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In a recent article arXiv:2402.05172 [hep-ph], the authors discuss the question "whether quantum statistics can help distinguish between Dirac and Majorana neutrinos." The paper contains, among other things, an unsubstantiated critique of the results derived in our papers arXiv:2106.11785 [hep-ph] and arXiv:2307.05654 [hep-ph]. One of the criticisms is related to our expression for differential decay rate for the back-to-back neutrino-antineutrino configuration in the decay $B^0 \to \mu^- \, \mu^+ \, \nu_\mu \, \overline{\nu}_\mu$. We show that the claim is wrong and point out how the correct result was obtained. The second criticism is related to the implementation of the anti-symmetrization as dictated by quantum statistics for Majorana neutrinos and antineutrinos (which are identical, by definition). Any direct observation of the neutrinos, as done in Ref. \cite{Akhmedov:2024}, would project the neutrinos into distinguishable helicity states, thus nullifying all observable effects of quantum statistics. They have missed the point that our procedure holds when the neutrino and antineutrino remain undetected by the detector. In the back-to-back kinematic configuration, one can infer the neutrino energies without directly detecting their identities. This smartly ensures that the quantum statistical effects are not erased. Their overriding assertion that our papers arXiv:2106.11785 [hep-ph] and arXiv:2307.05654 [hep-ph] are incorrect fails to recognize that in both arXiv:2106.11785 [hep-ph] and arXiv:2307.05654 [hep-ph] we also point out generic conditions under which the practical Dirac-Majorana confusion theorem holds. "Clearly there is no confusion over confusion theorem.", Comment: 8 pages, 0 figures

In arXiv:2305.14140 [hep-ph] the authors analyze the radiative leptonic decay $\ell^- \to \nu_\ell \, \overline{\nu}_{\ell'} \, \ell^{\prime -} \, \gamma$ to distinguish between Dirac and Majorana nature of neutrinos. They utilize the back-to-back kinematics for this purpose, a special kinematic configuration which we first proposed in our paper arXiv:2106.11785 [hep-ph]. Here we point out how and why their analysis of the back-to-back configuration is incorrect. This makes their conclusion and comments invalid and untenable., Comment: 4 pages, 0 figures

The nature of a neutrino, whether it is a Dirac type or Majorana type, may be comprehensively probed using their quantum statistical properties. If the neutrino is a Majorana fermion, then by definition it is identical and indistinguishable from the corresponding antineutrino. When a Majorana neutrino and antineutrino are pair produced, the corresponding state has to obey the Pauli principle unlike in the Dirac case. We use this property to distinguish between the two cases using the process $B^0 \to \mu^-\,\mu^+\,\nu_\mu\,\bar{\nu}_\mu$. We show that the two cases differ dramatically in a special kinematic scenario where, in the rest frame of the parent $B$ meson, the muons fly away back-to-back (i.e. fly with 3-momenta of equal magnitudes but opposite directions), and so do the neutrino and antineutrino. Unlike any other scenario, we know the energies and magnitudes of $3$-momenta of both the neutrino and the antineutrino in this back-to-back configuration without even directly measuring them. This provides a way of avoiding the constraint imposed by the `practical Dirac-Majorana confusion theorem', as one need not fully integrate over neutrino and antineutrino in this case. As a true signature of the universal principle of quantum statistics which does not depend on the size of the mass of the particle but its spin, the difference between Dirac and Majorana cases in this special kinematic configuration does survive independent of the neutrino mass as long as neutrino mass is nonzero. The analysis presented here is applicable immediately to several other processes with the same final state as in the case of $B^0$ decay without any major change., Comment: 30 pages, 5 figures, updated references, typos corrected, for published version visit the DOI

We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple leading-order exponential form given so far in the literature., Comment: 10 pages, 3 figures; v2: small editorial changes, correction of misprints v3: revised version, added new material by V. Kotesovec that confirms our conclusions

