Residue-squaring iterative diagonalization method for perturbation problems

We present a new method for the evaluation of the change in eigenvalues due to a perturbation of strength λ. It is a fast converging iterative method which, at thenth step, gives results accurate to order (2 ⁿ⁺¹?1) in λ. Unlike the Rayleigh-Schrödinger perturbation theory in quantum mechanics, which becomes prohibitively cumbersome when carried to higher orders, the present method involves a routine which remains stralghtforward at all stages.     

Construction of model hamiltonians in framework of Rayleigh-Schrödinger perturbation theory

A theory of the model Hamiltonians within the framework of Rayleigh-Schrödinger perturbation theory is elaborated. The approach of a model Hamiltonian is based on the assumption that if it is diagonalized in a chosen model space it will yield eigenvalues of the original Hamiltonian in the entire Hilbert space. The theory of the model Hamiltonians may be fruitful as a theoretical background for the study of effective Hamiltonians and as natural extension of the standard Rayleigh-Schrödinger perturbation theory.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

Action-variable perturbation theory

A quantum mechanical perturbation theory based upon the recently introduced quantum action variable is developed and illustrated. Unlike Rayleigh-Schrödinger or other asymptotic theories, this theory can provide a natural solution for the bound states of any potential.     

On the anharmonic oscillator in quantum mechanics and in field theory

A modified Rayleigh-Schrödinger perturbation method is used to derive explicit expansions for the eigenvalues and eigensolutions of the anharmonic oscillator. We then point out the dual relationship between the anharmonic oscillator and the Schrödinger equation for a Yukawa potential. Finally we consider an application of the method to a field-theoretic Hamiltonian, since the anharmonic oscillator plays a dominant role in many field-theoretic models.     

Singular perturbation potentials

This is a perturbative analysis of the eigenvalues and eigenfunctions of Schrödinger operators of the form ?Δ + A + λV, defined on the Hilbert space L²(Rⁿ), where $\text{Δ = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{?}{}^2\text{?X}{}_{\mathrm{i}}{}^2$, A is a potential function and V is a positive perturbative potential function which diverges at some finite point, conventionally the origin. λ is a small real or complex parameter. The emphasis is on one-dimensional or separable problems, and in particular the typical example is the “spiked harmonic oscillator” Hamiltonian, $\text{?d}{}^2\text{dx}{}^2\text{+ x}{}^2\text{+}\text{l(l + 1)}\text{x}{}^2\text{+ λ|x|}{}^{\mathrm{?\alpha }}$, where α is a positive constant. When this kind of perturbation is very singular, the first-order Rayleigh-Schrödinger perturbative correction, (u₀, Vu₀), where u₀ is the unperturbed eigenfunction, diverges. This analysis constructs explicitly calculable terms in a modified perturbation series to a finite order, by using linear operator theory in concert with approximation methods for differential equations. Along the way a connection between a W-K-B type approximation and Bessel functions is exploited.     

Calculation of the energy levels of excited vibrational states of the HD16O molecule by summing divergent series of the Rayleigh-Schrödinger perturbation theory. The shift of zero-order levels

The Rayleigh-Schrödinger perturbation theory is applied to calculation of the energy levels of excited vibrational states of the HD¹⁶O molecule. The model of coupled anharmonic oscillators is considered, with the anharmonic part of potential energy being taken into account as the perturbation. The calculations are carried out for the vibrational states that correspond to three-to seven-fold vibrational excitations. Since the perturbation series diverge in the case of strong resonance interactions and their approximations by the Padé and Padé-Hermite methods do not yield sufficiently correct results, a calculation technique is applied that allows the zero-order approximation to be modified. The zero-order Hamiltonian is modified by shifting the vibrational frequencies, which decreases the mixing of states. The new Rayleigh-Schrödinger series can be summed using the quadratic Padé-Hermite approximation method.     

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by ${e^{\epsilon x}}We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schr?dinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schr?dinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schr?dinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by e^ex{e^{\epsilon x}} uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with x₀ ? I{\xi_0\in I}, then uniformly in I,

