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In the present work neural network has been developed for the prediction of Vertical Total Electron Content (VTEC) over Bhopal (23.2°N, 77.4°E & MLAT 14.2°N), an equatorial anomaly crest region. The study is based on the VTEC data recorded by a multi-frequency multi-constellation GNSS receiver installed at Bhopal during May 2016 to August 2017. Neural Network (NN) consists of artificial neurons that are used to learn data patterns adjusting weights between input, hidden and output layers by computational modelling. In order to obtain the optimal number of neurons in the hidden layer, the performance of four systems of neural network have been tested. Out of which, the fourth neural network has been used to predict VTEC with 7 input neurons in the present study. The number of hidden layer neurons is a factor that affects the performance of the trained networks. The input layer of the network includes the year, day of the year, hour of the day, geographic latitude, geographic longitude, SSN and IRI NmF2. We have compared the TEC obtained from the neural network prediction (NN TEC), GPS derived TEC (GPS TEC) and TEC from IRI-2016 model (IRI TEC) and analysis has been made. We observed that the Root Mean Square Error (RMSE) decreases with increase in the number of hidden layer neurons. The RMSE comes out to be lowest (1.96 TECU), when the number of hidden layer neurons are 41.The TEC predicted from the NN model and estimated from GPS receiver have same trend at 01:00 and 07:00 UT. The modelled NN TEC values are in good agreement with GPS TEC values for all seasons and 0.99 correlation coefficient was calculated for summer months, but IRI TEC always underestimates GPS TEC in all seasons. The NN model excellently predicted the TEC values of August 2017 by putting the credentials of August 2016 and only 0.55–32.16% relative errors were estimated between them whereas it reaches up to 93.86% with IRI TEC. Thus, the result shows that the proposed NN model can predict the diurnal and seasonal GPS TEC more accurate than IRI over anomaly crest region Bhopal. However, the inclusion of IRI-NmF2 as input layer neuron increases the network performance.

Nighttime enhancement in the Vertical Total Electron Content (VTEC) of the ionosphere near equatorial ionization anomaly crest region in the Indian ionospheric sector has been studied during May 2016 to August 2017, a solar minimum period. VTEC data is obtained from a multi-frequency multi-constellation global navigation satellite system (GNSS) receiver installed at Bhopal (23.2° N, 77.4° E & MLAT 14.2° N). Only those enhancements which show VTEC enhancement ≥10% compared to the background has been considered in the study. Out of the total 105 cases of enhancement recorded during this period, 86 events occurred during pre-midnight and 19 during post-midnight hours. Enhancements in VTEC occurred in all seasons; more frequent however during summer than winter and equinox months. Some of the nighttime enhancements in VTEC resulted in the scintillations in the satellite signals as well. Peak amplitude of the enhancement maximizes in summer whereas its half amplitude duration maximizes during equinox month. The number of occurrences of nighttime enhancement in VTEC shows negative correlation with peak amplitude while they are positively correlated with half amplitude duration. We have found that the peak amplitude of nighttime enhancements in VTEC shows strong positive correlation while percentage occurrence shows strong negative correlation with solar and magnetic activities respectively. Other characteristics like half amplitude duration, time of peak and occurrence time of nighttime enhancement in VTEC show positive correlation with solar activity while they are insignificant with magnetic activity. The peak amplitude of pre-midnight enhancements in VTEC is highly correlated with the strength of equatorial electrojet strength. The results are compared with the earlier observations and possible mechanism is also discussed.

We consider a simple harmonic oscillator with non-Hermitian term and study it classically and quantum mechanically. We conclude that this version of oscillator, which breaks parity and time reversal, displays all the features possessed by the usual harmonic oscillator. In particular, we calculate its spectrum, adiabatic invariance and Wigner functions, and show that there is a consistency between the classical and quantum descriptions.

