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- Asymptotic-behavior of Jost functions near resonance points for Wigner-Vonneumann type potentialsKlaus, Martin (AIP Publishing, 1991-01)
Show more In this work are considered radial Schrodinger operators - psi" + V(r)-psi = E-psi, where V(r) = a sin br/r + W(r) with W(r) bounded, W(r) = O(r-2) at infinity (a,b real). The asymptotic behavior of the Jost function and the scattering matrix near the resonance point E(o) = b2/4 are studied. If \a\ > \b\, then this point may be an eigenvalue embedded in the continuous spectrum. The leading behavior of the Jost function for all values of a and b were determined. Somewhat surprisingly, situations were found where the Jost function becomes singular as E-->E(o) even if E(o) is an embedded eigenvalue. Moreover, it is found that the scattering matrix is always discontinuous at E(o) except in a few special cases. It is also shown that the asymptotics for the Jost function and the scattering matrix hold under weaker assumptions on W(r), in particular an angular momentum term l(l + 1)r-2 may be incorporated into W(r). The results are also applied to a whole line problem with a potential V(x) such that V(x) = 0 for x < 0 and V(x) of Wigner-von Neumann type for x > 0, and the behavior of the transmission and reflection coefficients as E-->E(o) is also studied.Show more - Asymptotics of the scattering coefficients for a generalized Schrödinger equationAktosun, T.; Klaus, Martin (AIP Publishing, 1999-08)
Show more The generalized Schrodinger equation d(2)psi/dx(2) + F(k)psi=[ikP(x) + Q(x)]psi is considered, where P and Q are integrable potentials with finite first moments and F satisfies certain conditions. The behavior of the scattering coefficients near zeros of F is analyzed. It is shown that in the so-called exceptional case, the values of the scattering coefficients at a zero of F may be affected by P(x). The location of the k-values in the complex plane where the exceptional case can occur is studied. Some examples are provided to illustrate the theory. (C) 1999 American Institute of Physics. [S0022-2488(99)03007-8].Show more - Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time VersionsAmaya, Austin J. (Virginia Tech, 2012-04-26)
Show more Given a full-range simply-invariant shift-invariant subspace*M*of the vector-valued*L*space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function^{2}*W*so that*M*may be represented as the image of of the Hardy space*H*on the disc under multiplication by^{2}*W*. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces*(M,M*which together form a direct-sum decomposition of^{Ã })*L*. In the case where the pair^{2}*(M,M*are finite-dimensional perturbations of the Hardy space^{Ã })*H*and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function^{2}*W*; this realization was parameterized in terms of zero-pole data computed from the pair*(M,M*. Later work by Ball-Raney extended this analysis to the case of nonrational functions^{Ã })*W*where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs*(M,M*of the^{Ã })*L*spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans.^{2}Show more - The Born-Oppenheimer approximation in scattering theoryKargol, Armin (Virginia Tech, 1994-05-05)
Show more We analyze the Schrödinger equation i𝜖 ¬^{2}â /â tÎ¨ = H(𝜖)Î¨, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬^{-4}and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬^{2}). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:Â»Î¨, i.e., we find Î¨(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems.Show more - Born-Oppenheimer Corrections Near a Renner-Teller CrossingHerman, Mark Steven (Virginia Tech, 2008-07-03)
Show more We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner-Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of Îµ, where Îµ4 is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter bË , which is equivalent to the parameter originally used by Renner. For 0 < bË < 1, we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near bË = 0.925. We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for 0 < bË < 0.925 and non-degenerate for 0.925 < bË < 1.Show more - Born-Oppenheimer Expansion for Diatomic Molecules with Large Angular MomentumHughes, Sharon Marie (Virginia Tech, 2007-10-29)
Show more Semiclassical and Born-Oppenheimer approximations are used to provide uniform error bounds for the energies of diatomic molecules for bounded vibrational quantum number n and large angular momentum quantum number l. Specifically, results are given when (l + 1) < κ𝛜⁻³/². Explicit formulas for the approximate energies are also given. Numerical comparisons for the H+₂ and HD+ molecules are presented.Show more - The Cauchy problem for the Diffusive-Vlasov-Enskog equationsLei, Peng (Virginia Tech, 1993-04-05)
Show more In order to better describe dense gases, a smooth attractive tail arising from a Coulomb-type potential is added to the hard core repulsion of the Enskog equation, along with a velocity diffusion. By choosing the diffusing term of Fokker-Planck type with or without dynamical friction forces. The Cauchy problem for the Diffusive-Vlasov-Poisson-Enskog equation (DVE) and the Cauchy problem for the Fokker-Planck-Vlasov-Poisson-Enskog equation (FPVE) are addressed.Show more - A class of weighted Bergman spaces, reducing subspaces for multiple weighted shifts, and dilatable operatorsLiang, Xiaoming (Virginia Tech, 1996-08-15)
