terms of
quantisation conditions obtained from the complex WKB method. In particular, we
consider the relation of the quantisation conditions to the reality and
positivity properties of the eigenvalues. The methods are also used to examine
further the pattern of eigenvalue degeneracies observed by Dorey et al. in
[1,2].Comment: 22 pages, 13 figures. Added references, minor revision
Excited state g-functions from the Truncated Conformal Space
In this paper we consider excited state g-functions, that is, overlaps
between boundary states and excited states in boundary conformal field theory.
We find a new method to calculate these overlaps numerically using a variation
of the truncated conformal space approach. We apply this method to the Lee-Yang
model for which the unique boundary perturbation is integrable and for which
the TBA system describing the boundary overlaps is known. Using the truncated
conformal space approach we obtain numerical results for the ground state and
the first three excited states which are in excellent agreement with the TBA
results. As a special case we can calculate the standard g-function which is
the overlap with the ground state and find that our new method is considerably
more accurate than the original method employed by Dorey et al.Comment: 21 pages, 6 figure
All Hermitian Hamiltonians Have Parity
It is shown that if a Hamiltonian is Hermitian, then there always exists
an operator P having the following properties: (i) P is linear and Hermitian;
(ii) P commutes with H; (iii) P^2=1; (iv) the nth eigenstate of H is also an
eigenstate of P with eigenvalue (-1)^n. Given these properties, it is
appropriate to refer to P as the parity operator and to say that H has parity
symmetry, even though P may not refer to spatial reflection. Thus, if the
Hamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses
time-reversal symmetry), then it immediately follows that H has PT symmetry.
This shows that PT symmetry is a generalization of Hermiticity: All Hermitian
Hamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric
Hamiltonians of this form are Hermitian
On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
In the context of two particularly interesting non-Hermitian models in
quantum mechanics we explore the relationship between the original Hamiltonian
H and its Hermitian counterpart h, obtained from H by a similarity
transformation, as pointed out by Mostafazadeh. In the first model, due to
Swanson, h turns out to be just a scaled harmonic oscillator, which explains
the form of its spectrum. However, the transformation is not unique, which also
means that the observables of the original theory are not uniquely determined
by H alone. The second model we consider is the original PT-invariant
Hamiltonian, with potential V=igx^3. In this case the corresponding h, which we
are only able to construct in perturbation theory, corresponds to a complicated
velocity-dependent potential. We again explore the relationship between the
canonical variables x and p and the observables X and P.Comment: 9 pages, no figure
PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics
A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted
by means of a similarity transformation to a physically equivalent Hermitian
Hamiltonian. This raises the following question: In which form of the quantum
theory, the non-Hermitian or the Hermitian one, is it easier to perform
calculations? This paper compares both forms of a non-Hermitian
quantum-mechanical Hamiltonian and demonstrates that it is much harder to
perform calculations in the Hermitian theory because the perturbation series
for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For
the Hermitian version of the theory, dimensional continuation is used to
regulate the divergent graphs that contribute to the ground-state energy and
the one-point Green's function. The results that are obtained are identical to
those found much more simply and without divergences in the non-Hermitian
PT-symmetric Hamiltonian. The contribution to the
ground-state energy of the Hermitian version of the theory involves graphs with
overlapping divergences, and these graphs are extremely difficult to regulate.
In contrast, the graphs for the non-Hermitian version of the theory are finite
to all orders and they are very easy to evaluate.Comment: 13 pages, REVTeX, 10 eps figure
Classical Trajectories for Complex Hamiltonians
It has been found that complex non-Hermitian quantum-mechanical Hamiltonians
may have entirely real spectra and generate unitary time evolution if they
possess an unbroken \cP\cT symmetry. A well-studied class of such
Hamiltonians is (). This paper
examines the underlying classical theory. Specifically, it explores the
possible trajectories of a classical particle that is governed by this class of
Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate
structure that depends sensitively on the value of the parameter and
on the initial conditions. A system for classifying complex orbits is
presented.Comment: 24 pages, 34 figure
Calculation of the Hidden Symmetry Operator for a \cP\cT-Symmetric Square Well
It has been shown that a Hamiltonian with an unbroken \cP\cT symmetry also
possesses a hidden symmetry that is represented by the linear operator \cC.
This symmetry operator \cC guarantees that the Hamiltonian acts on a Hilbert
space with an inner product that is both positive definite and conserved in
time, thereby ensuring that the Hamiltonian can be used to define a unitary
theory of quantum mechanics. In this paper it is shown how to construct the
operator \cC for the \cP\cT-symmetric square well using perturbative
techniques.Comment: 10 pages, 2 figure
On O(1) contributions to the free energy in Bethe Ansatz systems: the exact g-function
We investigate the sub-leading contributions to the free energy of Bethe
Ansatz solvable (continuum) models with different boundary conditions. We show
that the Thermodynamic Bethe Ansatz approach is capable of providing the O(1)
pieces if both the density of states in rapidity space and the quadratic
fluctuations around the saddle point solution to the TBA are properly taken
into account. In relativistic boundary QFT the O(1) contributions are directly
related to the exact g-function. In this paper we provide an all-orders proof
of the previous results of P. Dorey et al. on the g-function in both massive
and massless models. In addition, we derive a new result for the g-function
which applies to massless theories with arbitrary diagonal scattering in the
bulk.Comment: 28 pages, 2 figures, v2: minor corrections, v3: minor corrections and
references adde
Design-for-test structure to facilitate test vector application with low performance loss in non-test mode.
A switching based circuit is described which allows application of voltage test vectors to internal nodes of a chip without the problem of backdriving. The new circuit has low impact on the performance of an analogue circuit in terms of loss of bandwidth and allows simple application of analogue test voltages into internal nodes. The circuit described facilitates implementation of the forthcoming IEEE 1149.4 DfT philosophy [1]
Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncated
to N dimensions, and its up-down symmetrized special cases with J=[N/2] real
couplings are considered. In the strongly non-Hermitian regime the secular
equation gets partially factorized at all N. This enables us to reveal a
fine-tuned alignment of the dominant couplings implying an asymptotically
sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity
domain in which all the spectrum of energies remains real and observable.Comment: 28 pp., 4 tables, 1 figur
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