α-decay half-lives ( T 1 / 2 α -values) of even–even and superheavy nuclei (SHN) are determined employing interaction potential involving Coulomb and proximity potentials. The obtained results are compared with the experimental ones and to assess the precision of the present model in reproducing the experimental half-lives, a comparison is drawn with the other methods as well. We require disintegration energy (Q α -values) as an input in order to predict the ( T 1 / 2 α -values) of an experimentally unknown SHN. So, utilizing periodic-orbit theory within microscopic-macroscopic formalism, we have calculated the binding energies and thereafter, Q α -values of known SHN. The results are found to be in good agreement with the experimental ones. Subsequently, we predict the logarithmic values of T 1 / 2 α ’s for even–even unknown SHN whose atomic numbers lie in 118 ⩽ Z ⩽ 126 regime and compare our results with those calculated employing other theoretical methods. Also, the study of spontaneous fission half-lives and branching ratios leads us to predict the SHN which can be identified through α-decay chains in the laboratories. We believe that this work can play a significant role in the experiments leading to the synthesis and identification of new superheavy elements.

We study the phenomenon of neutrino spin-flavor oscillations due to solar magnetic fields. This allows us to examine how significantly the electron neutrinos produced in the solar interior undergo a resonant spin-flavor conversion. We construct analytical models for the solar magnetic field in all the three regions of the Sun. Neutrino spin-flavor oscillations in these magnetic fields are examined by studying the level crossing phenomenon and comparing the two cases of zero and non-zero vacuum mixing respectively. Results from the Borexino experiment are used to place an upper limit on the magnetic field in the solar core. Related phenomena such as effects of matter on neutrino spin transitions and differences between Dirac and Majorana transitions in the solar magnetic fields are also discussed., Comment: Published in Res.Astron.Astrophys

By proposing a model Hamiltonian in the first quantised form we investigate the chaotic dynamics of $$\alpha $$ -helical proteins by taking into account the quadrupole–quadrupole-type interaction. The dynamics is studied by deriving Hamilton’s equations of motion and by plotting the time-series evolution and phase-space trajectories. Chaotic trajectories are observed in the phase-space plots. The effect of the interaction parameters on the stability of proteins is also discussed.

Coloring the faces of 2-dimensional square lattice with $k$ distinct colors such that no two adjacent faces have the same color is considered by establishing connection between the $k$ coloring problem and a generalized vertex model. Associating the colors with $k$ distinct species of particles with infinite repulsive force between nearest neighbors of the same type and zero chemical potential $\mu$ associated with each species, the number of ways $[W(k)]^N$ for large $N$ is related to the entropy of the {\it{hard square lattice gas}} at close packing of the lattice, where $N$ is the number of lattice sites. We discuss the evaluation of $W(k)$ using transfer matrix method with non-periodic boundary conditions imposed on at least one dimension and show the characteristic Toeplitz block structure of the transfer matrix. Using this result, we present some analytical calculations for non-periodic models that remain finite in one dimension. The case $k=3$ is found to approach the exact result obtained by Lieb for the residual entropy of ice with periodic boundary conditions. Finally, we show, by explicit calculation of the contribution of subgraphs and the series expansion of $W(k)$, that the genenralized Pauling type estimate(which is based on mean field approximation) dominates at large values of $k$. We thus also provide an alternative series expansion for the chromatic polynomial of a regular square graph., Comment: 16 pages, 8 figures, Some major revisions done

The exact probability distributions of the amplitudes of eigenfunctions, $\Psi(x, y)$, of several integrable planar billiards are analytically calculated and shown to possess singularities at $\Psi = 0$; the nature of this singularity is shape-dependent. In particular, we prove that the distribution function for a rectangular quantum billiard is proportional to the complete elliptic integral, ${\bf K}(1 - \Psi ^2)$, and demonstrate its universality, modulo a weak dependence on quantum numbers. On the other hand, we study the low-lying states of nonseparable, integrable triangular billiards and find the distributions thereof to be described by the Meijer G-function or certain hypergeometric functions. Our analysis captures a marked departure from the Gaussian distributions for chaotic billiards in its survey of the fluctuations of the eigenfunctions about $\Psi = 0$.