We study the number $P(n)$ of partitions of an integer $n$ into sums of distinct squares and derive an integral representation of the function $P(n)$. Using semi-classical and quantum statistical methods, we determine its asymptotic average part $P_{as}(n)$, deriving higher-order contributions to the known leading-order expression [M. Tran {\it et al.}, Ann.\ Phys.\ (N.Y.) {\bf 311}, 204 (2004)], which yield a faster convergence to the average values of the exact $P(n)$. From the Fourier spectrum of $P(n)$ we obtain hints that integer-valued frequencies belonging to the smallest Pythagorean triples $(m,p,q)$ of integers with $m^2+p^2=q^2$ play an important role in the oscillations of $P(n)$. Finally we analyze the oscillating part $\delta P(n)=P(n)-P_{as}(n)$ in the spirit of semi-classical periodic orbit theory [M. Brack and R. K. Bhaduri: {\it Semiclassical Physics} (Bolder, Westview Press, 2003)]. A semi-classical trace formula is derived which accurately reproduces the exact $\delta P(n)$ for $n > \sim 500$ using 10 pairs of `orbits'. For $n > \sim 4000$ only two pairs of orbits with the frequencies 4 and 5 -- belonging to the lowest Pythagorean triple (3,4,5) -- are relevant and create the prominent beating pattern in the oscillations. For $n > \sim 100,000$ the beat fades away and the oscillations are given by just one pair of orbits with frequency 4., Comment: 38 pages, 19 figures. Version v2: Figs. 14-18 improved; correction of small misprints, small editorial changes. Version v3: better description of beat mechanism on p. 26, figures improved. Version v4: Figures further improved and captions corrected. Final version to be published in Phys. Rev. E

The Dirac leptonic CP phase {\delta}_CP is one of the crucial unknown parameters in neutrino oscillation physics. In this paper we explore the possibility of using low energy atmospheric neutrino events to probe {\delta}_CP . We show that at sub GeV energies, when the events are binned as a function of the energy and direction of the final state leptons, a consistent distinction between various true {\delta}_CP values is obtained. We also show that at these energies there is no sensitivity to the mass ordering/hierarchy, so that {\delta}_CP can be measured without hierarchy ambiguity. In addition a preliminary \c{hi}^2 analysis of the sensitivity to {\delta}_CP using atmospheric neutrinos assuming a generic detector with perfect separation between charged current {\nu}_{\mu} , \bar{{\nu}}_{\mu} , {\nu}_e and \bar{{\nu}}_e events is given., Comment: 18 pages, 9 figures, revised to include contrast between low and high energy atmospheric neutrino information and expanded

In this paper, we discuss P(n), the number of ways in which a given integer n may be written as a sum of primes. In particular, an asymptotic form P_as(n) valid for n towards infinity is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This directly gives the asymptotic number of prime partitions P_as(n). The leading term in the asymptotic expression grows exponentially as sqrt[n/ln(n)] and agrees with previous estimates. We calculate the next-to-leading order term in the exponent, porportional to ln[ln(n)]/ln(n), and show that an earlier result in the literature for its coefficient is incorrect. Furthermore, we also calculate the next higher order correction, proportional to 1/ln(n) and given in Eq.(43), which so far has not been available in the literature. Finally, we compare our analytical results with the exact numerical values of P(n) up to n \sim 8 10^6. For the highest values, the remaining error between the exact P(n) and our P_as(n) is only about half of that obtained with the leading-order (LO) approximation. But we also show that, unlike for other types of partitions, the asymptotic limit for the prime partitions is still quite far from being reached even for n \sim 10^7., Comment: LaTeX; 18 pp. 6 figures (color) - v4: revised presentation of text mainly in beginnings of Sect. I and II.A and in the conclusions Sect. IV. Final version published in Phys. Rev. E

We show that the hadronic "heat capacity" calculated as a function of temperature may be used to infer the possible presence of different scales underlying the dynamical structure of hadronic resonances using the phenomenon of Schottky anomaly. We first demonstrate this possibility with well known meson spectrum in various channels and comment on the possibility of using this method as a diagnostic to distinguish the exotic states., Comment: 21 pages, 21 figures

We consider the case of light dark matter ($\sim 10$ GeV). We discuss a simple $Z_2$ model of scalar self-interacting dark matter, as well as a related model of unstable long-lived dark matter which can explain the anomalous Kolar events observed decades ago., Comment: 7 pages, 2 figures

