geometrical phase in the cyclic evolution of non hermitian systems the authors derive by a biorthonormal state approach the analogy of berry s phase factor for open non conservative systems for both adiabatic and non adiabatic evolution in the latter case a non unitary evolution operator method is exploited an application is given to the optical supermode propagation in the free electron laser

data processing inequalities based on a certain structured class of information measures with application to estimation theory we study data processing inequalities that are derived from a certain class of generalized information measures where a series of convex functions and multiplicative likelihood ratios is nested alternately while these information measures can be viewed as a special case of the most general zakai ziv generalized information measure this special nested structure calls for attention and motivates our study specifically a certain choice of the convex functions leads to an information measure that extends the notion of the bhattacharyya distance or the chernoff divergence while the ordinary bhattacharyya distance is based on the weighted geometric mean of two replicas of the channel s conditional distribution the more general information measure allows an arbitrary number of such replicas we apply the data processing inequality induced by this information measure to a detailed study of lower bounds of parameter estimation under additive white gaussian noise awgn and show that in certain cases tighter bounds can be obtained by using more than two replicas while the resulting lower bound may not compete favorably with the best bounds available for the ordinary awgn channel the advantage of the new lower bound relative to the other bounds becomes significant in the presence of channel uncertainty like unknown fading this different behavior in the presence of channel uncertainty is explained by the convexity property of the information measure

fake airy functions and the asymptotics of reflectionlessness two classes of analytic refractive index profile p2 z in whose reflection coefficients r are zero for all values of a parameter in are studied as in to 0 the aim is to understand why r 0 rather than r varies as exp 1 in as for generic profiles the authors find that reflectionlessness is a consequence of the fact that transition points of p2 zeros or poles in the complex z plane form tight clusters whose size vanishes with in which can be regarded neither as coalesced nor well separated expansion near a cluster yields the local wave not as the usual airy function whose stokes phenomenon generates reflection but as bessel functions of half integer order fake airy functions which are exactly trigonometric functions with no stokes phenomenon and so no reflection

a general class of lower bounds in parameter estimation a general class of bayesian lower bounds on moments of the error in parameter estimation is formulated and it is shown that the cramer rao the bhattacharyya the bobrovsky zakai and the weiss weinstein lower bounds are special cases in the class the bounds can be applied to the estimation of vector parameters and any given function of the parameters the extension of these bounds to multiple parameter is discussed

adiabatic passage for a lossy two level quantum system by a complex time method using a complex time method with the formalism of stokes lines we establish a generalization of the davis dykhne pechukas formula which gives in the adiabatic limit the transition probability of a lossy two state system driven by an external frequency chirped pulse shaped field the conditions that allow this generalization are derived we illustrate the result with the dissipative allen eberly and rosen zener models

threshold region performance of maximum likelihood direction of arrival estimators this paper presents a performance analysis of the maximum likelihood ml estimator for finding the directions of arrival doas with a sensor array the asymptotic properties of this estimator are well known in this paper the performance under conditions of low signal to noise ratio snr and a small number of array snapshots is investigated it is well known that the ml estimator exhibits a threshold effect i e a rapid deterioration of estimation accuracy below a certain snr or number of snapshots this effect is caused by outliers and is not captured by standard techniques such as the crame spl acute r rao bound and asymptotic analysis in this paper approximations to the mean square estimation error and probability of outlier are derived that can be used to predict the threshold region performance of the ml estimator with high accuracy both the deterministic ml and stochastic ml estimators are treated for the single source and multisource estimation problems these approximations alleviate the need for time consuming computer simulations when evaluating the threshold region performance for the special case of a single stochastic source signal and a single snapshot it is shown that the ml estimator is not statistically efficient as snr spl rarr spl infin due to the effect of outliers

hannay berry phase and the restricted three vortex problem abstract asymptotic expansions for the hannay berry phase are computed for a class of hamiltonian systems in perturbed action angle form with slowly varying parameters where the parameters recur on a slow time scale a multi scale perturbation method is used which in principle can be carried to any desired order of accuracy as a specific application we use the method to examine the restricted three vortex problem a completely integrable configuration of two co rotating point vortices a distance d apart and a tracer particle rotating around one of the vortices with distance r in the asymptotic limit r d e we compute the particle trajectory through o e and find that the hannay berry phase is given by 2 3 r 0 d o 1 d 2 where r 0 is the conserved action

a fresh look at the bayesian bounds of the weiss weinstein family minimal bounds on the mean square error mse are generally used in order to predict the best achievable performance of an estimator for a given observation model in this paper we are interested in the bayesian bound of the weiss weinstein family among this family we have bayesian cramer rao bound the bobrovsky mayerwolf zakai bound the bayesian bhattacharyya bound the bobrovsky zakai bound the reuven messer bound and the weiss weinstein bound we present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator mmsee and on a constrained optimization problem with this approach we obtain a useful theoretical framework to derive new bayesian bounds for that purpose we propose two bounds first we propose a generalization of the bayesian bhattacharyya bound extending the works of bobrovsky mayer wolf and zakai second we propose a bound based on the bayesian bhattacharyya bound and on the reuven messer bound representing a generalization of these bounds the proposed bound is the bayesian extension of the deterministic abel bound and is found to be tighter than the bayesian bhattacharyya bound the reuven messer bound the bobrovsky zakai bound and the bayesian cramer rao bound we propose some closed form expressions of these bounds for a general gaussian observation model with parameterized mean in order to illustrate our results we present simulation results in the context of a spectral analysis problem