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit., 15 pages, 8 figures; Accepted for publication at Annals of Physics

We show that neutrino spin and spin-flavor transitions involve nonvanishing geometric phases. The geometric character of neutrino spin rotation is explored by studying the neutrino spin trajectory in the projective Hilbert space representation and its relation to the geometric phase. Analytical expressions are derived for noncyclic geometric phases. Several calculations are performed for different cases of rotating and nonrotating magnetic fields in the context of solar neutrinos and neutrinos produced inside neutron stars. Also the effects of adiabaticity, critical magnetic fields and cross boundary effects in the case of neutrinos emanating out of neutron stars are examined., Revised. PRD version

We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches to counting the nodal domains---using trace formulae, graph theory, and difference equations---are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wavefunctions of a "particle in a box'" as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity---and far-reaching implications---of the problem.

For planar integrable billiards, the eigenstates can be classified with respect to a quantity determined by the quantum numbers. Given the quantum numbers as $m, n$, the index which represents a class is $c = m\, \mbox{mod}\, k\,n$ for a natural number, $k$. We show here that the entire tower of states can be generated from an initially given state by application of the operators introduced here. Thus, these operators play the same role for billiards as raising and lowering operators in angular momentum algebra., Comment: 6 pages, 1 figure

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, $\nu $ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of $m\mod kn$, given a particular $k$, for a set of quantum numbers, $m, n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations., Comment: 13 pages, 5 figures

The idea of adiabatic invariance is presented in the context of simple classical mechanical models. The adiabatic invariant jumps across the separatrix — an attempt has been made to bring out the basic ideas underlying the Melnikov-Arnold integral. This becomes important as soon as a perturbation to a regular, stable system makes it dynamically unstable.

Semiclasically derived shell structure and pairing correlation energies have been included in the recently developed formula by Royer et al to evaluate the binding energies of several well known α−radioactive nuclei after appending the effects of deformation as well. Exploiting single-particle level density for spherically symmetric harmonic oscillator potential with spin–orbit interactions within the microscopic-macroscopic framework, Q α -values and the penetration probabilities are determined. This method has been extended to several recently detected superheavy elements, experimentally unknown ones and, also to four α−decay chains in Z = 120 isotopes. The obtained results have been compared with the available experimental and theoretical ones. A stability analysis for such nuclei has been done on the basis of the obtained half-lives (T 1/2-values) as well.

Data on nuclear level densities extracted from transmission data or gamma energy spectrum store the basic statistical information about nuclei at various temperatures. Generally this extracted data goes through model fitting using computer codes like CASCADE. However, recently established semiclassical methods involving no adjustable parameters to determine the level density parameter for magic and semi-magic nuclei give a good agreement with the experimental values. One of the popular ways to paramaterize the level density parameter which includes the shell effects and its damping was given by Ignatyuk. This damping factor is usually fitted from the experimental data on nuclear level density and it comes around 0.05 $MeV^{-1}$. In this work we calculate the Ignatyuk damping factor for various nuclei using semiclassical methods., Comment: 9 Figures, 1 Table

In this paper the classical evolution of a particle is studied which bounces back and forth in a 1D vibrating cavity such that the reflection from the wall does not change the speed of the particle. A peculiar behaviour of the particle motion can be seen where the time evolution of the motion shows superposition of linear and oscillatory behaviour. In particular, the parameter range is found in which the particle oscillates between the walls in steady state as if the wall was static and it is showed that for these parameter ranges the particle settles to this steady state for all initial conditions. It is proposed that this phenomenon can be used to bunch charged particles in short pulses where the synchronization proposed in our model should work against the space charge effect in the charged particle bunch.

The ionospheric Total Electron Content (TECs), derived by dual frequency signals from the Global Positioning System (GPS) recorded near the Indian equatorial anomaly region, Bhopal (23.2°N, 77.4°E, Geomagnetic 14.2°N) were analyzed for the period of January, 2005 to February, 2008. The work deals with monthly, diurnal, solar and magnetic activity variations on night-time enhancement in TEC. From a total of 157 night-time enhancements, 75 occur during pre-midnight and 82 post-midnight hours. The occurrence of night-time enhancement in TEC is utmost during summer months, followed by equinox and winter months. The occurrence of night-time enhancement in TEC decreases with increase in solar and magnetic activities. We observed that peak size and half amplitude duration are positively correlated, while time of occurrence of night-time enhancement in TEC and time of peak enhancement are negatively correlated with solar activity. The peak size, half amplitude duration, time of peak enhancement and time of occurrence of night-time enhancement in TEC shows negative correlation with magnetic activity. The results have been compared with the earlier ones and discussed in terms of possible source mechanism responsible for the enhancement at anomaly crest region.