The upcoming 50 kt magnetized iron calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO) is designed to study the atmospheric neutrinos and antineutrinos separately over a wide range of energies and path lengths. The primary focus of this experiment is to explore the Earth matter effects by observing the energy and zenith angle dependence of the atmospheric neutrinos in the multi-GeV range. This study will be crucial to address some of the outstanding issues in neutrino oscillation physics, including the fundamental issue of neutrino mass hierarchy. In this document, we present the physics potential of the detector as obtained from realistic detector simulations. We describe the simulation framework, the neutrino interactions in the detector, and the expected response of the detector to particles traversing it. The ICAL detector can determine the energy and direction of the muons to a high precision, and in addition, its sensitivity to multi-GeV hadrons increases its physics reach substantially. Its charge identification capability, and hence its ability to distinguish neutrinos from antineutrinos, makes it an efficient detector for determining the neutrino mass hierarchy. In this report, we outline the analyses carried out for the determination of neutrino mass hierarchy and precision measurements of atmospheric neutrino mixing parameters at ICAL, and give the expected physics reach of the detector with 10 years of runtime. We also explore the potential of ICAL for probing new physics scenarios like CPT violation and the presence of magnetic monopoles., Comment: 139 pages, Physics White Paper of the ICAL (INO) Collaboration, Contents identical with the version published in Pramana - J. Physics

We report on a detailed simulation study of the hadron energy resolution as a function of the thickness of the absorber plates for the proposed Iron Calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO). We compare the hadron resolutions obtained with absorber thicknesses in the range 1.5--8 cm for neutrino interactions in the energy range 2--15 GeV, which is relevant to hadron production in atmospheric neutrino interactions. We find that at lower energies, the thickness dependence of energy resolution is steeper than at higher energies, however there is a thickness-independent contribution that dominates at the lower thicknesses discussed in this work. As a result, the gain in hadron energy resolution with decreasing plate thickness is marginal. We present the results in the form of fits to a function with energy-dependent exponent.

Third and higher order quantum virial coefficients require the solution of the corresponding quantum many-body problem. Nevertheless, in an earlier paper (Phys. Rev. Lett. 108, 260402 (2012)) we proposed that the higher-order cluster integrals of a dilute unitary fermionic gas may be approximated in terms of the two-body cluster, together with an appropriate suppression factor. Although not exact, this ansatz gave a fair agreement up to fugacity z=6 with the experimentally obtained equation of state. The objective of the present note is to give some physical arguments in favor of this ansatz., Comment: 3 pages, 3 figures

The possibility of the unexplained Kolar events, recorded in the 70's and 80's, being due to the decays of dark matter particles of mass in the range of 5--10 GeV is pointed out., Comment: 6 pages; 1 figure added

The results of a Monte Carlo simulation study of the hadron energy response for the magnetized Iron CALorimeter detector, ICAL, proposed to be located at the India-based Neutrino Observatory (INO) is presented. Using a GEANT4 modeling of the detector ICAL, interactions of atmospheric neutrinos with target nuclei are simulated. The detector response to hadrons propagating through it is investigated using the hadron hit multiplicity in the active detector elements. The detector response to charged pions of fixed energy is studied first, followed by the average response to the hadrons produced in atmospheric neutrino interactions using events simulated with the NUANCE event generator. The shape of the hit distribution is observed to fit the Vavilov distribution, which reduces to a Gaussian at high energies. In terms of the parameters of this distribution, we present the hadron energy resolution as a function of hadron energy, and the calibration of hadron energy as a function of the hit multiplicity. The energy resolution for hadrons is found to be in the range 85% (for 1GeV) -- 36% (for 15 GeV)., Comment: 14 pages, 10 figures (24 eps files)

We propose a phenomenological approach for the equation of state of a unitary Fermi gas. The universal equation of state is parametrised in terms of Fermi-Dirac integrals. This reproduces the experimental data over the accessible range of fugacity and normalised temperature, but cannot describe the superfluid phase transition found in the MIT experiment \cite{ku}. The most sensitive data for compressibility and specific heat at phase transition can, however, befitted by introducing into the grand partition function a pair of complex conjugate zeros lying in the complex fugacity plane slightly off the real axis., Comment: LaTeX file (revtex-4-1), 10 pp., 4 figures (7 *.eps files), final version to be published in Pramana