\fracSL(x0 + a/L, x0 + b/L)SL(x0, x0)? \fracsin(pr(x0)(a - b))pr(x0)(a - b),\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)}\rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},      10. The E.S.R. spectrum of the tropolonyl radical      William T. Dixon  Wendy E.J. Foster  D. Murphy 《Molecular physics》2013,111(6):1709-1710 The many-body diagrammatic Rayleigh-Schrödinger perturbation theory (RSPT) is used for the calculation of ionization potentials of open-shell systems with one unpaired electron. This theoretical approach is tested on the simple examples of NO2 and NF2 molecules described by the INDO semi-empirical hamiltonian. The first- and second-order results are presented.      11. Calculation of vibrational energy levels of water molecule by summing divergent perturbation theory series      A. D. Bykov  K. V. Kalinin 《Optics and Spectroscopy》2011,111(3):367-375 The Rayleigh-Schrödinger perturbation theory is applied to a calculation of vibrational energy levels of the H2O molecule for isolated states and the states involved in the anharmonic Fermi and Darling-Dennison resonances. It is shown that in spite of the rapid divergence of the perturbation theory series caused by the resonances, the use of the summation methods of Padé, Padé-Borel, and Padé-Hermite and the moments method allows one to obtain quite satisfactory results.      12. Klein paradox in the Breit equation      W. Królikowski  A. Turski  J. Rzewuski 《Zeitschrift fur Physik C Particles and Fields》1979,1(4):369-375 We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r 0)=E there appears a Klein paradox atr=r 0. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm (1)→∞ andm (2)→∞, where we neglect the terms ～1/m (1) and 1/m (2). In Appendix I we show that in the Breit equation the oscillations accumulating atr=r 0 in the case ofm (1)≠m (2) are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r 0 in the case ofm (1)=m (2).      13. Quantization of the potential amplitude in the one-dimensional Schrödinger equation      B. Ya. Balagurov 《Journal of Experimental and Theoretical Physics》2004,99(4):856-874 A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ?n(x) and eigenvalues αn corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ?n(x) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.      14. Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian      Alexander Pushnitski  Georgi Raikov  Carlos Villegas-Blas 《Communications in Mathematical Physics》2013,320(2):425-453       15. Zonal Schrödinger operators on then-sphere: Inverse spectral problem and rigidity      David Gurarie 《Communications in Mathematical Physics》1990,131(3):571-603 We study the Direct and Inverse Spectral Problems for a class of Schrödinger operatorsH=–+V onS n withzonal (axisymmetric) potentials. Spectrum ofH is known to consist of clusters of eigenvalues {km=k(k+n-1)+km:mk}. The main result of the work is to derive asymptotic expansion of spectral shifts {km} in powers ofk –1, and to link coefficients of the expansion to certain transforms ofV. As a corollary we solve the Inverse Problem, get explicit formulae for the Weinsteinband-invariants of cluster distribution measures, and establishlocal spectral rigidity for zonal potential. The latter provides a partial answer to a long standing Spectral Rigidity Hypothesis of V. Guillemin.      16. Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails      A. Astrauskas 《Journal of statistical physics》2012,146(1):98-117 We study the asymptotic structure of the first K largest eigenvalues λ k,V and the corresponding eigenfunctions ψ(?;λ k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t 3) and some additional regularity conditions as t→∞. For z∈V, denote by λ 0(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ V +ξ(z)δ z in l 2(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ 0(?) in V. We first show that the eigenvalues λ k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ k,V ?B V )a V , where the normalizing constants a V >0 and B V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.      17. Diagrammatic non-degenerate RSPT for evaluation of ground-state energy of simple open-shell molecular systems      V. Kvasnička  S. Biskupič  V. Laurinc 《Molecular physics》2013,111(6):1345-1353 The diagrammatic non-degenerate Rayleigh-Schrödinger perturbation theory and the coupled cluster approach are applied to the evaluation of the ground state energy of a simple open-shell molecular system described in the zeroth order approximation by a single determinant wavefunction. The corresponding hamiltonian is built up by the creation and annihilation operators introduced over an orthonormal set of restricted Hartree-Fock one-particle functions. The perturbation H1 is composed of one- and two-particle terms, the one-particle term depending in an explicit way upon the type of restricted Hartree-Fock theory used. The efficiency of the elaborated theory is illustrated for the BeH molecule.      18. Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential      George D. Raikov 《Letters in Mathematical Physics》1991,21(1):41-49 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.      19. Convergence of rayleigh-schrödinger perturbation theory in calculations of multiphoton processes      L. Pan  K. T. Taylor  Charles W. Clark 《辐射效应与固体损伤》2013,168(2):725-741 Abstract We have calculated the high-order ac Stark Shift, multiphoton ionization rates, and nonlinear susceptibilities for high harmonic generation for the hydrogen atom in a radiation field. The calculations are done in the framework of Rayleigh-Schrödinger perturbation theory applied to a complex-rotated Hamiltonian. Our intention is to investigate the limitations of perturbation theory in calculations of multiphoton processes. Comparisons are made with results from nonperturbative calculations. For some frequencies the results of lowest-order perturbation theory are found to disagree with nonperturbative calculations even at moderate to low intensities (I ～ 1011 W cm?2) and in the absence of resonances. We find that the high-order perturbation expansion theory is not a reliable predictor of the behavior of hydrogen atoms in radiation fields with intensities greater than ～ 1012 W cm?2.      20. The long-time behaviour of the electric-field autocorrelation function in a finite photon gas      H. P. Baltes  E. R. Hilf  M. Pabst 《Applied Physics A: Materials Science & Processing》1974,3(1):21-29 We investigate an autocorrelation function of a soluble three-dimensional system, namely the temporal coherence functionC E(t)∝<E(0)E(t)> of the thermal radiation field in a cube-shaped cavity for the stochastic electrical fieldE. In the thermodynamic limit,C E(t) relaxes exponentially at intermediate times, but a “long-tail” behaviourC 0(t)=At?4 withA<0 is predominant for long times. In the case of a finite, but not too small, cavity lengthL obeyingΛ=hc/k BT?L and at timest withct?L, C E(t) is described by an asymptotic expansion in powers ofL ?1 using generalized Riemann zeta functions. Surface-and shape-effects enhance the long-tail. In the case of very small cavities withL«Λ, we calculate an expansion ofC E(t) in terms of exp(?L ?1) and cosines. An oscillatory, but not strictly periodic, long-time behaviour is observed in this case.

