Self-Energy and Action Principle in Relativistic Schrödinger Theory

The mathematical framework of Relativistic Schrödinger Theory (RST) is generalized in order to include the self-interactions of the particles as an integral part of the theory (i.e. in a non-perturbative way). The extended theory admits a Lagrangean formulation where the Noether theorems confirm the existence of the conservation laws for charge and energy–momentum which were originally deduced directly from the dynamical equations. The generalized RST dynamics is applied to the case of some heavy helium-like ions, ranging from germanium (Z=32) to bismuth (Z=83), in order to compute the interaction energy of the two electrons in their ground-state. The present inclusion of the electron self-energies into RST yields a better agreement of the theoretical predictions with the experimental data.     

Zitterbewegung and Quantum Jumps in Relativistic Schrödinger Theory

Within the general framework of the relativistic Schrödinger theory, a new waveequation is identified which stands between Dirac's four-component spinorequation and the scalar one-component Klein–Gordon equation. It is atwo-component, first-order wave equation in pseudo-Riemannian spacetime which onone hand can take account of the Zitterbewegung (similar to the Dirac theory),but on the other hand describes spinless particles (just like the Klein–Gordontheory). In this way it is demonstrated that spin and Zitterbewegung areindependent phenomena despite the fact that both effects refer to a certain kindof internal motion. An extra variable for the internal motion can be introduced(similarly as in the Dirac theory) so that the new wave equation is reduced tothe Klein–Gordon case when the internal variable takes its trivial value and theinternal motion is not excited. The internal degree of freedom admits the occurenceof quasi-pure states (i.e., a special subset of the mixtures), which undergo atransition to a pure state in finite time. If the initial configuration is already apure state, this transition occurs in the form of a sudden jump to the final purestate. The coupling of the new wave field to gravity via the Einstein equationsmakes the Zitterbewegung manifest through the corresponding trembling of theextension of a Friedmann–Robertson–Walker universe.     

Positive and Negative Mixtures in Relativistic Schrödinger Theory

The general formalism of relativistic Schrödinger theory (RST) is specialized to a scalar two-particle system with electromagnetic interactions (scalar helium atom). The set of dynamically allowed field configurations splits up into positive and negative mixtures and pure states. The static and spherically symmetric solutions are constructed by means of first-order perturbation theory for the case of an attractive Coulomb potential. The corresponding energy levels for the positive and negative mixtures resemble the emergence of ortho and para states in the conventional quantum theory. The associated energy eigenvalues predicted by the RST seem to undergo a certain kind of mixture degeneracy as the RST analog of the conventional exchange degeneracy. The charge densities of the positive mixtures assimilate, whereas the densities of the negative mixtures recede from one another. Thus, positive (negative) mixtures strongly resemble the bosonic (fermionic) matter of the conventional theory when the Pauli principle is applied.     

Relativistic Schrödinger Theory and the Hartree–Fock Approach

Within the framework of Relativistic Schrödinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the exchange interactions of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory

Relation between the vacuum eigenvalues of CFT Q-operators and spectral determinants of one-dimensional Schrödinger operator with homogeneous potential, recently conjectured by Dorey and Tateo for special value of Virasoro vacuum parameter p, is proven to hold, with suitable modification of the Schrödinger operator, for all values of p.     

Schrödinger Maps and Anti-Ferromagnetic Chains

We study the problem of restricting Schrödinger maps onto Lagrangian submanifolds. The restriction is imposed by an infinite constraining potential. We show that, in the limit, solutions of the Schrödinger map equations converge to solutions of generalized wave map equations. This result is applied to the anti-ferromagnetic systems where we prove rigorously that the dynamics is governed by wave map equations into .     