on the real and complex geometric phases some of the existing definitions of the real and complex geometric phases are unified and generalized the relationship of the real to the complex geometric phase is discussed it is shown why these two phases are in general completely independent in particular a simple explicit example is given which shows that the real geometric phase is not in general the real part of the complex geometric phase

a minimum discrepancy estimator in parameter estimation in statistical estimation theory a satisfactory estimator should be able to embody a large portion of the available information which may be known a priori or provided by the data hence the loss of information is minimum when this estimator is employed in previous work an estimator criterion based on the discrepancy between the estimator s error covariance and its information lower bound was proposed conceptually this criterion is a measure of the loss of information carried by a parameter estimator based on the bayesian approach a minimum discrepancy estimator mde was derived under the linearity assumption it was however pointed out that the minimal information loss could not be guaranteed by the linear version moreover some good asymptotic properties were not obtainable therefore in this paper the existence and uniqueness conditions of the general mde are studied under certain regularity conditions the mde can be obtained by solving a fredholm equation of the second kind furthermore it is shown to be consistent and asymptotically efficient as a result the mde is ensured to have the minimum loss of information in finite samples and no loss of information when sample size tends to infinity examples indicate that if the prior information is vague the mde is superior to the minimum variance estimator mve in terms of information loss if the prior distribution is suitably chosen the mde is superior to the maximum likelihood estimator mle on the basis of deficiency

the ziv zakai renyi bound for joint source channel coding shannon s capacity and rate distortion function combined with the separation principle provide tight bounds for the minimum possible distortion in joint source channel coding these bounds however are usually achievable only in the limit of a large block length in their 1973 paper ziv and zakai introduced a family of alternative capacity and rate distortion functions based on functionals satisfying the data processing inequality which potentially give tighter bounds for systems with a small block length there is a considerable freedom as to how to choose those functionals and the ways of finding the best possible functionals yielding the best bounds for a given source channel combination are not specified we examine recently conjectured high snr asymptotic expressions for the ziv zakai bounds based on the renyi divergence functional we derive nonasymptotic bounds on the ziv zakai renyi rate distortion function and capacity for a broad class of sources and additive noise channels which hold for arbitrary snr and prove the conjectured asymptotic expressions in the limit of a small distortion high snr the results lead to new bounds on the best achievable distortion in finite dimensional joint source channel coding examples are presented where the new bounds achieve significant improvement upon shannon s original bounds

discrete cal pt symmetric models of scattering one dimensional scattering mediated by non hermitian hamiltonians is studied a schematic set of models is used which simulates two point interactions at a variable strength and distance the feasibility of the exact construction of the amplitudes is achieved via the discretization of the coordinate by direct construction it is shown that in all our models the probability is conserved this feature is tentatively attributed to the space and time reflection symmetry also known as symmetry of our specific hamiltonians

unitary transformations depending on a small parameter we formulate a unitary perturbation theory for quantum mechanics inspired by the liedeprit formulation of canonical transformations the original hamiltonian is converted into a solvable one by a transformation obtained through a magnus expansion this ensures unitarity at every order in a small parameter a comparison with the standard perturbation theory is provided we work out the scheme up to order ten with some simple examples

relations between belini tartara chazan zakai ziv and wax ziv lower bounds a class of lower bounds on the mean square error in parameter estimation is presented that are based on the belini tartara lower bound 1974 the wax ziv lower bound 1977 is shown to be a special case in the class these bounds often are significantly tighter than the chazan zakai ziv lower bound 1975 when the parameter to be estimated is subject to ambiguity and threshold effects

data processing theorems and the second law of thermodynamics we draw relationships between the generalized data processing theorems of zakai and ziv 1973 and 1975 and the dynamical version of the second law of thermodynamics a k a the boltzmann h theorem which asserts that the shannon entropy h xt pertaining to a finite state markov process xt is monotonically nondecreasing as a function of time t provided that the steady state distribution of this process is uniform across the state space which is the case when the process designates an isolated system it turns out that both the generalized data processing theorems and the boltzmann h theorem can be viewed as special cases of a more general principle concerning the monotonicity in time of a certain generalized information measure applied to a markov process this gives rise to a new look at the generalized data processing theorem which suggests to exploit certain degrees of freedom that may lead to better bounds for a given choice of the convex function that defines the generalized mutual information indeed we demonstrate an example of a certain setup of joint source channel coding where this idea yields an improved lower bound on the distortion relative to both the 1973 ziv zakai lower bound and the lower bound obtained from the ordinary data processing theorem

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

time dependent hamiltonians with 100 evolution speed efficiency the evolution speed in projective hilbert space is considered for hermitian hamiltonians and for non hermitian nh ones based on the hilbert schmidt norm and the spectral norm of a hamiltonian resource related upper bounds on the evolution speed are constructed these bounds are valid also for nh hamiltonians and they are illustrated for an optical nh hamiltonian and for a nh pt symmetric matrix hamiltonian furthermore the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective hilbert space a recipe for the construction of time dependent hamiltonians which ensure 100 speed efficiency is given generally these efficient hamiltonians are nh but there is a hermitian efficient hamiltonian as well finally the extremal case of a nh non diagonalizable hamiltonian with vanishing energy difference is shown to produce a 100 efficient evolution with minimal resources consumption