The E.S.R. spectrum of the tropolonyl radical

The many-body diagrammatic Rayleigh-Schrödinger perturbation theory (RSPT) is used for the calculation of ionization potentials of open-shell systems with one unpaired electron. This theoretical approach is tested on the simple examples of NO₂ and NF₂ molecules described by the INDO semi-empirical hamiltonian. The first- and second-order results are presented.     

Calculation of vibrational energy levels of water molecule by summing divergent perturbation theory series

The Rayleigh-Schrödinger perturbation theory is applied to a calculation of vibrational energy levels of the H₂O molecule for isolated states and the states involved in the anharmonic Fermi and Darling-Dennison resonances. It is shown that in spite of the rapid divergence of the perturbation theory series caused by the resonances, the use of the summation methods of Padé, Padé-Borel, and Padé-Hermite and the moments method allows one to obtain quite satisfactory results.     

Klein paradox in the Breit equation

We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r ₀)=E there appears a Klein paradox atr=r ₀. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm ⁽¹⁾→∞ andm ⁽²⁾→∞, where we neglect the terms ～1/m ⁽¹⁾ and 1/m ⁽²⁾. In Appendix I we show that in the Breit equation the oscillations accumulating atr=r ₀ in the case ofm ⁽¹⁾≠m ⁽²⁾ are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r ₀ in the case ofm ⁽¹⁾=m ⁽²⁾.     

Quantization of the potential amplitude in the one-dimensional Schrödinger equation

A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ?_n(x) and eigenvalues α_n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ?_n(x) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.     

Zonal Schrödinger operators on then-sphere: Inverse spectral problem and rigidity

We study the Direct and Inverse Spectral Problems for a class of Schrödinger operatorsH=–+V onS _n withzonal (axisymmetric) potentials. Spectrum ofH is known to consist of clusters of eigenvalues {_km=k(k+n-1)+_km:mk}. The main result of the work is to derive asymptotic expansion of spectral shifts {_km} in powers ofk ^–1, and to link coefficients of the expansion to certain transforms ofV. As a corollary we solve the Inverse Problem, get explicit formulae for the Weinsteinband-invariants of cluster distribution measures, and establishlocal spectral rigidity for zonal potential. The latter provides a partial answer to a long standing Spectral Rigidity Hypothesis of V. Guillemin.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.     

Diagrammatic non-degenerate RSPT for evaluation of ground-state energy of simple open-shell molecular systems

The diagrammatic non-degenerate Rayleigh-Schrödinger perturbation theory and the coupled cluster approach are applied to the evaluation of the ground state energy of a simple open-shell molecular system described in the zeroth order approximation by a single determinant wavefunction. The corresponding hamiltonian is built up by the creation and annihilation operators introduced over an orthonormal set of restricted Hartree-Fock one-particle functions. The perturbation H₁ is composed of one- and two-particle terms, the one-particle term depending in an explicit way upon the type of restricted Hartree-Fock theory used. The efficiency of the elaborated theory is illustrated for the BeH molecule.     

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.     

Convergence of rayleigh-schrödinger perturbation theory in calculations of multiphoton processes

Abstract

We have calculated the high-order ac Stark Shift, multiphoton ionization rates, and nonlinear susceptibilities for high harmonic generation for the hydrogen atom in a radiation field. The calculations are done in the framework of Rayleigh-Schrödinger perturbation theory applied to a complex-rotated Hamiltonian. Our intention is to investigate the limitations of perturbation theory in calculations of multiphoton processes. Comparisons are made with results from nonperturbative calculations. For some frequencies the results of lowest-order perturbation theory are found to disagree with nonperturbative calculations even at moderate to low intensities (I ～ 10¹¹ W cm^?2) and in the absence of resonances. We find that the high-order perturbation expansion theory is not a reliable predictor of the behavior of hydrogen atoms in radiation fields with intensities greater than ～ 10¹² W cm^?2.     

The long-time behaviour of the electric-field autocorrelation function in a finite photon gas

We investigate an autocorrelation function of a soluble three-dimensional system, namely the temporal coherence functionC _E(t)∝<E(0)E(t)> of the thermal radiation field in a cube-shaped cavity for the stochastic electrical fieldE. In the thermodynamic limit,C _E(t) relaxes exponentially at intermediate times, but a “long-tail” behaviourC ₀(t)=At^?4 withA<0 is predominant for long times. In the case of a finite, but not too small, cavity lengthL obeyingΛ=hc/k _BT?L and at timest withct?L, C _E(t) is described by an asymptotic expansion in powers ofL ^?1 using generalized Riemann zeta functions. Surface-and shape-effects enhance the long-tail. In the case of very small cavities withL«Λ, we calculate an expansion ofC _E(t) in terms of exp(?L ^?1) and cosines. An oscillatory, but not strictly periodic, long-time behaviour is observed in this case.