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1-12 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards., 7 pages, 2 figures

We investigate a quantum many-body system with particles moving on a circle and subject to two-body and three-body potentials. In this new class of models, that extrapolates from the celebrated Calogero-Sutherland model and a system with interactions among nearest and next-to-nearest neighbors, the interactions can be tuned as a function of range. We determine the exact ground state energy and wavefunction and obtain a part of the excitation spectrum., Comment: 8 pages, 1 figure

We argue that mass parameters appearing in the treatment of large-amplitude collective motion, be it fission or heavy-ion reactions, originate as a consequence of their relation with Lyapunov exponents coming from the classical dynamics, and, fractal dimension associated with diffusive modes coming from hydrodynamic description.

GPS satellites data obtained at Bhopal (23.16° N, 77.36° E, geomagnetic latitude 14.23° N) India were analyzed to study the TEC changes during several geomagnetic storms (−300 nT

We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity P and time-reversal invariance T. In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log-log plots.

It is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian of the quantum system for an arbitrary pseudo-unitary (and hence \( \mathcal{P}\mathcal{T} \)-) quantum evolution. The result generalizes the time-energy uncertainty principle for pseudo-unitary quantum evolutions. Further, employing random matrix theory developed for pseudo-Hermitian systems, time correlation functions are studied in the framework of linear response theory. The results given here provide a quantum brachistochrone problem where the system will evolve in a thermodynamic environment with spectral complexity that can be modelled by random matrix theory.

We construct the expression for the ground state eigenfunction of the Schrodinger equation for a particle inside an arbitrary planar triangle under the influence of a potential.

We calculate the electric dipolarizability of 7 Li nucleus within the cluster model and estimate a value of about 0.0188 fm 3 . We also discuss the possibility of observing this in the scattering of 7 Li from a 208 Pb target at energies about 30 MeV.

We investigate if physical laws can impose limit on computational time and speed of a quantum computer built from elementary particles. We show that the product of the speed and the running time of a quantum computer is limited by the type of fundamental interactions present inside the system. This will help us to decide as to what type of interaction should be allowed in building quantum computers in achieving the desired speed., Comment: Latex file, 7 pages, no figures

We characterise the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems., 26 pages, 13 figures

Integrability of a square billiard is spontaneously broken as it rotates about its centre [1]. The system becomes quasi-integrable where the invariant tori are broken with respect to a certain parameter, λ = 2E/ω where E is the energy of the particle inside the billiard and ω is the angular frequency of rotation of billiard. We study the system classically and quantum mechanically in view of obtaining a correspondence in the two descriptions. In a finite set of levels, the spectral statistics shows a transition from Poisson to Wigner distribution as the system turns chaotic with increase in λ. At the same time, it is also observed that a very long sequence of energy levels will have Poisson-distributed fluctuations. The amplitude distribution of the eigenfunctions also corroborates a region where transition from order to chaos is observed. In all, we have treated the system comprehensively and believe that the results found have significant applications to the physics of rotating nuclei and clusters.

Nuclei have complex energy-level sequence with statistical properties in agreement with canonical random matrix theory. This agreement appears when the one-particle one-hole states are mixed completely with two-particle two-hole states. In the transition, there is a new universality which we present here, bringing about a relation between dynamics and statistics. We summarize also the role of chaos in thermalization and dissipation in isolated systems like nuclei. The methods used to bring forth this understanding emerge from random matrix theory, semiclassical physics, and the theory of dynamical systems.