everything you always wanted to know about training guidelines derived using the affine precoding framework and the crb in this paper affine precoding is used to investigate the tradeoffs that exist while using the transmitter resources on training versus information symbols the channel input is a training vector superimposed on a linearly precoded vector of symbols a block fading frequency selective multi input multi output mimo channel is considered to highlight the tradeoffs between training and data symbols the fisher information matrix fim is derived under two circumstances the random parameter vector to be estimated contains 1 only fading channel coefficients and 2 unknown data symbols as well as the channel coefficients while strategy 1 corresponds to the receiver structure in which the channel is estimated initially and the channel measurement is utilized to retrieve the data symbols strategy 2 corresponds to the structure in which channel and symbol estimations are performed jointly the interesting outcome of the study in this paper is that minimizing the channel cramer rao bound crb for strategies 1 and 2 under a total average transmit power constraint leads to different affine precoder design guidelines

a class of weiss weinstein bounds and its relationship with the bobrovsky mayer wolf zakai bounds a fairly general class of bayesian large error lower bounds of the weiss weinstein family essentially free from regularity conditions on the probability density functions support and for which a limiting form yields a generalized bayesian cramer rao bound bcrb is introduced in a large number of cases the generalized bcrb appears to be the bobrovsky mayer wolf zakai bound bmzb interestingly enough a regularized form of the bobrovsky zakai bound bzb applicable when the support of the prior is a constrained parameter set is obtained modified weiss weinstein bound and bzb which limiting form is the bmzb are proposed in expectation of an increased tightness in the threshold region some of the proposed results are exemplified with a reference problem in signal processing the gaussian observation model with parameterized mean and uniform prior

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

four easy pieces this paper is a mini summary of four new advances in the theory of pt symmetric quantum mechanics it describes some new calculations that were completed in 2005 the first advance concerns the classical coordinate space trajectories in some pt symmetric theories depending on the initial conditions one can find arbitrarily long periodic pt symmetric classical trajectories the longer trajectories originate from smaller regions of initial conditions there is an interesting resemblance to the so called period doubling route to chaos the second advance concerns the perturbative construction of the c operator for a pt symmetric square well the result has notable similarities to and differences from that for the pt symmetric cubic oscillator the third advance is a detailed comparison of a pt symmetric hamiltonian and the corresponding hermitian hamiltonian it is argued that a perturbative construction of the hermitian hamiltonian leads to an almost intractable theory the fourth advance concerns reflectionless potentials and pt symmetry and a possible connection with problems in cosmology

superadiabatic tracking of quantum evolution the adiabatic quantum evolution of a two state system without energy level crossings is an example of the stokes phenomenon in the latter a small subdominant exponential is an asymptotic expansion appears when a stokes line is crossed truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth compact and universal manner across the stokes line in quantum evolution this corresponds to a smooth transition universal in form between superadiabatic basis states high order wkb approximate solutions of the time dependent schrodinger equation the authors give a numerical demonstration of this previously predicted universality by constructing for two hamiltonians the superadiabatic quantum bases asymptotic to the actual evolving state universality when a stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term

geometric phase bundle classification and group representation the line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the borel weil bott theorem of the representation theory the remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the borel weil bott theorem to berry s adiabatic phase this enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical lie groups these charges signify the topological content of the phase they can be explicitly computed in this paper the problem of the determination of the parameter space of the hamiltonian is also addressed it is shown that in general the parameter space is either a flag manifold or one of its submanifolds a simple topological argument is presented to indicate the relation between the riemannian structure on the parameter spa

a lower bound on the mean square error in random parameter estimation corresp a new lower bound on mean square error in parameter estimation is presented the bound is tighter than the cramer rao and bobrovsky zakai lower bounds it requires no bias or regularity assumptions it is computationally simple and it can be applied to estimates of vector parameters or functions of the parameters

a comment on the weiss weinstein bound for constrained parameter sets the weiss weinstein bound wwb provides a lower limit on the mean squared error mse achievable by an estimator of an unknown random parameter in this correspondence it is shown that some previously proposed simplified versions of the bound do not always hold for constrained parameters i e parameters whose distribution has finite support these simplifications can produce results which are no longer lower bounds on the mse sufficient conditions are provided for the reductions to be valid

non adiabatic berry phase for periodic hamiltonians berry s 1984 formulation of the topological phase for periodic hamiltonians is extended to the case of non adiabatic evolution by proper choice of initial states this restores a full parallel with the non adiabatic formalism of aharonov and anandan 1987 for periodic orbits in a projective hilbert space the berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by aharonov and anandan and by page 1987

general classes of performance lower bounds for parameter estimation part ii bayesian bounds in this paper a new class of bayesian lower bounds is proposed derivation of the proposed class is performed via projection of each entry of the vector function to be estimated on a hilbert subspace of l2 this hilbert subspace contains linear transformations of elements in the domain of an integral transform applied on functions used for computation of bounds in the weiss weinstein class the integral transform generalizes the traditional derivative and sampling operators used for computation of existing performance lower bounds such as the bayesian cramer rao bayesian bhattacharyya and weiss weinstein bounds it is shown that some well known bayesian lower bounds can be derived from the proposed class by specific choice of the integral transform kernel a new lower bound is derived from the proposed class using the fourier transform kernel the proposed bound is compared with other existing bounds in terms of signal to noise ratio snr threshold region prediction in the problem of frequency estimation the bound is shown to be computationally manageable and provides better prediction of the snr threshold region exhibited by the maximum a posteriori probability map and minimum mean square error mmse estimators