Schrödinger equation in multiply connected spaces and phase optics

The magnetic vector potential\(\vec A\) in a field free spaceR ₀ cannot be removed by gauge transformations in general, ifR ₀ is multiply connected.Aharonov andBohm ¹ have noticed recently that\(\vec A\) therefore should have more physical meaning than only to give the magnetic field by differentiation. They could show that\(\vec A\) inR ₀ may influence the phase ofSchrödinger'sψ-function in an observable manner. We want to point out here that this influence can be expressed in a simple, general form: “A closed magnetic field line operates uponψ like ae Φ/?-phase-shifter placed on any area bounded by the field line.” Surface like phase shifters are familiar in phase optics. There exists a narrow relationship between electron scattering at magnetic fields and some special problems of phase optics. An electron phase contrast microscope is discussed.     

Spectral Theory for Periodic Schrödinger Operators with Reflection Symmetries

Let H=–+V be defined on ^d with smooth potential V, such that In addition we assume that where This is a periodic Schrödinger operator with additional reflection symmetries. We investigate the associated Floquet operators H ^q , q[0,1] ^d . In particular we show that the associated lowest eigenvalues _q are simple if q=(q ₁ ,q ₂ ,,q _d ) satisfies q _j 1/2 for each j=1,2,,d. Supported by Ministerium für Bildung, Wissenschaft und Kunst der Republik ÖsterreichSupported by the European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT)     

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.     

Thermodynamic Gravity and the Schrödinger Equation

We adopt a formulation of the Mach principle that the rest mass of a particle is a measure of it’s long-range collective interactions with all other particles inside the horizon. As a consequence, all particles in the universe form a ‘gravitationally entangled’ statistical ensemble and one can apply the approach of classical statistical mechanics to it. It is shown that both the Schrödinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to the discreetness and finiteness of the Machian ensemble.     

Collapse in the nonlinear Schrödinger equation

A new class of exact solutions with a singularity at finite time (collapse) is obtained for the nonlinear Schrödinger equation.     

On Three Magnetic Relativistic Schr?dinger Operators and Imaginary-Time Path Integrals

Three magnetic relativistic Schr?dinger operators are considered corresponding to the classical relativistic Hamiltonian symbol with magnetic vector and electric scalar potentials. We discuss their difference in general and their coincidence in the case of constant magnetic fields, as well as whether they are covariant under gauge transformation. Then results are surveyed on path integral representations for their respective imaginary-time relativistic Schr?dinger equations, i.e. heat equations, by means of the probability path space measure coming from the Lévy process concerned.     

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L¹L estimates for the time evolution of Hamiltonians H=–+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is (1+|x|)|V(x)|dx<, whereas in d=3 it is |V(x)|C(1+|x|)^–3–.Supported by the NSF grant DMS-0070538 and a Sloan fellowship.     

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.     

Peregrine solitons and gradient catastrophes in discrete nonlinear Schrödinger systems

In the present work, we examine the potential robustness of extreme wave events associated with large amplitude fluctuations of the Peregrine soliton type, upon departure from the integrable analogue of the discrete nonlinear Schrödinger (DNLS) equation, namely the Ablowitz–Ladik (AL) model. Our model of choice will be the so-called Salerno model, which interpolates between the AL and the DNLS models. We find that rogue wave events are drastically distorted even for very slight perturbations of the homotopic parameter connecting the two models off of the integrable limit. Our results suggest that the Peregrine soliton structure is a rather sensitive feature of the integrable limit, which may not persist under “generic” perturbations of the limiting integrable case.     

The Schrödinger and Pauli-Dirac Oscillators in Noncommutative Phase Space

We investigate the non-relativistic Schrödinger and Pauli-Dirac oscillators in noncommutative phase space using the five-dimensional Galilean covariant framework. The Schrödinger oscillator presented the correct energy spectrum whose non isotropy is caused by the noncommutativity with an expected similarity between this system and the particle in a magnetic field. A general Hamiltonian for the 3-dimensional Galilean covariant Pauli-Dirac oscillator was obtained and it presents the usual terms that appears in commutative space, like Zeeman effect and spin-orbit terms. We find that the Hamiltonian also possesses terms involving the noncommutative parameters that are related to a type of magnetic moment and an electric dipole moment.