5,749 research outputs found
The attractive nonlinear delta-function potential
We solve the continuous one-dimensional Schr\"{o}dinger equation for the case
of an inverted {\em nonlinear} delta-function potential located at the origin,
obtaining the bound state in closed form as a function of the nonlinear
exponent. The bound state probability profile decays exponentially away from
the origin, with a profile width that increases monotonically with the
nonlinear exponent, becoming an almost completely extended state when this
approaches two. At an exponent value of two, the bound state suffers a
discontinuous change to a delta-like profile. Further increase of the exponent
increases again the width of the probability profile, although the bound state
is proven to be stable only for exponents below two. The transmission of plane
waves across the nonlinear delta potential increases monotonically with the
nonlinearity exponent and is insensitive to the sign of its opacity.Comment: submitted to Am. J. of Phys., sixteen pages, three figure
Quadratic invariants for discrete clusters of weakly interacting waves
We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M Ă N matrix with entries 1, â1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ⥠N â M* â„ N â M, where M* is the number of linearly independent rows in Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the CharneyâHasegawaâMima wave model, and by showing a classification of small (up to three-triad) clusters
A family of Nikishin systems with periodic recurrence coefficients
Suppose we have a Nikishin system of measures with the th generating
measure of the Nikishin system supported on an interval \Delta_k\subset\er
with for all . It is well known that
the corresponding staircase sequence of multiple orthogonal polynomials
satisfies a -term recurrence relation whose recurrence coefficients,
under appropriate assumptions on the generating measures, have periodic limits
of period . (The limit values depend only on the positions of the intervals
.) Taking these periodic limit values as the coefficients of a new
-term recurrence relation, we construct a canonical sequence of monic
polynomials , the so-called \emph{Chebyshev-Nikishin
polynomials}. We show that the polynomials themselves form a sequence
of multiple orthogonal polynomials with respect to some Nikishin system of
measures, with the th generating measure being absolutely continuous on
. In this way we generalize a result of the third author and Rocha
\cite{LopRoc} for the case . The proof uses the connection with block
Toeplitz matrices, and with a certain Riemann surface of genus zero. We also
obtain strong asymptotics and an exact Widom-type formula for the second kind
functions of the Nikishin system for .Comment: 30 pages, minor change
Searching for cavities of various densities in the Earth's crust with a low-energy electron-antineutrino beta-beam
We propose searching for deep underground cavities of different densities in
the Earth's crust using a long-baseline electron-antineutrino disappearance
experiment, realized through a low-energy beta-beam with highly-enhanced
luminosity. We focus on four cases: cavities with densities close to that of
water, iron-banded formations, heavier mineral deposits, and regions of
abnormal charge accumulation that have been posited to appear prior to the
occurrence of an intense earthquake. The sensitivity to identify cavities
attains confidence levels higher than and for exposures
times of 3 months and 1.5 years, respectively, and cavity densities below 1 g
cm or above 5 g cm, with widths greater than 200 km. We
reconstruct the cavity density, width, and position, assuming one of them known
while keeping the other two free. We obtain large allowed regions that improve
as the cavity density differs more from the Earth's mean density. Furthermore,
we demonstrate that knowledge of the cavity density is important to obtain
O(10%) error on the width. Finally, we introduce an observable to quantify the
presence of a cavity by changing the orientation of the electron-antineutrino
beam, with which we are able to identify the presence of a cavity at the
to C.L.Comment: 7 pages, 5 figures; matches published versio
Two-phase stretching of molecular chains
While stretching of most polymer chains leads to rather featureless
force-extension diagrams, some, notably DNA, exhibit non-trivial behavior with
a distinct plateau region. Here we propose a unified theory that connects
force-extension characteristics of the polymer chain with the convexity
properties of the extension energy profile of its individual monomer subunits.
Namely, if the effective monomer deformation energy as a function of its
extension has a non-convex (concave up) region, the stretched polymer chain
separates into two phases: the weakly and strongly stretched monomers.
Simplified planar and 3D polymer models are used to illustrate the basic
principles of the proposed model. Specifically, we show rigorously that when
the secondary structure of a polymer is mostly due to weak non-covalent
interactions, the stretching is two-phase, and the force-stretching diagram has
the characteristic plateau. We then use realistic coarse-grained models to
confirm the main findings and make direct connection to the microscopic
structure of the monomers. We demostrate in detail how the two-phase scenario
is realized in the \alpha-helix, and in DNA double helix. The predicted plateau
parameters are consistent with single molecules experiments. Detailed analysis
of DNA stretching demonstrates that breaking of Watson-Crick bonds is not
necessary for the existence of the plateau, although some of the bonds do break
as the double-helix extends at room temperature. The main strengths of the
proposed theory are its generality and direct microscopic connection.Comment: 16 pges, 22 figure
Effects of Eye-phase in DNA unzipping
The onset of an "eye-phase" and its role during the DNA unzipping is studied
when a force is applied to the interior of the chain. The directionality of the
hydrogen bond introduced here shows oscillations in force-extension curve
similar to a "saw-tooth" kind of oscillations seen in the protein unfolding
experiments. The effects of intermediates (hairpins) and stacking energies on
the melting profile have also been discussed.Comment: RevTeX v4, 9 pages with 7 eps figure
Dynamic force spectroscopy of DNA hairpins. II. Irreversibility and dissipation
We investigate irreversibility and dissipation in single molecules that
cooperatively fold/unfold in a two state manner under the action of mechanical
force. We apply path thermodynamics to derive analytical expressions for the
average dissipated work and the average hopping number in two state systems. It
is shown how these quantities only depend on two parameters that characterize
the folding/unfolding kinetics of the molecule: the fragility and the
coexistence hopping rate. The latter has to be rescaled to take into account
the appropriate experimental setup. Finally we carry out pulling experiments
with optical tweezers in a specifically designed DNA hairpin that shows
two-state cooperative folding. We then use these experimental results to
validate our theoretical predictions.Comment: 28 pages, 12 figure
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