a general class of outage error probability lower bounds in bayesian parameter estimation in this paper a new class of lower bounds on the outage error probability in bayesian parameter estimation is proposed the outage error probability is an important criterion in parameter estimation that provides meaningful information even in the presence of large errors and is useful for prediction of the system operation region computation of the minimum outage error probability is usually not tractable and thus lower bounds on this probability can be very useful for performance analysis the proposed class of lower bounds on the outage error probability is derived using holder s inequality several bounds in the proposed class are presented it is shown that the ziv zakai lower bound on the outage error probability can be obtained from a subclass in the proposed class of bounds the proposed class of bounds is utilized to derive a new class of bayesian bounds on the mean square error it is shown that for unimodal posterior probability density functions the tightest lower bound on the probability of outage error in the proposed class attains the minimum probability of outage error the proposed bounds are exemplified in linear gaussian model parameter estimation and time delay estimation

application of the weiss weinstein bound to a two dimensional antenna array the weiss weinstein ww bound is applied to an 8 8 element uniform antenna array for a single source impinging on the array the ww bound for this problem is evaluated and compared to simulations of the music algorithm and classical beamforming the simulations show that for high signal to noise ratio snr the ww bound is met by both algorithms however the threshold for music and classical beamforming occurs at a higher snr than the bound predicts this suggests the possibility of other spectrum estimation algorithms with much lower thresholds than those currently used today

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

analysis of berry s phase by the evolution operator method berry 1984 discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase aharonov and anandan 1987 on the other hand showed that an eigenstate transporting round a closed circuit in the projective hilbert space suffices to generate the topological phase the authors employ the evolution operator method to study the propagation of an eigenstate they show that berry s phase can be represented as a closed path integral in the hilbert space and is independent of the choice for the base mod n t the treatments of berry and aharonov and anandan are shown to correspond to two different choices of the base therefore their two approaches are unified they have acquired a more general viewpoint on the origin of berry s phase

a lower bound on the bayesian mse based on the optimal bias function a lower bound on the minimum mean squared error mse in a bayesian estimation problem is proposed in this paper this bound utilizes a well known connection to the deterministic estimation setting using the prior distribution the bias function which minimizes the cramer rao bound can be determined resulting in a lower bound on the bayesian mse the bound is developed for the general case of a vector parameter with an arbitrary probability distribution and is shown to be asymptotically tight in both the high and low signal to noise ratio snr regimes a numerical study demonstrates several cases in which the proposed technique is both simpler to compute and tighter than alternative methods

spiked potentials and quantum toboggans even if the motion of a quantum quasi particle proceeds along a left right symmetric symmetric curved path in complex plane the spectrum of bound states may remain physical i e real and bounded below a generalization is outlined first we show how the topologically less trivial tobogganic contours may be allowed to live on several sheets of a riemann surface second the specification of a scattering regime is formulated for such a class of models

superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum during the last years several theoretical and experimental works have been devoted to different aspects of the adiabatic regime of quantum dynamics we consider here two aspects of this physically important regime the construction of approximate evolutions of the schriidinger equation called superadiabatic evolutions and the explicit computation of the probability of a transition between two nondegenerate energy levels isolated in the spectrum of the hamiltonian from the mathematical point of view the adiabatic limit for time dependent systems is of the same kind as the semiclassical limit for stationary systems it is a singular limit which cannot be tackled by usual perturbation methods recently a mathematical theory of the two considered aspects has been developed when the time dependent hamiltonian is analytic in time the transition probability for two level systems has been studied in refs l 3 and was shown to be exponentially small in the adiabaticity parameter for general systems in ref 4 on the other hand the existence of superadiabatic evolutions has been shown in ref 5 however no general formula for the transition probability between isolated levels in the adiabatic limit can be found in the literature it is the goal of this paper to establish such a formula for general unbounded hamiltonians the paper is divided into two parts we first construct a superadiabatic evolution which approximates the true evolution of the schrijdinger equation up to exponentially small correction terms with respect to the parameter controlling the adiabatic regime once the superadiabatic evolution is defined we show that the computation of the transition probability of the initial problem reduces to an effective two dimensional problem which we analyze as in ref 1 the concept of superadiabatic evolution can be traced back up to a paper of garrido 6 who considered evolutions different from the usual adiabatic evolution which he used to obtain higher order corrections to the adiabatic theorem of quantum mechanics but it is berry7v8 who emphasized the importance of such evolutions in the present problem and introduced the

quantum canonical transformations and exact solution of the schrodinger equation time dependent unitary transformations are used to study the schrodinger equation for explicitly time dependent hamiltonians of the form h t r t j where r is an arbitrary real vector valued function of time and j is the angular momentum operator the solution of the schrodinger equation for the most general hamiltonian of this form is shown to be equivalent to the special case r 1 0 t this corresponds to the problem of a driven two level atom for the spin half representation of j it is also shown that by requiring the magnitude of r to depend on its direction in a particular way one can solve the schrodinger equation exactly in particular it is shown that for every hamiltonian of the form h t r t j there is another hamiltonian with the same eigenstates for which the schrodinger equation is exactly solved the application of the results to the exact solution of the parallel transport equation and exact holomony calculation for su 2 principal bundles yang mills gauge theory is also pointed

variation of the action in the classical time dependent harmonic oscillator an exact result from the exact solution of certain time dependent classical harmonic oscillators in one dimension the authors investigate the behaviour of the adiabatic invariant j t for a subclass of such potentials delta j identical to j infinity j infinity 0 whatever the regime they show that this does not necessarily imply a breakdown of the commonly accepted asymptotic exponential law for delta j