We present a systematic theory of dissipation in finite Fermi systems like nuclei and metallic clusters. This theory is based on the application of semiclassical methods and random matrix theory to linear response of many-body systems. The theory is developed in the approximation wherein the many-body system can be treated as a single particle in an effective, time-dependent mean-field. We find semiclassical expressions for energy dissipation relevant in one-body dissipation in heavy nuclei. We also show that this energy dissipation, related to damping of collective excitations, is irreversible. The irreversibility is proved by our development of a quantum diffusion equation. It may be noted that the quantum diffusion equation is derived from the von Neumann equation and makes no assumption about the initial form of the density operator. It is shown that, in the semiclassical limit, the quantum diffusion equation reduces to the classical Smoluchowski equation. Further, we show that the dissipation is a purely quantal phenomenon as it is related to the geometric phase acquired by a single-particle wavefunction as the system evolves in a slow-varying mean-field. It is explicitly shown that the dissipation rate is related to the nature of dynamics and the spectrum of the classical Liouvillian operator. Finally, we present an expression for the viscosity tensor encountered in nuclear fission in terms of periodic orbits of the single particle in an adiabatically deforming nucleus., LaTeX2e, 34 pages

We present a random matrix ensemble where real, positive semi-definite matrix elements, x, are log-normal distributed, exp[−log2(x)]. We show that the level density varies with energy, E, as 2/(1+E) for large E, in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here). An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed and its connection with level dynamics is brought out. It is further proved that Dyson’s Coulomb gas analogy breaks down whenever the confining potential is given by a transcendental function for which there exist orthogonal polynomials.

For N impenetrable particles in one dimension where only the nearest and next-to-nearest neighbours interact, we obtain the complete spectrum both on a line and on a circle. Further, we establish a mapping between these N-body problems and the short-range Dyson model introduced recently to model intermediate spectral statistics in some systems using which we compute the two-point correlation function and prove the absence of long-range order in the corresponding many-body theory. Further, we also show the absence of off-diagonal long-range order in these systems.

We show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the Wigner - Dyson family of random matrix theory and the Poisson distribution. Some of the distributions find explanation in a recent work of Bogomolny, Gerland, and Schmit. For the irrational billiards, despite ergodicity, we get the same distribution for the examples considered - once again, distinct from the Wigner - Dyson distributions. All the results are obtained numerically by a method that allows us to reach very high energies.

Treating a many-body Fermi system in terms of a single particle in a deforming mean field. We relate adiabatic geometric phase to susceptibility for the noncyclic case, and to its derivative for the cyclic case. Employing the semiclassical expression of susceptibility, the expression for geometric phase for chaotic quantum system immediately follows. Exploiting the well-known association of the absorptive part of susceptibility with dissipation, our relations may provide a quantum mechanical origin of the damping of collective excitations in Fermi systems., Comment: 4 pages, LaTeX

We present a theory where the statistical mechanics for dilute ideal gases can be derived from random matrix approach. We show the connection of this approach with Srednicki approach which connects Berry conjecture with statistical mechanics. We further establish a link between Berry conjecture and random matrix theory, thus providing a unified edifice for quantum chaos, random matrix theory, and statistical mechanics. In the course of arguing for these connections, we observe sum rules associated with the outstanding counting problem in the theory of braid groups. We are able to show that the presented approach leads to the second law of thermodynamics.

We present a treatment of many-body Fermionic systems that facilitates an expression of the well-known quantities in a series expansion of the Planck's constant. The ensuing semiclassical result contains to a leading order of the response function the classical time correlation function of the observable followed by the Weyl-Wigner series, on top of these terms are the periodic-orbit correction terms. The treatment given here starts from linear response assumption of the many-body theory and in its connection with semiclassical theory, it makes no assumption of the integrability of classical dynamics underlying the one-body quantal system. Applications of the framework are also discussed., Comment: 18 pages, TeX

Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present applications to problems in statistical mechanics where new results have become possible. We have found it important to mention the precise directions where advances could be made if further results become available.

We present a complete discussion of the dynamical scenario supported by the cranking model for high- j orbital where we consider the effects of the nonlinear terms. We carry out a semiclassical quantization about the various fixed points of the model and obtain the alignment of particle angular momentum with the rotation axis under various conditions. Further, we establish the occurrence of second-order phase transition accompanied by a change of sign of aligned angular momentum. Possible applications of the model to deformed and superdeformed nuclei are considered.

