Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

Appearance of a purely singular continuous spectrum in a class of random schrödinger operators

We consider a discrete Schrödinger operator on l²() with a random potential decaying at infinity as ¦n¦^–1/2. We prove that its spectrum is purely singular. Together with previous results, this provides simple examples of random Schrödinger operators having a singular continuous component in its spectrum.     

Two Points Blow-up in Solutions of the Nonlinear Schrödinger Equation with Quartic Potential on \mathbb{R}

We consider the blow-up problem for the nonlinear Schrödinger equation with quartic self-interacting potential on . We exhibit a class of initial data leading to the blow-up solutions which have at least two L ²-concentration points.     

Symmetries of Schrödinger Operator with Point Interactions

The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling W are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.     

A New Estimate for the Groundstate Energy of Schrödinger Operators

A new estimate for the groundstate energy of Schrödinger operators on L²(ⁿ) (n 1) is presented. As a corollary, it is shown that the groundstate energy of the Schrödinger operator with a scalar potential V is more than the classical lower bound ess.inf_x__V(x) if V is essentially bounded from below in a certain manner (enhancement of the groundstate energy due to quantization). As an application, it is proven that the groundstate energy of the Hamiltonian of the hydrogen-like atom is enhanced under a class of perturbations given by scalar potentials (vanishing at infinity).     

Weakly coupled states on branching graphs

We consider a Schrödinger particle on a graph consisting of N links joined at a single point. Each link supports a real locally integrable potential V _j ; the self-adjointness is ensured by the type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as x ^–1– along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrödinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the coupling constant may be interpreted in terms of a family of squeezed potentials.     

Resonance energies of meta-stable dtµ*

Using the coupled rearrangement channel method, we have calculated resonance energies for meta-stable states of the molecular ion dtµ^* associated with the adiabatic 3 potential. The vacuum polarization effect was taken into account by direct inclusion of the Uehling potential in our three-body Hamiltonian. Comparing with the solution of the pure Coulombic Schrödinger equation a shift of approximately +0.1 eV is found. Thus the infinite series of states of the Coulombic Schrödinger equation becomes truncated. Eleven states remain semi-bound, five of them with binding energy smaller than the dissociation energy of the D₂ molecule, facilitating formation of dtµ^* in tµ(2s)-D₂ scattering by means of the Vesman-mechanism.     

Degeneracy of Resonances, Jordan Blocks, and Gamow-Jordan Eigenfunctions

We show that a degeneracy of resonances is associated with a second rank pole in the scattering matrix and a Jordan chain of generalized eigenfunctions of the radial Schrödinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in complex resonance energy eigenfunctions. In this biorthonormal basis, any operator f(H _r which is a regular function of the Hamiltonian is represented by a nondiagonal complex matrix with a Jordan block of rank 2.     

Anderson Localization for Time Quasi-Periodic Random Schrödinger and Wave Equations

We prove that at large disorder, with large probability and for a corresponding set of Diophantine frequencies of large measure, Anderson localization in ^d is stable under localized time quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. This paper paves the way for the construction of time quasi-periodic KAM type of solutions of non linear random Schrödinger equations in [BW].Wei-Min Wang thanks A. Soffer and T. Spencer for many useful conversations and for initiations to the subject. She also thanks M. Combescure and J. Sjöstrand for helpful discussions on the quasi-energy operator formulation of time dependent Schrödinger equations. The support of NSF grant DMS 9729992 is gratefully acknowleged.     

On the theory of eigenfunctions expansion of the Schrödinger operator

It is shown that the generalized eigenfunctions of the Schrödinger operator with singular potentials actins in L ₂(₃) are ordinary functions with determined asymptotic behaviour at infinity.     

Integrability in the theory of Schrödinger operator and harmonic analysis

The algebraic integrability for the Schrödinger equation in ⁿ and the role of the quantum Calogero-Sutherland problem and root systems in this context are discussed. For the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding Sutherland operator is found. As an application the explicit formula for the zonal spherical functions on the symmetric spacesSU _2n ^* /Sp_n (type A II in Cartan notations) is presented.     

A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.     

Singular perturbations of discrete spectrum

Even solutions of the Schrödinger equation with retaining potential x² are constructed for singular perturbation potentials |x|^–. It is shown that the perturbation automatically entails an induced point potential, taking account of which the perturbation matrix elements and Rayleigh-Schrödinger series may be constructed when 1 < < 3/2. In the opposite case (3/2 2), although the solutions are analytic with respect to , not even diverging series can be obtained for the energy solutions without solution of the Schrödinger equation. The analogy with quantum field theory is explored.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 58–64, March, 1988.     

Super-extensions of energy dependent Schrödinger operators

We consider the energy dependent super Schrödinger linear problem which is a direct generalization of the purely even, energy dependent Schrödinger equation discussed in [1]. We show that the isospectral flows of that problem possess (N+1) compatible Hamiltonian structures. We also extend a generalised factorisation approach of [2] to this case and derive a sequence ofN modifications for the 2N component systems. Then ^th such modification possesses (N–n+1) compatible Hamiltonian structures.On leave of absence from Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-00-681 Warsaw, Poland (present address)     

Anderson localization for multi-dimensional systems at large disorder or large energy

We prove that discrete Schrödinger operators on ^d with a random-potential have almost-surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy. This is the first proof of localization for multi-dimensional Anderson models.Groupe de recherche 048 du CNRS     

Low energy asymptotics for Schrödinger operators with slowly decreasing potentials

Low energy behavior of Schrödinger operators with potentials which decay slowly at infinity is studied. It is shown that if the potential is positive then the zero energy is very regular and the resolvent is smooth near 0. This implies rapid local decay for the solutions of the Schrödinger equation. On the other hand, if the potential is negative then the resolvent has discontinuity at zero energy. Thus one cannot expect local decay faster than ordert ^–1 ast.     

Zariski closure of subgroups of the symplectic group and Lyapunov exponents of the Schrödinger operator on the strip

We consider the Schrödinger equation with a random potential

The Lagnese–Stellmacher Potentials Revisited

We give a new proof of a classical result of Lagnese and Stellmacher, characterizing all Huygens’ operators of the form , where q(x) depends on only one variable. The proof amounts to characterize the Schrödinger operators with a finite heat kernel expansion.     

Feynman path integrals and the trace formula for the Schrödinger operators

We study Schrödinger operators of the form on ^d , whereA ² is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If _n are the eigenvalues ofH we show that the theta function is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of (t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.Partially supported by the USP-Mathematisierung, University of Bielefeld (Forschungsprojekt Unendlich dimensionale Analysis)     

The final states in photoemission and field emission

For a semiinfinite medium and for a slab it is shown how the final state functions describing photoemission and field emission are to be defined if more than one solution of the Schrödinger equation for a fixed value of the energyE and the parallel componentk of the wave vectork exist. As by-product some interesting statements concerning complexband structure and the solutions of the Schrödinger equation for a semiinfinite medium or a crystalline slab are derived.