semiclassically weak reflections above analytic and non analytic potential barriers the coefficient r for reflection above a barrier v x is computed semiclassically i e as h cross to 0 employing an exact multiple reflection series whose mth term is a 2m 1 fold integral if v x is analytic all terms have the same semiclassical order exp h cross 1 the multiple integrals are evaluated exactly and the series summed if v x has a discontinuous nth derivative the term m 1 dominates semiclassically and gives r approximately h cross n if v x has all derivatives continuous but possesses an essential singularity on the real axis the term m 1 again dominates semiclassically and for v approximately exp mod x mod n gives r approximately exp h cross n n 1 with an oscillatory factor corresponding to transmission resonances the formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de broglie wavelength is less than 1 of the barrier width and mod r mod 2 approximately 10 1000

weiss weinstein lower bounds for markovian systems part 1 theory being essentially free from regularity conditions the weiss weinstein estimation error lower bound can be applied to a larger class of systems than the well known crameacuter rao lower bound thus this bound is of special interest in applications involving hybrid systems i e systems with both continuously and discretely distributed parameters which can represent in practice fault prone systems however the requirement to know explicitly the joint distribution of the estimated parameters with all the measurements makes the application of the weiss weinstein lower bound to markovian dynamic systems cumbersome a sequential algorithm for the computation of the crameacuter rao lower bound for such systems has been recently reported in the literature along with the marginal state distribution the algorithm makes use of the transitional distribution of the markovian state process and the distribution of the measurements at each time step conditioned on the appropriate states both easily obtainable from the system equations a similar technique is employed herein to develop sequential weiss weinstein lower bounds for a class of markovian dynamic systems in particular it is shown that in systems satisfying the crameacuter rao lower bound regularity conditions the sequential weiss weinstein lower bound derived herein reduces for a judicious choice of its parameters to the sequential crameacuter rao lower bound

the optical singularities of birefringent dichroic chiral crystals using a new formalism involving projection from the sphere of directions to the stereographic plane and associated complex variables explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic absorbing and chiral optically active this enables three types of polarization singularity to be classified and explored singular axes which are degeneracies where the two refractive indices are equal and which for a transparent non chiral crystal condense pairwise onto the optic axes c points where the polarization is purely circular right or left handed with topological index 1 or and whose positions are independent of the chirality and l lines where the polarization is purely linear dividing direction space into regions with right and left handedness a local model captures essential features of the general theory interference figures generated by slabs of crystal viewed directly or through a polarizer and or analyser enable the singularities to be displayed directly

scattering from a discrete quasi hermitian delta function potential scattering from a discrete quasi hermitian delta function potential is studied and the metric operator is found a generalized continuity relation in the physical hilbert space is derived and the probability current density is defined the reflection and transmission coefficients computed with this current are shown to obey the unitarity relation

uniform asymptotic expansions of integrals a selection of problems abstract on the occasion of the conference we mention examples of stieltjes work on asymptotics of special functions the remaining part of the paper gives a selection of asymptotic methods for integrals in particular on uniform approximations we discuss several standard problems and examples in which known special functions error functions airy functions bessel functions etc are needed to construct uniform approximations finally we discuss the recent interest and new insights in the stokes phenomenon an extensive bibliography on uniform asymptotic methods for integrals is given together with references to recent papers on the stokes phenomenon for integrals and related topics

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

stokes s phenomenon for superfactorial asymptotic series superfactorial series depending on a parameter z are those whose terms a n z grow faster than any power of n if the terms get smaller before they increase the function f z represented by sum 0 infty a n z will exhibit a stokes phenomenon similar to that occurring in asymptotic series whose divergence is merely factorial across stokes lines in the z plane where the late terms all have the same phase a small exponential switches on in the remainder when the series is truncated near its least term the jump is smooth and described by an error function whose argument has a slightly more general form than in the factorial case this result is obtained by a method which is heuristic but applies to superfactorial series where borel summation fails several examples are given including an analytical interpretation of the sum and a numerical test of the error function formula for the function represented by f z sum 0 infty exp n 2 a 2nz where a gg 1

a pessimistic approximation for the fisher information measure the problem of determining the intrinsic quality of a signal processing system with respect to the inference of an unknown deterministic parameter theta is considered while the fisher information measure f theta forms a classical tool for such a problem direct computation of the information measure can become difficult in various situations for the estimation theoretic performance analysis of nonlinear measurement systems the form of the likelihood function can make the calculation of the information measure f theta challenging in situations where no closed form expression of the statistical system model is available the analytical derivation of f theta is not possible at all based on the cauchy schwarz inequality we derive an alternative information measure s theta it provides a lower bound on the fisher information f theta and has the property of being evaluated with the mean the variance the skewness and the kurtosis of the system model at hand these entities usually exhibit good mathematical tractability or can be determined at low complexity by real world measurements in a calibrated setup with various examples we show that s theta provides a good conservative approximation for f theta and outline different estimation theoretic problems where the presented information bound turns out to be useful

a new barankin bound approximation for the prediction of the threshold region performance of maximum likelihood estimators it is well known that the ml estimator exhibits a threshold effect i e a rapid deterioration of estimation accuracy below a certain signal to noise ratio snr or number of snapshots this effect is caused by outliers and is not captured by standard tools such as the cramer rao bound crb the search of the snr threshold value where the crb becomes unreliable for prediction of maximum likelihood estimator variance can be achieved with the help of the barankin bound bb as proposed by many authors the major drawback of the bb in comparison with the crb is the absence of a general analytical formula which compels one to resort to a discrete form usually the mcaulay seidman bound msb requesting the search of an optimum over a set of test points in this paper we propose a new practical bb discrete form that provides for a given set of test points an improved snr threshold prediction in comparison with existing approximations msb abel bound mcaulay hofstetter bound at the expense of the computational complexity increased by a factor les p 1 3 where p is the number of unknown parameters we have derived its expression for the general gaussian observation model to be used in place of existing approximations

