Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1 ^N , where the coefficients a _ij ∈ C ^∞(? _x ⁿ × ? _ξ ⁿ ? C(0, 1] satisfy the estimates |? _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ? C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ? ⁿ + i $ \mathcal{B} We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1^N, where the coefficients a _ij ∈ C ^∞(ℝ _x ⁿ × ℝ _ξ ⁿ ⊗ C(0, 1] satisfy the estimates |ϖ _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ⩽ C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ℝ ⁿ + i , where is a bounded domain in ℝ ⁿ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let H _h ^s (ℝ ⁿ , ℂ ^N ) be the space of distributions with values in ℂ ^N which is equipped with the norm , let Ω ⊂ ℝ ⁿ be a bounded open set, let v ∈ C ^∞(ℝ ⁿ ), let ▿v(x) ∈ for any x ∈ Ω, and let . Let u _h (∈ H _h ^s (ℝ ⁿ ,‒ ^N )) be a solution of the equation Op _h (α)u = 0. In this case, for every ϕ ∈ C ₀^∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h ₀ > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h ₀]:

((1))

We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schr?dinger, Dirac, and square-root Klein-Gordon operators. To the memory of Professor V. A. Borovikov     

On the vacuum status in Weyl–Dirac theory

A hydrodynamic model of the Weyl–Dirac theory in the non-relativistic approach is established. Any microparticle is permanently interacting with the ‘subquantum level’ through the quantum potential, which depends only on the imaginary part of a complex speed. The complex speed fields indicate a possible connection between the Weyl–Dirac theory and Scale Relativity Theory. In such conjecture, some properties of the vacuum states result: the vacuum states behave as a superconducting state, they act as an energy accumulator etc.     

Relativistic Lagrangians for the Lorentz–Dirac equation

We present two types of relativistic Lagrangians for the Lorentz–Dirac equation written in terms of an arbitrary world-line parameter. One of the Lagrangians contains an exponential damping function of the proper time and explicitly depends on the world-line parameter. Another Lagrangian includes additional cross-terms consisting of auxiliary dynamical variables and does not depend explicitly on the world-line parameter. We demonstrate that both the Lagrangians actually yield the Lorentz–Dirac equation with a source-like term.     

On the eigenvalue problem for the schrödinger operator with a confining potential

A new method for the computation of the eigenvalues of the Schrödinger operator proposed recently is applied here to the case of a confining potential. It is shown that one can draw conclusions similar to those in the case of short-range potentials discussed in previous work.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.     

Quasiclassical approximation for the resonant Kapitza–Dirac scattering

In a momentum space, the resonant Kapitza–Dirac effect is described by the difference Schrödinger equation the step of which is the two-photon recoil momentum. For the interferometry of a multipath atom, the case of intense counter-propagating waves is of interest, when the number of generated momenta can reach to some tens. After that, in an intermediate stage of calculations, the discrete momentum distribution is regarded as a continuous one, and the Taylor expansion is applicable to it. This approximation preserves the spectrum of a diffraction well, in particular, the formation of a pair of almost monopulse states from the initial Gaussian distribution.     

On the Schrödinger equation and the eigenvalue problem

If _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ⁿ , H. Weyl's asymptotic formula asserts that , hence . We prove that for any domain and for all . A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on ⁿ (n3) in terms of is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1     

On the Born-Oppenheimer approximation for excited states of polyatomic Schrödinger operators

In this Letter, excited states of polyatomic molecular Schrödinger operators are investigated with the help of the Born-Oppenheimer approximation. The ratio of electronic and nuclear mass plays the role of a semi-classical parameter h ². Asymptotic series of eigenvalues at the bottom of the spectrum are constructed up to any order in h. Mathematically, this leads to the discussion of the semi-classical limit of pseudo-differential operators with the principal symbol po(x,) = ² + , where has a degenerate minimum (a whole manifold).     

On lower bound estimates for the energy spectrum of the Fr?hlich polaron model

Method of intermediate problems was applied to investigation of the energy spectrum of the Fr?hlich polaron model. It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy can be derived for arbitrary values of the electron-phonon interaction constant. These bound estimates allow for explicit numerical evaluation at all orders. In conjunction with numerous well-known upper bound estimates for the energy spectral curve of the Fr?hlich polaron as a function of the interaction constant and the polaron total momentum, the aforesaid improvable lower bound estimates might provide one with virtually precise magnitude for the energy of the slow-moving polaron. Possible generalization to the case of the Fr?hlich polaron in external magnetic field is outlined.     