It is suggested that for a fermi gas at unitarity, the two-body bond plays a special role. We propose an equation of state using an ansatz relating the interaction part of the $l$-body cluster to its two-body counterpart. This allows a parameter-free comparison with the recently measured equation of state by the ENS group. The agreement between the two over a range of fugacity ($z<5$ for a homogeneous gas, and $z<10$ for the trapped gas) leads us to perform the calculations of more sensitive quantities measured recently by the MIT group., Comment: 4 pages, 6 figures To be published in Physical Review Letters

We follow up on the analysis of Mecozzi and Bellini (arXiv:1110:1253v1) where they showed, in principle, the possibility of superluminal propagation of neutrinos, as indicated by the recent OPERA result. We refine the analysis by introducing wave packets for the superposition of energy eigenstates and discuss the implications of their results with realistic values for the mixing and mass parameters in a full three neutrino mixing scenario. Our analysis shows the possibility of superluminal propagation of neutrino flavour in a very narrow range of neutrino parameter space. Simultaneously this reduces the number of observable events drastically. Therefore, the OPERA result cannot be explained in this frame-work., Comment: 10 pages revtex with 2 figures. Important changes have been made; in particular, it has been revised to include a discussion on the nature of the measurement and its impact on the result

We demonstrate that the (s-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$., Comment: LaTeX (revtex 4), 18pp., 4 Figs., already published in Phys. Rev. A but here a note with a new referece is added on p. 14

Consider two identical atoms in a spherical harmonic oscillator interacting with a zero-range interaction which is tuned to produce an s-wave zero-energy bound state. The quantum spectrum of the system is known to be exactly solvable. We note that the same partial wave quantum spectrum is obtained by the one-dimensional scale-invariant inverse square potential. Long known as the Calogero-Sutherland-Moser (CSM) model, it leads to Fractional Exclusion Statistics (FES) of Haldane and Wu. The statistical parameter is deduced from the analytically calculated second virial coefficient. When FES is applied to a Fermi gas at unitarity, it gives good agreement with experimental data without the use of any free parameter., Comment: 11 pages, 3 figures, To appear in J. Phys. B. Atomic, Molecular and Optical Physics

We consider N particles interacting pair-wise by an inverse square potential in one dimension (Calogero-Sutherland-Moser model). When trapped harmonically, its classical canonical partition function for the repulsive regime is known in the literature. We start by presenting a concise re-derivation of this result. The equation of state is then calculated both for the trapped and the homogeneous gas. Finally, the classical limit of Wu's distribution function for fractional exclusion statistics is obtained and we re-derive the classical virial expansion of the homogeneous gas using this distribution function., Comment: 9 pages; added references to some earlier work on this problem; this has led to a significant shortening of the paper and a changed title

We investigate the particle and kinetic-energy densities for a system of $N$ fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers $N$, we show that they practically are surprisingly accurate also for moderate values of $N$., Comment: LaTeX, 34 pp., 12 figures; final version (v5) to be published in J. Phys. A. This version contains only the part of the previous version (v3) on local virial theorems; the semiclassical theory has appeared as arXiv:0912.4374v1 [math-ph] and is published in Phys. Rev. E 81, 011118 (2010)

We consider a few-particle system of trapped neutral fermionic atoms at ultra-low temperatures, with the attractive interaction tuned to Feshbach resonance. We calculate the energies and the spatial densities of the few-body systems using a generalisation of the extended Thomas-Fermi (ETF) method, and assuming the particles obey the Haldane-Wu fractional exclusion statistics (FES) at unitarity. This method is different from the scaled ETF version given by Chang and Bertsch (Phys. Rev. A76,021603(R) (2007)). Our semiclassical FES results are consistent with the Monte-Carlo calculations of the above authors, but can hardly be distinguished from their over all scaling of the ETF result at unitarity., Comment: 4 pages, 3 figures