delta function potential with a complex coupling we explore the hamiltonian operator where is the dirac delta function and z is an arbitrary complex coupling constant for a purely imaginary z h has a spectral singularity at for re z 0 h has a real positive continuous spectrum that is free from spectral singularities for this latter case we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive definite inner product that renders h self adjoint this allows us to address the intriguing question of the nonlocal aspects of the equivalent hermitian hamiltonian for the system in particular we compute the energy expectation values for various gaussian wave packets to show that the non hermiticity effect diminishes rapidly outside an effective interaction region

linear adiabatic theory exponential estimates a general adiabatic expansion is written down the basic result is that under appropriate smoothness conditions the adiabatic estimations can be pushed up to exponential order the results imply exponential estimates not only for hamiltonians analytic in a neighbourhood of the real axis but also for hamiltonians which in an appropriate sense beong to gevrey classes

weiss weinstein lower bounds for markovian systems part 2 applications to fault tolerant filtering characterized by sudden structural changes fault prone systems are modeled using the framework of systems with switching parameters or hybrid systems since a closed form mean square optimal filtering algorithm for this class of systems does not exist it is of particular interest to derive a lower bound on the state estimation error covariance the well known crameacuter rao bound is not applicable to fault prone systems because of the discrete distribution of the fault indicators which violates the regularity conditions associated with this bound on the other hand the weiss weinstein lower bound is essentially free from regularity conditions moreover a sequential version of the weiss weinstein bound suitable for markovian dynamic systems is presented by the authors in a companion paper in the present paper this sequential version is applied to several classes of fault prone dynamic systems the resulting bounds can be used to examine fault detectability and identifiability in these systems moreover it is shown that several recently reported lower bounds for fault prone systems are special cases of or closely related to the sequential version of the weiss weinstein lower bound

the role of the geometric phases in adiabatic population tracking for non hermitian hamiltonians the definition of instantaneous eigenstate populations for a dynamical non self adjoint system is not obvious the nadirect extension of the definition used for the self adjoint case leads to inconsistencies the resulting artifacts can induce a false inversion of population or a false adiabaticity we show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations an example is given which demonstrates both the anomalous effects and their removal by our approach

front formation in an active scalar equation we study the formation of thermal fronts in an active scalar equation that is similar to the euler equation for a particular initial condition an earlier candidate for finite time blowup the front forms in a generalized self similar way with constant hyperbolicity at the center the behavior belongs to a class of scenarios for which finite time blowup is impossible a systematic exploration of many different initial conditions reveals no evidence of singular solutions copyright ital 1999 ital the american physical society

on the removability of berry s phase the authors present a thorough analysis of the removability of the berry phase they show that for any system they can by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one however this extra shift in the dynamical phase retains its truly geometrical character the removability is thus only apparent and the geometrical phase can still be measured in interference experiments as indeed has already been done several times

universal oscillations of high derivatives differentiation generates oscillations for the n th derivative f n t of a function f t that is analytic in a strip including the real t axis the oscillations occupy a t interval that gets larger as n increases the oscillations are studied in detail using integral representations and large n asymptotics for functions with singularities poles or branch points in the complex t plane the oscillations of high derivatives are determined by the singularities for entire functions the oscillations originate in complex saddle points in a wide class of cases the oscillations are contained in a gaussian envelope in the t interval where f n t is largest with the envelope including about graphic 1 oscillations examples of the universal oscillations are given for f t with a simple pole competing branch points a single saddle competing pole and saddle and where the zeros are confined to half the real axis 1 embed inline graphic 1 gif

on the evolution of sharp fronts for the quasi geostrophic equation we consider the problem of the evolution of sharp fronts for the surface quasi geostrophic qg equation this problem is the analogue to the vortex patch problem for the two dimensional euler equation the special interest of the quasi geostrophic equation lies in its strong similarities with the three dimensional euler equation while being a two dimen sional model in particular an analogue of the problem considered here the evolution of sharp fronts for qg is the evolution of a vortex line for the three dimensional euler equation the rigorous derivation of an equation for the evolution of a vortex line is still an open problem the influence of the singularity appearing in the velocity when using the biot savart law still needs to be understood we present two derivations for the evolution of a periodic sharp front the first one heuristic shows the presence of a logarithmic singularity in the velocity while the second making use of weak solutions obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve finally using a nash moser argument as the main tool we obtain local existence and uniqueness of a solution for the derived equation in the c case 2004 wiley periodicals inc

lower bounds on parameter modulation estimation under bandwidth constraints the problem of modulating the value of a parameter onto a band limited signal to be transmitted over a continuous time additive white gaussian noise awgn channel and then estimating this parameter at the receiver is considered the performance is measured by the mean power alpha error mp alpha text e which is defined as the worst case alpha th order moment of the absolute estimation error the optimal exponential decay rate of the mp alpha text e as a function of the transmission time is investigated two upper converse bounds on the mp alpha text e exponent are derived on the basis of known bounds for the awgn channel of inputs with unlimited bandwidth the bounds are computed for typical values of the error moment and the signal to noise ratio snr and the snr asymptotics of the different bounds are analyzed the new bounds are compared with known converse and achievability bounds which were derived from channel coding considerations