Show more This thesis consists of four chapters. Chapter 1 contains the preliminaries. We give the background, notation and some results needed for this work, and we describe our main results of this thesis. In Chapter 2 we will introduce a class of weighted Bergman spaces. We then will discuss some properties about the multiplication operator, Mz , on them. We also characterize the dual spaces of these weighted Bergman spaces. In Chapter 3 we will characterize the reducing subspaces of multiple weighted shifts. The reducing subspaces of the Bergman and the Dirichlet shift of multiplicity N are portrayed from this characterization. In Chapter 4 we will introduce the class of super-isometrically dilatable operators and describe their elementary properties. We then will discuss an equivalent description of the invariant subspace lattice for the Bergman shift. We will also discuss the interpolating sequences on the bidisk. Finally, we will examine a special class of super-isometrically dilatable operators. One corollary of this work is that we will prove that the compression of the Bergman shift on two compliments of two invariant subspaces are unitarily equivalent if and only if the two invariant subspaces are equal.Show more - Conical Intersections and Avoided Crossings of Electronic Energy LevelsGamble, Stephanie Nicole (Virginia Tech, 2021-01-14)
Show more We study the unique phenomena which occur in certain systems characterized by the crossing or avoided crossing of two electronic eigenvalues. First, an example problem will be investigated for a given Hamiltonian resulting in a codimension 1 crossing by implementing results by Hagedorn from 1994. Then we perturb the Hamiltonian to study the system for the corresponding avoided crossing by implementing results by Hagedorn and Joye from 1998. The results from these demonstrate the behavior which occurs at a codimension 1 crossing and avoided crossing and illustrates the differences. These solutions may also be used in further studies with Herman-Kluk propagation and more. Secondly, we study codimension 2 crossings by considering a more general type of wave packet. We focus on the case of Schrödinger equation but our methods are general enough to be adapted to other systems with the geometric conditions therein. The motivation comes from the construction of surface hopping algorithms giving an approximation of the solution of a system of Schrödinger equations coupled by a potential admitting a conical intersection, in the spirit of Herman-Kluk approximation (in close relation with frozen/thawed approximations). Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase and its proof is based on a normal form approach.Show more - Continuity of the S matrix for the perturbed Hill's equationClemence, D. P.; Klaus, Martin (AIP Publishing, 1994-07)
Show more The behavior of the scattering matrix associated with the perturbed Hill's equation as the spectral parameter approaches an endpoint of a spectral band is studied. In particular, the continuity of the scattering matrix at the band edges is proven and explicit expressions for the transmission and reflection coefficients at those points are derived. All possible cases are discussed and our fall-off assumptions on the perturbation are weaker than those made by other authors.Show more - A Coupled Tire Structure-Acoustic Cavity ModelMolisani, Leonardo Rafael (Virginia Tech, 2004-05-20)
Show more Recent experimental results have shown that the vibration induced by the tire air cavity resonance is transmitted into the vehicle cabin and may be responsible for significant interior noise. The tire acoustic cavity is excited by the road surface through the contact patch on the rotating tire. The effect of the cavity resonance is that results in significant forces developed at the vehicle's spindle, which in turn drives the vehicle's interior acoustic field. This tire-cavity interaction phenomenon is analytically investigated by modeling the fully coupled tire-cavity systems. The tire is modeled as an annular shell structure in contact with the road surface. The rotating contact patch is used as a forcing function in the coupled tire-cavity governing equation of motion. The contact patch is defined as a prescribed deformation that in turn is expanded in its Fourier components. The response of the tire is then separated into static (i.e. static deformation induced by the contact patch) and dynamic components due to inertial effects. The coupled system of equations is solved analytically in order to obtain the tire acoustic and structural responses. The model provides valuable physical insight into the patch-tire-acoustic interaction phenomenon. The influence of the acoustic cavity resonance on the spindles forces is shown to be very important. Therefore, the tire cavity resonance effect must be reduced in order to control the tire contribution to the vehicle interior. The analysis and modeling of two feasible approaches to control the tire acoustic cavity resonances are proposed and investigated. The first approach is the incorporation of secondary acoustic cavities to detune and damp out the main tire cavity resonance. The second approach is the addition of damping directly into the tire cavity. The techniques presented in this dissertation to suppress the adverse effects of the acoustic cavity in the tire response, i.e. forces at the spindle, show to be very effective and can be easily applied in practice.Show more - Coupling constant behavior of eigenvalues of Zakharov-Shabat systemsKlaus, Martin; Mityagin, Boris (AIP Publishing, 2007-12)