We first give a short review of the ``local-current approximation'' (LCA), derived from a general variation principle, which serves as a semiclassical description of strongly collective excitations in finite fermion systems starting from their quantum-mechanical mean-field ground state. We illustrate it for the example of coupled translational and compressional dipole excitations in metal clusters. We then discuss collective electronic dipole excitations in C$_{60}$ molecules (Buckminster fullerenes). We show that the coupling of the pure translational mode (``surface plasmon'') with compressional volume modes in the semiclasscial LCA yields semi-quantitative agreement with microscopic time-dependent density functional (TDLDA) calculations, while both theories yield qualitative agreement with the recent experimental observation of a ``volume plasmon''., Comment: LaTeX, 12 pages, 5 figures (8 *.eps files); Contribution to XIV-th Nuclear Physics Workshop at Kazimierz Dolny, Poland, Sept. 26-29, 2007

We consider a gas of neutral fermionic atoms at ultra-low temperatures, with the attractive interaction tuned to Feshbach resonance. We calculate, the variation of the chemical potential and the energy per particle as a function of temperature by assuming the system to be an ideal gas obeying the Haldane-Wu fractional exclusion statistics. Our results for the untrapped gas compare favourably with the recently published Monte Carlo calculations of two groups. For a harmonically trapped gas, the results agree with experiment, and also with other published work., Comment: 4 pages, 1 postscript figure

We study in detail the sensitivity of neutrino oscillation probabilities to the fundamental neutrino parameters and their possible determination through experiments. The first part of the paper is devoted to the broad theme of isolating regions in the neutrino (and anti-neutrino) energy and propagation length that are sensitive to the oscillation parameters. Such a study is relevant to neutrinos both from the Earth's atmosphere or from a neutrino factory. For completeness we discuss the sensitivity, however small, to the parameters involved in a three-generation framework, and to the Earth matter density profile. We then study processes relevant to atmospheric neutrinos which are sensitive to and allow precision measurements of the mixing angle theta_23 and mass-squared difference delta_32 apart from the mixing angle theta_13. Crucial to this analysis is charge identification; detectors having this capability can isolate these matter effects. In particular, we address the issue of using matter effects to determine whether the mixing angle theta_23 is maximal, and, if not, to explore how well its octant can be determined. When realistic detector resolutions are included, we find that deviations of about 15% (20%) from a maximal value of sin^2 theta_23=1/2 can be measured at 95% (99%) CL provided theta_13 is non-zero, sin^2 theta_13 >= 0.015, and the neutrino mass ordering is normal, with fairly large exposures of 1000 kton-years., Comment: 37 pages Latex file, 30 eps figure files; minor typos fixed

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for the restricted or coloured partitions p_k^s(n), which is the number of partitions of an integer n into the summand of s^{th} powers of integers such that each power of a given integer may occur utmost k times. While the method is not rigorous, it reproduces the well known asymptotic results for s=1 apart from yielding more general results for arbitrary values of s., Comment: 13 Pages, 3 Postscript figures, title changed and minor corrections

It is by now established that neutrinos mix, have (different) non-zero masses, and therefore oscillate. The oscillation parameters themselves, however, are not all well-known. An open problem is that of the neutrino mass hierarchy. We study the possibility of determining the neutrino mass hierarchy with atmospheric neutrinos using an iron calorimeter detector capable of charge identification such as the proposed MONOLITH and ICAL/INO detectors. We find that such detectors are sensitive to the sign of the mass-squared difference, \delta_{32} = m_3^2 - m_2^2, provided the as-yet unknown mixing angle between the first and third generations, \theta_{13}, is greater than 6 degrees (\sin^2 2\theta_{13} > 0.04). A result with a significance greater than 90% CL requires large exposures (more than 500 kton-years) as well as good energy and angular resolution of the detected muons (better than 15%), especially for small \theta_{13}. Hence obtaining definitive results with such a detector is difficult, unless \theta_{13} turns out to be large. In contrast, such detectors can establish a clear oscillation pattern in atmospheric neutrinos in about 150 kton-years, therefore determining the absolute value of \delta_{32} and \sin^2 2 \theta_{23} to within 10%., Comment: 36 pages revtex with 14 eps figures; new section on statistical significance when detector resolution is included

