Inverse spectral problems for Sturm--Liouville operators with matrix-valued potentials

We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.Comment: 39 pages, uses iopart.cls, iopams.sty and setstack.sty by IO

Casimir interaction between normal or superfluid grains in the Fermi sea

We report on a new force that acts on cavities (literally empty regions of space) when they are immersed in a background of non-interacting fermionic matter fields. The interaction follows from the obstructions to the (quantum mechanical) motions of the fermions caused by the presence of bubbles or other (heavy) particles in the Fermi sea, as, for example, nuclei in the neutron sea in the inner crust of a neutron star or superfluid grains in a normal Fermi liquid. The effect resembles the traditional Casimir interaction between metallic mirrors in the vacuum. However, the fluctuating electromagnetic fields are replaced by fermionic matter fields. We show that the fermionic Casimir problem for a system of spherical cavities can be solved exactly, since the calculation can be mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks. Finally we generalize the map method to other Casimir systems, especially to the case of a fluctuating scalar field between two spheres or a sphere and a plate under Dirichlet boundary conditions.Comment: 8 pages, 2 figures, submitted to the Proceedings of QFEXT'05, Barcelona, Sept. 5-9, 200

Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.Comment: 20 pages, LaTe

A priori estimates for the Hill and Dirac operators

Consider the Hill operator $Ty=-y''+q'(t)y$ in $L^2(\R)$, where $q\in L^2(0,1)$ is a 1-periodic real potential. The spectrum of $T$ is is absolutely continuous and consists of bands separated by gaps \g_n,n\ge 1 with length |\g_n|\ge 0. We obtain a priori estimates of the gap lengths, effective masses, action variables for the KDV. For example, if \m_n^\pm are the effective masses associated with the gap \g_n=(\l_n^-,\l_n^+), then |\m_n^-+\m_n^+|\le C|\g_n|^2n^{-4} for some constant $C=C(q)$ and any $n\ge 1$. In order prove these results we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping and the identities. Moreover, we obtain the similar estimates for the Dirac operator

Darboux transformations for quasi-exactly solvable Hamiltonians

We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of the first type give singular intermediate potentials and the ones of the second type give complex-valued intermediate potentials while final potentials are meaningful in all cases. These developments are illustrated on the so-called radial sextic oscillator.Comment: 11 pages, Late

On the stability of periodically time-dependent quantum systems

The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e., conditions under which it holds sup_{t in R} | (psi(t),H(t) psi(t)) |<\infty where psi(t) denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next we show, under certain assumptions, that if the spectrum of the monodromy operator U(T,0) is pure point then there exists a dense subspace of initial conditions for which the mean value of energy is uniformly bounded in the course of time. Further we show that if the propagator admits a differentiable Floquet decomposition then || H(t) psi(t) || is bounded in time for any initial condition psi(0), and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.Comment: 39 page

Symplectic Structures for the Cubic Schrodinger equation in the periodic and scattering case

We develop a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data. As an example we consider the cubic nonlinear Schr\"{o}dinger equation.Comment: This is expanded and corrected versio

Charge Symmetry Breaking in 500 MeV Nucleon-Trinucleon Scattering

Elastic nucleon scattering from the 3He and 3H mirror nuclei is examined as a test of charge symmetry violation. The differential cross-sections are calculated at 500 MeV using a microsopic, momentum-space optical potential including the full coupling of two spin 1/2 particles and an exact treatment of the Coulomb force. The charge-symmetry-breaking effects investigated arise from a violation within the nuclear structure, from the p-nucleus Coulomb force, and from the mass-differences of the charge symmetric states. Measurements likely to reveal reliable information are noted.Comment: 5 page

Dressing chain for the acoustic spectral problem

The iterations are studied of the Darboux transformation for the generalized Schroedinger operator. The applications to the Dym and Camassa-Holm equations are considered.Comment: 16 pages, 6 eps figure