Show more We consider the eigenvalues of the non-self-adjoint Zakharov-Shabat systems as the coupling constant of the potential is varied. In particular, we are interested in eigenvalue collisions and eigenvalue trajectories in the complex plane. We identify shape features in the potential that are responsible for the occurrence of collisions and we prove asymptotic formulas for large coupling constants that tell us where eigenvalues collide or where they emerge from the continuous spectrum. Some examples are provided which show that the asymptotic methods yield results that compare well with exact numerical computations. (c) 2007 American Institute of Physics.Show more - Coupling constant thresholds of perturbed periodic HamiltoniansFassari, S.; Klaus, Martin (AIP Publishing, 1998-09)
Show more We consider Schrodinger operators of the form H-lambda= -Delta + V + lambda W on L-2(R-v) (v=1, 2, or 3) with V periodic, W short range, and lambda a real non-negative parameter. Then the continuous spectrum of H-lambda has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of H-lambda that are functions of the parameter lambda. Let (a,b) be a gap and E(lambda)E(a,b) an eigenvalue of H-lambda. We study the asymptotic behavior of E(lambda) as lambda approaches a critical value lambda(0), called a coupling constant threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming E(lambda)down arrow a as lambda down arrow lambda(0), is E(lambda)-a similar to c(lambda - lambda(0))(alpha) for some alpha>0 and c not equal 0, or is there an expansion in some other quantity? As one expects from previous work in the case V=0, the answer strongly depends on v. (C) 1998 American Institute of Physics.Show more - Discrete dynamical systems in solving H-equationsChen, Jun (Virginia Tech, 1995-08-17)
Show more Three discrete dynamical models are used to solve the Chandrasekhar H-equation with a positive or negative characteristic function. Two of them produce series of continuous functions which converge to the solution of the H-equation. An iteration model of the nth approximation for the H-equation is discussed. This is a nonlinear n-dimensional dynamical system. We study not only the solutions of the nth approximation for the H-equation but also the mathematical structure and behavior of the orbits with respect to the parameter function, i.e. characteristic function. The dynamical system is controlled by a manifold. For n=2, stability of the fixed points is studied. The stable and unstable manifolds passing through the hyperbolically fixed point are obtained. Globally, the bounded orbits region is given. For parameter c in some region a periodic orbit of one dimension will cause periodic orbits in the higher dimensional system. For changing parameter c, the bifurcation points are discussed. For c ∈ (-5.6049, 1] the system has a series of double bifurcation points. For c ∈ (-8, -5.6049] chaos appears. For c in a window contained the chaos region, a new bifurcation phenomenon is found. For c ≤ -7 any periodic orbits appear. For c in the chaos region the behavior of attractor is discussed. Chaos occurs in the n-dimensional dynamical system.Show more - The effect of cumulative forward and single large-angle scattering of an electromagnetic wave in a random mediumBegum, Syeda Rasheda (Virginia Polytechnic Institute and State University, 1987)
Show more A general discussion on propagation of an electromagnetic wave in a random medium is presented. Emphasis is placed on the bistatic scattering problem. The first phase of the investigation is focused on a random continuum. This is an extension of work done by de Wolf [19] recently. He derives a formal expression for the enhancement factor of the electromagnetic flux at large angles (excluding backscatter) from an extended weakly random medium. Enhancement describes the factor by which the singly-scattered flux is modified when the effects of cumulative forward scatterings are taken into account before and after one large-angle scattering. Explicit results are calculated here for a two-dimensional geometry describing cylindrical scattering from a slab of width L filled with a uniformly turbulent dielectric described by a power-law spectrum in the inertial subrange. The results show that the enhancement factor is close to unity beyond the mean free path of the small-angle scatterings and it increases when the medium width L exceeds the mean free path of a large-angle scattering. This result is extended for a generalized power-law structure function of the dielectric permittivity fluctuation, which shows a possibility of using the cumulative forward and single large-angle scattered flux to detect the statistical properties of a random continuum. The negligence of the Fresnel terms in the expression of the scattered flux is justified by including those in the phase