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For $N$ bosons in a one dimensional harmonic oscillator potential, it is well known that the asymptotic (N -> infinity) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p^s(n), the latter being the number of partitions of n into a sum of s-th powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d^s(n) for distinct partitions. We find that the distinct square partitions d^2(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos-Lehner formula for restricted partitions for the $s=1$ case by our method, we generalize it to obtain a new formula for distinct restricted partitions., Comment: 17 pages including figure captions. 6 figures. To be submitted to J. Phys. A: Math. Gen

We review aspects of classical and quantum mechanics of many anyons confined in an oscillator potential. The quantum mechanics of many anyons is complicated due to the occurrence of multivalued wavefunctions. Nevertheless there exists, for arbitrary number of anyons, a subset of exact solutions which may be interpreted as the breathing modes or equivalently collective modes of the full system. Choosing the three-anyon system as an example, we also discuss the anatomy of the so called ``missing'' states which are in fact known numerically and are set apart from the known exact states by their nonlinear dependence on the statistical parameter in the spectrum. Though classically the equations of motion remains unchanged in the presence of the statistical interaction, the system is non-integrable because the configuration space is now multiply connected. In fact we show that even though the number of constants of motion is the same as the number of degrees of freedom the system is in general not integrable via action-angle variables. This is probably the first known example of a many body pseudo-integrable system. We discuss the classification of the orbits and the symmetry reduction due to the interaction. We also sketch the application of periodic orbit theory (POT) to many anyon systems and show the presence of eigenvalues that are potentially non-linear as a function of the statistical parameter. Finally we perform the semiclassical analysis of the ground state by minimizing the Hamiltonian with fixed angular momentum and further minimization over the quantized values of the angular momentum., Comment: 44 pages, one figure, eps file. References updated

We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive analytically a differential equation connecting these densities, which so far have been proven only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is identified uniquely. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation $\tau_{TF}[\rho]$ between kinetic and particle density is fulfilled also for the asymptotic quantum densities $\tau(r)$ and $\rho(r)$ including their leading-order oscillating terms., Comment: LaTeX, 22 pages with 6 figures (*.eps), to be submitted to J. Phys. A

It is well-known that the number fluctuation in the grand canonical ensemble, which is directly proportional to the compressibility, diverges for an ideal bose gas as T -> 0. We show that this divergence is removed when the atoms interact in one dimension through an inverse square two-body interaction. In two dimensions, similar results are obtained using a self-consistent Thomas-Fermi (TF) model for a repulsive zero-range interaction. Both models may be mapped on to a system of non-interacting particles obeying the Haldane-Wu exclusion statistics. We also calculate the number fluctuation from the ground state of the gas in these interacting models, and compare the grand canonical results with those obtained from the canonical ensemble., Comment: 11 pages, 1 appendix, 3 figures. Submitted to J. Phys. B: Atomic, Molecular & Optical

We have analysed the recent results from the observation of charged current \nu_e d \to e^- p p events from solar neutrinos by the Sudbury Neutrino Observatory SNO assuming neutrino oscillations with three active flavours. The data seem to prefer a low mass-squared difference and large mixing angle solution (the so-called LOW solution) in (12) parameter space. However, when combined with the Gallium charged current interaction data from Gallex and GNO, distinct (1\sigma) allowed regions corresponding to the large mixing angle (LMA) and small mixing angle (SMA) appear while the LOW solution is disfavoured upto 3\sigma standard deviation. The physical electron neutrino survival probability corresponding to these best fit solutions are then determined and analysed for their energy dependence., Comment: 16 pages Latex file, with 5 epsf figures; one reference added

We have identified some important and worthwhile physics opportunitites with a possible neutrino detector located in India. Particular emphasis is placed on the geographical advantage with a stress on the complimentary aspects with respect to other neutrino detectors already in operation., Comment: 9 pages; arXiv copy of published proceedings contribution

