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 × N matrix with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N − M* ≥ N − 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 Charney–Hasegawa–Mima 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 $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval \Delta_k\subset\er with $\Delta_k\cap\Delta_{k+1}=\emptyset$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $\Delta_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^{\infty}$, the so-called \emph{Chebyshev-Nikishin polynomials}. We show that the polynomials $P_{n}$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $\Delta_{k}$. In this way we generalize a result of the third author and Rocha \cite{LopRoc} for the case $p=2$. 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 $\{P_{n}\}_{n=0}^{\infty}$.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 $3\sigma$ and $5\sigma$ for exposures times of 3 months and 1.5 years, respectively, and cavity densities below 1 g cm$^{-3}$ or above 5 g cm$^{-3}$, 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 $2\sigma$ to $5\sigma$ 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