term and investigating the resulting effects. This investigation reveals that the inclusion of the Fresnel terms makes the scattered flux complex, an error arising from the truncation of the higher-order phase terms, which is not observed when the Fresnel terms are neglected. The second phase of the investigation involves discrete random media. A mathematical model is developed for the purpose of deriving an integral equation of the coherent field and autocorrelation function in the very general case of an electromagnetic wave propagating in a medium of densely packed nontenuous particles. All orders of N-tuple particle correlation function are included. The resulting equations are a generalization of those derived recently by Tsolakis et al. [23]; the four lowest-order terms of each equation include those of Tsolakis et al.'s equations which incorporate only binary correlation between particles. The mathematical model is then used to derive an expression for the scattered flux of an electromagnetic wave under a first-order cumulative forward and single large-angle scattering approximation. The resulting expression is valid at high frequencies under Twersky's approximation [5]. It is shown that the discrete scatterer case may be treated by an approach similar to the continuum case by using the new formulation.Show more - Eigenvalue Statistics for Random Block OperatorsSchmidt, Daniel F. (Virginia Tech, 2015-04-28)
Show more The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators.Show more - Exact behavior of Jost functions at low energyKlaus, Martin (AIP Publishing, 1988-01)
Show more For Schrödinger operators with central potential q(r) and angular momentuml, the behavior of the Jost function F l (k) as k→0 is investigated. It is assumed that ∫∞ 0 d r (1+r)σ‖q(r)‖<∞, where σ≥1. Situations where q is integrable with 1≤σ<2, but not with σ≥2 are of particular interest. For potentials satisfying q(r)∼q 0 r −2−ε (0<ε≤1) and l=0, the leading behavior of F 0(k) and the phase shift δ0(k) as k→0 is derived. Also comments are made on the differentiability properties of the Jost solutions with respect to the variable k at k=0. For σ=1 Levinson’s theorem is proved, thereby clarifying some questions raised recently by Newton [J. Math. Phys. 2 7, 2720 (1986)].Show more - Exponentially Accurate Error Estimates of Quasiclassical EigenvaluesToloza, Julio Hugo (Virginia Tech, 2002-12-11)
Show more We study the behavior of truncated Rayleigh-SchrÃ¶odinger series for the low-lying eigenvalues of the time-independent SchrÃ¶odinger equation, when the Planck's constant is considered in the semiclassical limit. Under certain hypotheses on the potential energy, we prove that, for any given small value of the Planck's constant, there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and actual eigenvalue is smaller than an exponentially small function of the Planck's constant. We also prove the analogous results concerning the eigenfunctions.Show more - Extended describing function method for small-signal modeling of resonant and multi-resonant convertersYang, Eric Xian-Qing (Virginia Tech, 1994)
Show more The extended describing function method is proposed as a systematic small-signal modeling approach to nonlinear switching circuits. This method offers significant simplification upon the previous work on using the multi-variable describing functions to treat the circuit nonlinearities. As an extension to the statespace averaging method, this modeling technique can incorporate any Fourier components for good model accuracy and provides continuous-time small-signal models for PWM topologies and various soft-switching resonant topologies. The proposed method is demonstrated using four resonant topologies and two multi-resonant topologies. These circuits are strongly oscillatory, and thus they cannot be modeled by means of traditional averaging techniques. By employing the proposed modeling method, the dynamics of the resonant converters are analyzed with emphasis on the nonlinear interaction between the switching frequency and the circuit natural resonant frequency. Equivalent circuit models are provided for more convenience of practical designs. Small-signal analysis is also performed for two challenging multi-resonant topologies with complex structure and operation. All of the theoretical models are verified experimentally and the predictions are well supported by the measurement data up to the Nyquist frequency.Show more - Factorization of scattering matrices due to partitioning of potentials in one-dimensional Schrödinger-type equationsAktosun, Tuncay; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1996-12)
Show more The one-dimensional Schrodinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero-energy transmission coefficient vanishes when the potential is nontrivial, but in the exceptional case this coefficient is nonzero, resulting in tunneling through the potential. It is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into generic pieces in infinitely many ways. The results remain valid when Dirac delta functions are included in the potential and other coefficients are added to the Schrodinger equation. For such Schrodinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem. (C) 1996 American Institute of Physics.Show more