An exactly solvable model of two-component interacting Fermi vapour in two dimension within Thomas Fermi approach has been proposed. We assume a realistic off-diagonal s-wave interaction between fermions in the two hyperfine states. The interaction is taken to be given by a screened Coulomb interaction which is both relevant and is analytically tractable. There are two distinct limits in the case of trapped fermionic systems: When the rotating frequency is less than the trap frequency a Thomas-Fermi approach may be used for a reasonably large number of particles. In the case of rapidly rotating fermions, an alternative variational approach is used, where the local density approximation may be applied after projecting the system on to the lowest degenerate level. Analytic expressions for the relevant physical quantities are obtained and their significance is discussed in both these cases., Comment: 25 latexed pages, 7 ps. figures

Signatures of neutrino and antineutrino signals from stellar collapse in heavy water detectors are contrasted with those in water detectors. The effects of mixing, especially due to the highly dense matter in the supernova core, are studied. The mixing parameters used are those sets allowed by current understanding of available neutrino data: from solar, atmospheric and laboratory neutrino experiments. Signals at a heavy water detector, especially the dominant charged current reactions on deuteron, are very sensitive to some of these sets of allowed mixing parameters. Theoretical uncertainties on supernova neutrino spectra notwithstanding, a combination of supernova measurements with water and heavy water detectors may be able to distinguish many of these mixing possibilities and thus help in ruling out many of them., Comment: 36 pages Latex file, with 13 postscript figures; important improvements in the analysis and more detailed presentation of results. To appear in Phys. Rev. D

Properties of confined mesoscopic systems have been extensively studied numerically over recent years. We discuss an analytical approach to the study of finite rotating fermionic systems in two dimension. We first construct the energy functional for a finite fermionic system within the Thomas-Fermi approximation in two dimensions. We show that for specific interactions the problem may be exactly solved. We derive analytical expressions for the density, the critical size as well as the ground state energy of such systems in a given angular momentum sector., Comment: Latex 15 pages, 3 ps. figures. Poster in SCES-Y2K, held at SAHA Institute of Nuclear Physics,Calcutta,October (2000)

A modified Gross-Pitaevskii approximation was introduced recently for bosons in dimension $d\le2$ by Kolomeisky {\it et al.} (Phys. Rev. Lett. {\bf 85} 1146 (2000)). We use the density functional approach with sixth-degree interaction energy term in the Bose field to reproduce the stationary-frame results of Kolomeisky {\it et al.} for a one-dimensional Bose-Einstein system with a repulsive interaction. We also find a soliton solution for an attractive interaction, which may be boosted to a finite velocity by a Galilean transformation. The stability of such a soliton is discussed analytically. We provide a general treatment of stationary solutions in one dimension which includes the above solutions as special cases. This treatment leads to a variety of stationary wave solutions for both attractive and repulsive interactions., Comment: Latex, 14 pages, No figure

We consider a small and fixed number of fermions (bosons) in a trap. The ground state of the system is defined at T=0. For a given excitation energy, there are several ways of exciting the particles from this ground state. We formulate a method for calculating the number fluctuation in the ground state using microcanonical counting, and implement it for small systems of noninteracting fermions as well as bosons in harmonic confinement. This exact calculation for fluctuation, when compared with canonical ensemble averaging, gives considerably different results, specially for fermions. This difference is expected to persist at low excitation even when the fermion number in the trap is large., Comment: 20 pages (including 1 appendix), 3 postscript figures. An error was found in one section of the paper. The corrected version is updated on Sep/05/2000

We study the effect of non-vanishing masses and mixings among neutrino flavours on the detection of neutrinos from stellar collapse by a water Cerenkov detector. We consider a frame-work in which there are four neutrino flavours, including a sterile neutrino, whose mass squared differences and mixings are constrained by the present understanding of solar, atmospheric and laboratory neutrino detection. We also include the effects of high density matter within the supernova core. Unlike in the three flavour scenario, we find that the number of events due to the dominant process involving electron-antineutrinos changes dramatically for some allowed mixing parameters. Furthermore, contributions from charged-current scattering off oxygen nuclei in the detector can be considerably enhanced due to flavour mixing. We also present a comparison between the two possible scenarios, namely, when only three active neutrino flavours are present and when they are accompanied by a fourth sterile neutrino., Comment: 25 pages, 5 EPS figures

