1,380 research outputs found
Quantum counterpart of spontaneously broken classical PT symmetry
The classical trajectories of a particle governed by the PT-symmetric
Hamiltonian () have been studied in
depth. It is known that almost all trajectories that begin at a classical
turning point oscillate periodically between this turning point and the
corresponding PT-symmetric turning point. It is also known that there are
regions in for which the periods of these orbits vary rapidly as
functions of and that in these regions there are isolated values of
for which the classical trajectories exhibit spontaneously broken PT
symmetry. The current paper examines the corresponding quantum-mechanical
systems. The eigenvalues of these quantum systems exhibit characteristic
behaviors that are correlated with those of the associated classical system.Comment: 11 pages, 7 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
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
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
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
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
Wall Crossing and Instantons in Compactified Gauge Theory
We calculate the leading weak-coupling instanton contribution to the
moduli-space metric of N=2 supersymmetric Yang-Mills theory with gauge group
SU(2) compactified on R^3 x S^1. The results are in precise agreement with the
semiclassical expansion of the exact metric recently conjectured by Gaiotto,
Moore and Neitzke based on considerations related to wall-crossing in the
corresponding four-dimensional theory.Comment: 24 pages, no 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
On the Uniqueness of the effective Lagrangian for N= 2 SQCD
The low energy effective Lagrangian for N= 2 SU(2) supersymmetric Yang-Mills
theory coupled to N_F<4 massless matter fields is derived from the BPS mass
formula using asymptotic freedom and assuming that the number of strong
coupling singularities is finite.Comment: 16 pages, LaTeX, 2 figures, title changed, sections on central charge
and superconformal anomaly extende
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