We present a Gaussian ensemble of random cyclic matrices on the real field and study their spectral fluctuations. These cyclic matrices are shown to be pseudo-symmetric with respect to generalized parity. We calculate the joint probability distribution function of eigenvalues and the spacing distributions analytically and numerically. For small spacings, the level spacing distribution exhibits either a Gaussian or a linear form. Furthermore, for the general case of two arbitrary complex eigenvalues, leaving out the spacings among real eigenvalues, and, among complex conjugate pairs, we find that the spacing distribution agrees completely with the Wigner distribution for Poisson process on a plane. The cyclic matrices occur in a wide variety of physical situations, including disordered linear atomic chains and Ising model in two dimensions. These exact results are also relevant to two-dimensional statistical mechanics and $\nu$-parametrized quantum chromodynamics., Comment: 4 pages, 3 figures

The level density parameter is calculated on the basis of exact semiclassical 'trace formulae' for magic and semi-magic nuclei. This powerful formula entails the level density of a quantum system in terms of the detailed properties of periodic orbits of the underlying classical system. Keeping the description within the mean-field approximation of a many-body Fermionic system, we present the well-known dependence of the level density parameter a on the mass number A which summarizes the shell effects in nuclei. To understand the melting of shell effects with temperature, we present calculations leading us to the behaviour of a vs A at non-zero temperatures for nearly one hundred nuclei. We see all the points huddling together along to the -line as the temperature increases. With no adjustable parameters, we also find that our results for a are within ten to fifteen percent of the values deduced from the experiments.

We discover that the energy-integral of time-delay is an adiabatic invariant in quantum scattering theory and corresponds classically to the phase space volume. The integral thus found provides a quantization condition for resonances, explaining a series of results recently found in non-relativistic and relativistic regimes. Further, a connection between statistical quantities like quantal resonance-width and classical friction has been established with a classically deterministic quantity, the stability exponent of an adiabatically perturbed periodic orbit. This relation can be employed to estimate the rate of energy dissipation in finite quantum systems., Comment: 8 pages

Employing the concept of time-delay, a relation is found which counts the number of quantal resonances supported by a potential. Several simple and advanced illustrations include a treatment of square-well, Dirac delta barrier, an interesting physical situation from neutron reflectometry, and the Delta resonance appearing in the scattering of \pi meson from proton., Comment: 9 pages, 5 figures

We explore the usefulness of the existing relations between the $S$-matrix and time delay in characterizing baryon resonances in pion-nucleon scattering. We draw attention to the fact that the existence of a positive maximum in time delay is a necessary criterion for the existence of a resonance and should be used as a constraint in conventional analyses which locate resonances from poles of the $S$-matrix and Argand diagrams. The usefulness of the time delay plots of resonances is demonstrated through a detailed analysis of the time delay in several partial waves of $\pi N$ elastic scattering., Comment: 28 pages, 5 figures and 3 tables, title changed, references added, manuscript revised

Following the semiclassical formalism of Strutinsky et al., we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit $\eta\to1.0$. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the eigenvalues of sphere and an exact comparison with the quantum mechanical results is observed upto the second decimal place for $kR_{0}\geq{7}$. The contributions of the equatorial, the planar (in the axis of symmetry plane) and the non-planar(3-Dimensional) orbits are obtained from the same trace formula by using the appropriate conditions. The resulting eigenvalues compare very well with the quantum mechanical eigenvalues at normal deformation. It is interesting that the partial sum of equatorial orbits leads to eigenvalues with maximum angular momentum projection, while the summing of planar orbits leads to eigenvalues with $L_z=0$ except for L=1. The remaining quantum mechanical eigenvalues are observed to arise from the 3-dimensional(3D) orbits. Very few spurious eigenvalues arise in these partial sums. This result establishes the important role of 3D orbits even at normal deformations., Comment: 17 pages, 7 ps figures

Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudo-unitary group. Further, we develop a random matrix theory which is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices as the pseudo-unitary ensemble. We obtain exact results for the nearest-neighbour level spacing distribution for (2 X 2) PT-symmetric Hamiltonian matrices which has a novel form, s log (1/s) near zero spacing. This shows a level repulsion in marked distinction with an algebraic form in the Wigner surmise. We believe that this paves way for a description of varied phenomena in two-dimensional statistical mechanics, quantum chromodynamics, and so on., Comment: 9 pages, 2 figures, LaTeX, submitted to the Physical Review Letters on August 20, 2002