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

generalized swanson models and their solutions we analyse a class of non hermitian quadratic hamiltonians which are of the form where are real constants with and and are generalized creation and annihilation operators respectively thus these hamiltonians may be classified as generalized swanson models it is shown that the eigenenergies are real for a certain range of values of the parameters a similarity transformation mapping the non hermitian hamiltonian h to a hermitian one h is also obtained it is shown that h and h share identical energies as explicit examples the solutions of a couple of models based on the trigonometric rosen morse i and the hyperbolic rosen morse ii type potentials are obtained we also study the case where the non hermitian hamiltonian is symmetric

on the observability and asymmetry of adiabatic state flips generated by exceptional points in open quantum systems where the effective hamiltonian is not hermitian it is known that the adiabatic or instantaneous basis can be multivalued by adiabatically transporting an eigenstate along a closed loop in the parameter space of the hamiltonian it is possible to end up in an eigenstate different from the initial eigenstate this adiabatic flip effect is an outcome of the appearance of a degeneracy known as an exceptional point inside the loop we show that contrary to what is expected of the transport properties of the eigenstate basis the interplay between gain loss and non adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect

exponential error bounds on parameter modulation estimation for discrete memoryless channels we consider the problem of modulation and estimation of a random parameter u to be conveyed across a discrete memoryless channel upper and lower bounds are derived for the best achievable exponential decay rate of a general moment of the estimation error e u u 0 when both the modulator and the estimator are subjected to optimization these exponential error bounds turn out to be intimately related to error exponents of channel coding and to channel capacity while in general there is some gap between the upper and the lower bounds they asymptotically coincide both for very small and for very large values of the moment power this means that our achievability scheme which is based on simple quantization of u followed by channel coding is nearly optimum in both limits some additional properties of the bounds are discussed and demonstrated and finally an extension to the case of a multidimensional parameter vector is outlined with the principal conclusion that our upper and lower bounds asymptotically coincide also for a high dimensionality

fundamental limitations on the resolution of deterministic signals we exploit results from detection theory for deriving fundamental limitations on resolution the results are general and are not based on any specific resolution technique and therefore hold for any method and for any resolution success rate we show that for signals with additive white gaussian noise the resolution limit is approximated by a simple expression related to the signal to noise ratio the signal waveform and the resolution success rate as an example we discuss the resolution of two complex exponentials with closely spaced frequencies the theoretical result is compared with the empirical performance of model order selection methods such as the akaike information criterion and the minimum description length

universal transition prefactors derived by superadiabatic renormalization for time dependent two state quantum systems the transition probability is exponentially small in the adiabatic parameter in with the exponent determined by a transition point tc in the complex time plane the authors study the in independent prefactors associated with different sorts of transition point which need not correspond to complex degeneracies of the adiabatic energy unlike previous approaches the method they use does not make use of special functions it consists of applying first order perturbation theory to the schrodinger equation obtained by transforming to a series of superadiabatic bases clinging ever more closely to the evolving state if the original matrix elements share a leading singularity t tc r and their fractional deviation from this is t tc s the prefactor is 4 sin2 pi s 2 2r s 2 this is universal in the sense of being invariant under time reparametrization and quantum changes of frame

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

berry s phases for arbitrary spins nonlinearly coupled to external fields application to the entanglement of n 2 non correlated 1 2 spins we derive the general formula giving berry s phase for an arbitrary spin having both magnetic dipole and electric quadrupole couplings with external time dependent fields we assume that the effective electric and magnetic fields remain orthogonal during the quantum cycles this mild restriction has many advantages it provides simple symmetries leading to useful selection rules furthermore the hamiltonian parameter space coincides with the density matrix space for a spin s 1 this implies a mathematical identity between berry s phase and the aharonov anandan phase which is lost for s 1 we have found that new physical features of berry s phases emerge for integer spins 2 we provide explicit numerical results of berry s phases for s 2 3 4 for any spin one easily finds well defined regions of the parameter space where instantaneous eigenstates remain widely separated we have decided to stay within them and not deal with non abelian berry s phases we present a thorough and precise analysis of the non adiabatic corrections with separate treatment for periodic and non periodic hamiltonian parameters in both cases the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time dependence having a blackman pulse shape this has the effect of taming the non adiabatic oscillation corrections which could be generated by a linear ramping for realistic experimental conditions the non adibatic corrections can be kept below the 0 1 level for a class of quantum cycles involving as a sole periodic parameter the precession angle of the electric field around the magnetic field the corrections odd upon the reversal of the associated rotation velocity can be canceled exactly if the quadrupole to dipole coupling is chosen appropriately magic values the even ones are eliminated by taking the difference of the berry phases generated by two mirror cycles we end by indicating a possible application of the theoretical tools developed in this paper we propose a way to perform a holonomic entanglement of n non correlated 1 2 spins by performing adiabatic cycles governed by a hamiltonian given as a nonlinear function of the total spin operator s defined as the sum of the n individual spin operators the basic idea behind this proposal is the mathematical fact that any non correlated states can be expanded into eigenstates of s2 and sz the same eigenvalues appear several times in the decomposition when n 2 but all these states differ by their symmetry properties under the n spin permutations the case n 4 and sz 1 is treated explicitly and a maximum entanglement is achieved