On the spectra of Schrödinger operators

We give two formulas for the lowest point in the spectrum of the Schrödinger operatorL=–(d/dt)p(d/dt)+q, where the coefficientsp andq are real-valued, bounded, uniformly continuous functions on the real line. We determine whether or not is an eigenvalue forL in terms of a set of probability measures on the maximal ideal space of theC ^*-algebra generated by the translations ofp andq.Research supported in part by the National Science Foundation under Grant DMS-910496     

Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators

We provide lower bounds on the eigenvalue splitting for ?d ²/dx ²+V(x) depending only on qualitative properties ofV. For example, ifV is C^∝ on [a, b] andE _n ,E _n?1 are two successive eigenvalues of ?d ²/dx ²+V withu(a)=u(b)=0 boundary conditions, and if \(\lambda = \mathop {\max }\limits_{E \in (E_{n - 1} ,E_n );x \in (a,b)} |E - V(x)|^{1/2} \) , then $$E_n - E_{n - 1} \geqq \pi \lambda ^2 \exp \left[ { - \lambda (b - a)} \right]$$ . The exponential factor in such bounds are saturated precisely in tunneling examples. Our results arenot restricted toV's of compact support, but only require \(E_n< \mathop {\lim }\limits_{\overline {x \to \infty } } V(x)\) .     

On Spatially Homogeneous Solutions of a Modified Boltzmann Equation for Fermi–Dirac Particles

The paper considers a modified spatially homogeneous Boltzmann equation for Fermi–Dirac particles (BFD). We prove that for the BFD equation there are only two classes of equilibria: the first ones are Fermi–Dirac distributions, the second ones are characteristic functions of the Euclidean balls, and they can be simply classified in terms of temperatures: T>2/5T _F and T=2/5T _F , where T _F denotes the Fermi temperature. In general we show that the L ^∞-bound 0≤f≤ 1/ε derived from the equation for solutions implies the temperature inequality T≥2/5T _F , and if T>2/5T _F , then f trend towards Fermi–Dirac distributions; if T=2/5T _F , then f are the second equilibria. In order to study the long-time behavior, we also prove the conservation of energy and the entropy identity, and establish the moment production estimates for hard potentials.     

Laser Running Waves for the Kapitza–Dirac Diffraction in the Atom Interferometer

The scheme of multi-momentum atom interferometer based on the resonant Kapitza–Dirac diffraction is considered. The conclusion is made that the arms of counter propagating waves forming the standing wave, which scatters the atoms, should have the equal length.     

On the interaction of the Dirac oscillator with the Aharonov–Casher system in topological defect backgrounds

In this paper, we study the influence of the Aharonov–Casher effect [Y. Aharonov, A. Casher, Phys. Rev. Lett. 53 (1984) 319] on the Dirac oscillator in three different scenarios of general relativity: the Minkowski spacetime, the cosmic string spacetime and the cosmic dislocation spacetime. In this way, we solve the Dirac equation and obtain the energy levels for bound states and the Dirac spinors for positive-energy solutions. We show that the relativistic energy levels depend on the Aharonov–Casher geometric phase. We also discuss the influence of curvature and torsion on the relativistic energy levels and the Dirac spinors due to the topology of the cosmic string and cosmic dislocation spacetimes.     

On the dressing method for the generalised Zakharov–Shabat system

The dressing procedure for the generalised Zakharov–Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ‘dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.     

On the “Equivalence” of the Maxwell and Dirac Equations

It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor in the representation of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell's and Dirac's equations into each other.     

Solutions of the Dirac–Fock Equations for Atoms¶and Molecules

The Dirac-Fock equations are the relativistic analogue of the well-known Hartree-Fock equations. They are used in computational chemistry, and yield results on the inner-shell electrons of heavy atoms that are in very good agreement with experimental data. By a variational method, we prove the existence of infinitely many solutions of the Dirac-Fock equations "without projector", for Coulomb systems of electrons in atoms, ions or molecules, with Z h 124, N h 41, N h Z. Here, Z is the sum of the nuclear charges in the molecule, N is the number of electrons.     

Quantizer–dequantizer operators as a tool for formulating the quantization procedure

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.     

On the Magnetic Shield for a Vlasov–Poisson Plasma

We study the screening of a bounded body \(\Gamma \) against the effect of a wind of charged particles, by means of a shield produced by a magnetic field which becomes infinite on the border of \(\Gamma \). The charged wind is modeled by a Vlasov–Poisson plasma, the bounded body by a torus, and the external magnetic field is taken close to the border of \(\Gamma \). We study two models: a plasma composed by different species with positive or negative charges, and finite total mass of each species, and another made of many species of the same sign, each having infinite mass. We investigate the time evolution of both systems, showing in particular that the plasma particles cannot reach the body. Finally we discuss possible extensions to more general initial data. We show also that when the magnetic lines are straight lines, (that imposes an unbounded body), the previous results can be improved.