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.     

PT-Symmetry, Canonical Decomposition, Perturbation Theory

Let H be any PT-symmetric Schrödinger operator of the type H=- ² +x ² +igW(x), where W is a real polynomial, odd under reflection of all coordinates, gR, acting on L ² ( R ^d ). The proof is outlined of the following statements: PH is self-adjoint and its eigenvalues coincide with the eigenvalues of (H*H). Moreover the eigenvalues of (H*H), known as the singular values of H, can be computed via perturbation theory by Borel summability.     

On the spectrum of Schrödinger operators with a random potential

We investigate the spectrum of Schrödinger operatorsH of the type:H =–+q _i ()f(x–x _i + _i ())(q _i () and _i () independent identically distributed random variables,i ^d ). We establish a strong connection between the spectrum ofH and the spectra of deterministic periodic Schrödinger operators. From this we derive a condition for the existence of forbidden zones in the spectrum ofH . For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly.     

A general class of (essentially) iso-spectral perturbations

We consider (essentially) iso-spectral perturbations of operators of the formH _A=A ^* A withA being a densely defined closed linear operator from a Hilbert space to another Hilbert space . We perturbH _A by perturbingA asA+B withB being a linear operator from to . Two classes ofB are defined so as to obtain (essentially) iso-spectral perturbations ofH _A. The abstract results are applied to Schrödinger operators. Our approach gives also a mathematical unification for the so-called factorization method in quantum mechanics.     

Lower bounds on the width of Stark-Wannier type resonances

We prove that the Schrödinger operator –d ²/dx²+Fx+W(x) onL ²(R) withW bounded and analytic in a strip has no resonances in a region ImE–exp(–C/F).     

Commutators of Schrödinger Hamiltonians and applications in scattering theory

We give results on the behaviour at infinity of commutators of the form [(H), f(Q)], where H is a Schrödinger operator and Q denotes the position operator in [(H),f(Q)]. These results are applied to obtain propagation properties and asymptotic completeness below the three-body threshold for N-body systems.     

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 spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.     

Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L ^m/2( ^m ) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|^– , (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.     

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.     

Quantum theory of single events: Localized De Broglie wavelets,Schrödinger waves,and classical trajectories

For an arbitrary potential V with classical trajectoriesx=g(t), we construct localized oscillating three-dimensional wave lumps (x, t,g) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbitg(t), slightly modified due to the potential V, and (x, t,g) satisfies the Schrödinger equation. The field energy, momentum, and angular momentum calculated as integrals over all space are equal to the particle energy, momentum, and angular momentum. The relation to coherent states and to Schrödinger waves is also discussed.     

On the integrated density of states for crystals with randomly distributed impurities

In the present paper, we discuss spectral properties of a periodic Schrödinger operator which is perturbed by randomly distributed impurities; such operators occur as simple models for crystals (or semi-conductors) with impurities. While the spectrum itself is independent of the concentrationp of impurities, for 0<p<1, we focus our attention on the limiting behavior of the integrated density of states _p of the random Schrödinger operator, inside a spectral gap of the periodic operator, asp0. Denoting byU ₀ the set of eigenvalues (in the gap) of the reference problem having precisely one impurity (located at the origin, say), we show that the integrated density of states concentrates around the points ofU ₀, in the sense that _p (U ) is of orderp, for any fixed -neighborhoodU ofU ₀, while _p (K)C·p ², for any compact subsetK of the gap which does not intersectU .Research partially supported by Deutsche Forschungsgemeinschaft     

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation (u/q _i )²+V(q)=E consists of finding new curvilinear coordinates x _i (q) in which the transformed equation admits a complete separated solution u(x)=u _(i)(x _i ;). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.     

Spouge's Conjecture on Complete and Instantaneous Gelation

We investigate the stochastic counterpart of the Smoluchowski coagulation equation, namely the Marcus–Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluster in an arbitrarily small time, which is known as a complete and instantaneous gelation phenomenon. Indeed, Spouge (also Domilovskii et al. for a special case) conjectured that K(i, j)=(ij) , >1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function (i, j), increasing in both i and j such that _j=1 1/(j(i, j))< for all i, and K(i, j)ij(i, j) for all i, j, then complete and instantaneous gelation occurs. Evidently, this implies that any kernels K(i, j)ij(log(i+1)log(j+1)) , >1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim _i+j ij/K(i, j)=0 and _{i, j=1} 1/K(i, j)=, then gelation time T _g satisfies 0<T _g<. Moreover, the gelation is complete after T _g.     

Internal Lifschitz singularities for one dimensional Schrödinger operators

The integrated density of states of the periodic plus random one-dimensional Schrödinger operator ;f0,q _i ()0, has Lifschitz singularities at the edges of the gaps inSp(H ). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.     

Invariant subspaces of clustering operators

A class of clustering operators is defined which is a generalization of a transfer matrix of a Gibbs lattice field with an exponential decay of correlations. It is proved that for small values of the clustering operator has invariant subspaces which are similar tok-particle subspaces of the Fock space. The restriction of the clustering operator onto these subspaces resembles the operator exp(-H _k, whereH _k is thek- particle Schrödinger Hamiltonian in nonrelativistic quantum mechanics. The spectrum of eachH _k,k1, is contained in the interval (C ₁^k,C ₂^k). These intervals do not intersect with each other.     

Localization for random potentials supported on a subspace

We consider the lattice Schrödinger operator acting onl ² ( ^d ) with random potential (independent, identically distributed random variables), supported on a subspace of dimension 1 v <d. We use the multiscale analyses scheme to prove that this operator exhibits exponential localization at the edges of the spectrum for any disorder or outside the interval [-2d, 2d] for sufficiently high disorder.     

Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field

A rigorous path integral representation of the solution of the Cauchy problem for the pure-imaginary-time Schrödinger equation _t (t, x)=–[H–mc ²](t,x) is established.H is the quantum Hamiltonian associated, via the Weyl correspondence, with the classical Hamiltonian [(cp–eA(x))²+m ² c ⁴]^1/2+e(x) of a relativistic spinless particle in an electromagnetic field. The problem is connected with a time homogeneous Lévy process.     

Expectation Values of Observables in Time-Dependent Quantum Mechanics

Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H ₀(t) + W(t), where H ₀(t) commutes for all twith a complete set of time-independent projectors . Consider the observable A=_j P _jjwhere _j j , >0, for jlarge. Assuming that the matrix elements of W(t) behave as for p>0 large enough, we prove estimates on the expectation value for large times of the type where >0 depends on pand . Typical applications concern the energy expectation H₀(t) in case H ₀(t) H ₀or the expectation of the position operator x²(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H ₀(t) is a time-dependent multiplicative potential.