reflectionlessness kurtosis and top curvature of potential barriers apart from the rectangular barrier other barriers having a single maximum generally display reflectivity r e as a smoothly decreasing function of energy we conjecture that symmetric potential barriers with a single maximum entail zeros or sharp minima in r e provided they have either their coefficient of kurtosis lying in the range 1 8 3 0 or their top curvature as zero or both

an inviscid regularization for the surface quasi geostrophic equation inspired by recent developments in berdina like models for turbulence we propose an inviscid regularization for the surface quasi geostrophic sqg equations we are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the sqg equations the new regularization yields a necessary and sufficient condition satisfied by the regularized solution when a regularization parameter tends to 0 for the solution of the original sqg equations to develop a singularity in finite time as opposed to the commonly used viscous regularization the inviscid equations derived here conserve a modified energy therefore the new regularization provides an attractive numerical procedure for finite time blowup testing in particular we prove that if the initial condition is smooth then the regularized solution remains as smooth as the initial data for all times 2007 wiley periodicals inc

parallel transport along a space curve and related phases the authors investigate geometrical properties of a space curve and of its spherical images they show that a space curve is characterised by two phase like quantities and comment on the relation of these quantities to the berry phase

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

slow non hermitian cycling exact solutions and the stokes phenomenon for non hermitian hamiltonians with an isolated degeneracy exceptional point a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of bessel functions floquet solutions returning exactly to their initial states up to a constant are found as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous hamiltonian adiabatically i e for slow cycles the multipliers of exponentially subdominant eigenstates can vary wildly unlike those driven by hermitian operators which change little these variations are explained as an example of the stokes phenomenon of asymptotics improved superadiabatic approximations tame the variations of the multipliers but do not eliminate them

full asymptotic expansion of transition probabilities in the adiabatic limit the authors consider a two level quantum mechanical system driven by an analytic time dependent hamiltonian of the form h in t in the adiabatic limit in or 0

on optimum parameter modulation estimation from a large deviations perspective we consider the problem of jointly optimum modulation and estimation of a real valued random parameter conveyed over an additive white gaussian noise awgn channel where the performance metric is the large deviations behavior of the estimator namely the exponential decay rate as a function of the observation time of the probability that the estimation error would exceed a certain threshold our basic result is in providing an exact characterization of the fastest achievable exponential decay rate among all possible modulator estimator transmitter receiver pairs where the modulator is limited only in the signal power but not in bandwidth this exponential rate turns out to be given by the reliability function of the awgn channel we also discuss several ways to achieve this optimum performance and one of them is based on quantization of the parameter followed by optimum channel coding and modulation which gives rise to a separation based transmitter if one views this setting from the perspective of joint source channel coding this is in spite of the fact that in general when error exponents are considered the source channel separation theorem does not hold true we also discuss several observations modifications and extensions of this result in several directions including other channels and the case of multidimensional parameter vectors one of our findings concerning the latter is that there is an abrupt threshold effect in the dimensionality of the parameter vector below a certain critical dimension the probability of excess estimation error may still decay exponentially but beyond this value it must converge to unity

energy spectra of steady two dimensional turbulent flows more than 30 years ago the energy spectrum of twodimensional turbulence was theoretically predicted yet the evidence for this spectrum has remained weak in some cases classical theory 1 3 predicts the energy spectrum based on a cascade of either energy or enstrophy once the relevant physical quantities have been identified simple dimensional analysis leads to the power spectrum as a function of wave number k however the spectra measured in many numerical simulations decay substantially faster with k than predicted 4 the range of wave numbers larger than the inverse system size and up to the dissipative damping is called inertial range in this paper we assume power laws for the inertial range as a working hypothesis since possible deviations from a power behavior cannot be determined from numerical data with confidence a general review of two dimensional turbulence and the slope of the energy spectrum can be found in refs 5 6 following we summarize recent results on the inertial range spectrum

a time dependent born oppenheimer approximation with exponentially small error estimates we present the construction of an exponentially accurate time dependent born oppenheimer approximation for molecular quantum mechanics

reflectionless potentials and pt symmetry large families of hamiltonians that are non hermitian in the conventional sense have been found to have all eigenvalues real a fact attributed to an unbroken pt symmetry the corresponding quantum theories possess an unconventional scalar product the eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane for a special class of such systems it is possible to impose the pt symmetric boundary conditions on the real axis which lies on the edges of the wedges the pt symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless

spectral singularity and reflectionlessness in non hermitian complex ginocchio potential recently interest in scattering from complex non hermitian potentials has been renewed both theoretically and experimentally when a potential becomes complex several new features arise which may not be predicted in this regard we study the non hermitian version of the exactly solvable versatile ginocchio potential we find that when it is non hermitian and non absorptive there exists at most one spectral singularity ss the well known reflectionlessness of this potential gives way to deep minima in reflectivity we find the co existence of an ss with deep oscillations in reflectivity remarkably the complex pt symmetric version of this potential does not entail any ss in this case the reflectionlessness at discrete energies persists with the handedness of reflectivity and non unitarity in the new parlance this would be called the invisibility of an optical potential but in this case uniquely it is from both sides left and right

on mmse estimation a linear model under gaussian mixture statistics in a bayesian linear model suppose observation y hx n stems from independent inputs x and n which are gaussian mixture gm distributed with known matrix h the minimum mean square error mmse estimator for x has analytical form however its performance measure the mmse itself has no such closed form because existing bayesian mmse bounds prove to have limited practical value under these settings we instead seek analytical bounds for the mmse both upper and lower this paper provides such bounds and relates them to the signal to noise ratio snr