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.     

The inverse scattering matrix for the Schrödinger equation when the potentialq(x) ∈L 1 1 with a singular term

When the potentialq(x) L ₁ ¹ with a singular term, the continuities of the scattering matrix of the Schrödinger equation are investigated. By means of the transformation approach, we arrive at the conclusion that the scattering matrix S(k) of such a potential is continuous for the wholek,- <k < .     

Relativistic quantum theory for charged spinless particles in external vector fields

Guided by a diagonalized form of the classical field-energy we construct a time-dependent canonical pair of Schrödinger fields _t (x) and _t (x) which diagonalizes the field-HamiltonianH _t . These Schrödinger fields in general belong to inequivalent representations of the canonical commutation relations for differentt's.The Heisenberg field is constructed by solving the Heisenberg equation of motion and its time-evolution turns out to be governed by a unitary operator, i.e. the Heisenberg fields at different times are unitarily equivalent.Scattering theory (including eventual incoming and/or outgoing bound-states) is finally constructed.     

A solution spectrum of the nonlinear schrödinger equation

The soliton solutions of the form=A/coshkx and=B tanhkx of the nonlinear Schrödinger equation have been considered with respect to many problems. In this paper, it is shown that the nonlinear Schrödinger equation also possesses a solution manifold that generalizes the above soliton functions and provides a discrete spectrum of eigenfunctions and eigenvalues. With the help of a slight modification of these eigenfunctions, it is possible to extend them to the relativistic case, where they become solutions of a nonlinear Klein-Gordon equation associated with a discrete mass spectrum.     

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.     

Solution of the radial Schrödinger equation by a modified “shooting” method

Based on a comparative analysis of various modifications of the shooting method, used for solving the radial Schrödinger equation, a new algorithm is proposed, which allows to increase considerably its solution accuracy. This is achieved, first, by rising the order of approximation of the Schrödinger equation fromh ⁴ (in the traditional Numerov-Cooley method) up toh ⁶ and, second, by using the variable integration steph. The computer program has been written in Fortran IV to execute this algorithm for calculating vibrational wave functions and related quantities for diatomic molecules.     

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.     

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.     

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.     

Physical foundations of quantum theory: Stochastic formulation and proposed experimental test

The time-dependent Schrödinger equation has been derived from three assumptions within the domain of classical and stochastic mechanics. The continuity equation isnot used in deriving the basic equations of the stochastic theory as in the literature. They are obtained by representing Newton's second law in a time-inversion consistent equation. Integrating the latter, we obtain the stochastic Hamilton-Jacobi equation. The Schrödinger equation is a result of a transformation of the Hamilton-Jacobi equation and linearization by assigning the arbitrary constant =2mD. An experiment is proposed to determine and to test a hypothesis of the theory directly. A mathematical apparatus is formulated from the Jacobian formalism to derive physical parameters from (x, t) and to obtain operators for the boundary cases of the theory. The operator formalisms are compared by means of a well-known solution in the quantum theory.     

Some problems of symmetry of the Schrödinger equations

The Schrödinger algebra sch₃ is examined as a subalgebra of the algebra k_1,4 of conformal transformations of the space R_{1, 4}. Orbits of the associated representations of the Schrödinger group are found in the algebra sch₃. It is proven that all nontrivial local differential symmetry operators of second order belong to the enveloping algebra U(sch₃) of the algebra sch₃, and the space of these operators is defined. All the absolute identities and identities on the solutions of the Schrödinger equation are obtained in the space of second-order operators of the algebra U(sch₃).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 120–123, April, 1991.     

Differential equations in the spectral parameter

We determine all the potentialsV(x) for the Schrödinger equation (– _x ² +V(x))=k² such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formB(k, _k )ø=(x)ø. For each suchV(x) we determine the algebra of all possible operatorsB and the corresponding functions (x)This research was partially supported by NSF grant DMS 84-03232 and ONR contract NOOO14-84-C-0159     

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.     

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.     

Instability of nonlinear bound states

We establish a sharp instability theorem for the bound states of lowest energy of the nonlinear Klein-Gordon equation,u _tt–u+f(u)=0, and the nonlinear Schrödinger equation, –iu _t–u+f(u)=0.Supported in part by NSF Grants MCS 81-21487 and MCS 82-01599     

Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques

Under some hypotheses of analyticity and integrability we show the existence and uniqueness of a strong regular solution of the Schrödinger equation using a natural generalisation to the complex case of the Feynman-Kac formula. This explicit representation allows us to study in certain cases the asymptotic behavior of the solution when the Planck constanth tends to zero. The same method can be used for the solution of more general Schrödinger equations.

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Membre du Laboratoire Associé au C.N.R.S., n° 224 Processus Stochastiques et Applications     

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.     

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r ^– in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case = 4 is elucidated. In general, whenever the parameter is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.     

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.