We study the effect of non-vanishing masses and mixings among neutrino flavours on the detection of neutrinos from stellar collapse by a water Cerenkov detector. We consider a realistic framework in which there are three neutrino flavours whose mass squared differences and mixings are constrained by the present understanding of solar and atmospheric neutrinos. We also include the effects of high dense matter within the supernova core. We find that the number of events due to the dominant process involving electron-antineutrinos may change dramatically for some allowed mixing parameters. Furthermore, contributions from charged-current scattering off oxygen atoms in the detector can be considerably enhanced due to flavour mixing; such events have a distinct experimental signature since they are backward-peaked. Hence, mixing has a non-trivial effect on the signature of neutrinos (and antineutrinos) from stellar collapse., Comment: 22 pages Latex file, with 6 postscript figures, minor changes made in text

We give a formulation of the single particle occupation probabilities for a system of identical particles obeying fractional exclusion statistics of Haldane. We first derive a set of constraints using an exactly solvable model which describes an ideal exclusion statistics system and deduce the general counting rules for occupancy of states obeyed by these particles. We show that the problem of negative probabilities may be avoided with these new counting rules., Comment: REVTEX 3.0, 14 pages

We consider N fermions in a two-dimensional harmonic oscillator potential interacting with a very short-range repulsive pair-wise potential. The ground-state energy of this system is obtained by performing a Thomas-Fermi as well as a self-consistent Hartree-Fock calculation. The two results are shown to agree even for a small number of particles. We next use the finite temperature Thomas-Fermi method to demonstrate that in the local density approximation, these interacting fermions are equivalent to a system of noninteracting particles obeying the Haldane-Wu fractional exclusion statistics. It is also shown that mapping onto to a system of N noninteracting quasi-particles enables us to predict the energies of the excited states of the N-body system., Comment: Latex(Revtex) + 2 postscript figures

Periodic orbit expressions for the density of states lead to spurious results when directly used to calculate quantities of thermodynamic interest. This is because the trace formula is usually valid only for large energies while the calculations make use of it at all energies. We present a prescription for circumventing this problem by isolating contributions which arise from the inaccurate low-energy behavior, and are spurious, from other subdominant contributions which are physical and are not contained in the Weyl series. The method is tested by analyzing the fermionic grand partition function for a disk billiard and may readily be extended to other dynamical systems., Comment: Substantial changes. There are now explicit asymptotic relations as well as a deeper analysis of the analytic structure of the various relations. We dispensed with the harmonic oscillator in order to have a deeper discussion of periodic orbits in billiards

The classical ground state of a D- dimensional many body system with two and three body interactions is studied as a function of the strength of the three body interaction. We prove exactly that beyond a critical strength of the three body interaction, the classical ground state of the system is one in which all the particles are on a line. The positions of the particles in this string configuration are uniquely determined by the zeros of the Hermite polynomials., Comment: 4 pages, RevTeX, no figure; version to appear in Physical Review Letters

We report on a study of a classical, finite system of confined particles in two dimensions with a two-body repulsive interaction. We first develop a simple analytical method to obtain equilibrium configurations and energies for few particles. When the confinement is harmonic, we prove that the first transition from a single shell occurs when the number of particles changes from five to six. The shell structure in the case of an arbitrary number of particles is shown to be independent of the strength of the interaction but dependent only on its functional form. It is also independent of the magnetic field strength when included. We further study the effect of the functional form of the confinement potential on the shell structure. Finally we report some interesting results when a three-body interaction is included, albeit in a particular model., Comment: Minor corrections, a few references added. To appear in J. Phys: Condensed Matter

We report on a study of a finite system of classical confined particles in two-dimensions in the presence of a uniform magnetic field and interacting via a two-body repulsive potential. We develop a simple analytical method of analysis to obtain ground state energies and configurations. We prove analytically the minimum energy configurations are independent of the nature of two body interaction and the magnetic field. In particular we prove that the first transition from a single shell occurs when the number of particles changes from five to six. These results are exact., Comment: 4 pages, two column, revtex file, no figures