PERTURBATION THEORY

1. INTRODUCTION

• PERTURBATION THEORY, ACTIVITY WITH TRADITIONAL ENCOURAGEMENTS:

  
  

  • ANALYTIC TRANSPARENCY

    (M. Z., Properties of two-variable expansions of scattering amplitudes. Czech. J. Phys. B 23 (1973) 685-95).
    
    

  • COMPUTATIONAL APPEAL

    (M. Z., New methods of solving the Bethe-Goldstone equation. Phys. Rev. C 12 (1975) 2077, and
    Reply on comment ... . Phys. Rev. C 14 (1976) 2321).
    
    

  • ALGEBRAIC BEAUTY

    (M. Z., Moshinsky brackets for light nuclei. Phys. Rev. C 15 (1977) 423.).
    
  
  

  • RECURRENT FEASIBILITY

    (M. Z., Recurrence relations for reaction matrices. Phys. Rev. C 18 (1978) 1078).
    
  
  

  • NUMERICAL EFFICIENCY

    (M. Z., Comparison of Brueckner theory with 'exact' results for He3 and He4 nuclei. Czech. J. Phys. B 30 (1980) 488-98).
    
  
  
  
  
  

• MOST PROMISING FRESH IDEAS ARE BECOMING AVAILABLE:

  
  

  • PADE' RESUMMATION

    at more points: M. Znojil,
    Three-point Pade resummation of perturbation series for anharmonic oscillators.
    Phys. Lett. A 177 (1993) 111-20.
    
    in the coordinate: F. M. Fernandez, R. Guardiola and M. Znojil,
    Riccati-Pade quantization and oscillators $V(r) = g r^{\alpha}$.
    Phys. Rev. A 48 (1993) 4170-4.
    
    
  
  

  • FACTORIZATIONS

    pre-conditioning: M. Znojil,
    A new form of re-arrangement of the Rayleigh-Schrodinger perturbation series.
    Cz. J. Phys. B 44 (1994) 545-56.
    
    an explicit account of asymptotics: M. Znojil,
    Bound-state method with elementary-product wavefunctions
    J. Phys. A 28 (1995) 6265-76.
    
    
  
  

  • ITERATION TECHNIQUES

    using fixed points: M. Znojil,
    Nonlinearized perturbation theories.
    J. Nonlin. Math. Phys. 3 (1996) 51 - 62.
    
    using continued fractions: M. Znojil,
    The most general iteration scheme for the Lippmann-Schwinger-type equations.
    Phys. Lett. A 211 (1996) 319 - 26.
    
    
    
  
  

2. INNOVATED OLDER PRESCRIPTIONS

• FORMULAE WITH NON-DIAGONAL PROPAGATORS

  
  

  • HILL-DETERMINANT PERTURBATION THEORIES

    fixed-point version: M. Znojil,
    Classification of oscillators in the Hessenberg-matrix representation.
    J. Phys. A: Math. Gen. 27 (1994) 4945-68.
    
    algorithm with continued fractions: M. Znojil,
    Double-well model V(r) = a r^2 + b r^4 + c r^6 with a < 0 and perturbation method with triangular propagators
    Phys. Lett. A 222 (1996) 291 - 8.
    
    
  
  

  • RUNGE-KUTTA DISCRETIZATION APPROACH

    perturbative return to continuous limit: M. Znojil,
    One-dimensional Schroedinger equation and its exact representation on a discrete lattice.
    Phys. Lett. A 223 (1996) 411 - 6.
    
    semi-relativistic test: M. Znojil,
    Harmonic oscillations in a quasi-relativistic regime.
    J. Phys. A 29 (1996) 2905 - 17.
    
    
  
  
  
  

• SIGNIFICANTLY MODIFIED OLDER RECIPES

  
  

  • PERTURBATION-SHOOTING ALGORITHMS

    one-dimensional case: M. Znojil,
    Comment on the letter "A new efficient method ..." by L. Skála and J. Cízek.
    J. Phys. A 29 (1996) 5253 - 6.
    
    review: M. Znojil,
    Perturbation theory for quantum mechanics in its Hessenberg-matrix representation
    Int. J. Mod. Phys. A 12 (1997) 299 - 304.
    
    
    

  • PERTURBATIVE - ITERATIVE SCHEMES

    low-order implementation: M. Znojil,
    Asymmetric bound states via the quadrupled Schroedinger equation.
    Phys. Lett. A 230 (1997) 283 - 7.
    
    
  
  

3. MORE SOPHISTICATED ZERO ORDER

• SOURCES OF ENHANCED FLEXIBILITY

  
  

  • RESOLUTION OF TECHNICAL SUBTLETIES

    evaluation of overlap integrals: M. Znojil,
    Comment on ``The nonsingular spiked harmonic oscillator; [J. Math. Phys. 34, 437 (1993)].
    J. Math. Phys. 34 (1993) 4914.
    
    asymmetry: M. Znojil,
    A generalized Morse asymmetric potential and multiplets of its non-numerical exact bound states.
    J. Phys. A: Math. Gen. 27 (1994) 7491-501.
    
    periodicity of boundary conditions: M. Znojil,
    Circular vectors and toroidal matrices,
    Rendiconti del Circolo Matematico di Palermo Serie II - Suppl. 39 (1996) 143-8.
    
    
    

  • THE ADMITTED PRESENCE OF SINGULARITIES

    Mathieu solvability: M. Znojil,
    Two-sided estimates of energies and the ``forgotten" exactly solvable potential $V(r)=-a^2 r^{-2}+b^2 r^{-4}$.
    Phys. Lett. A 189 (1994) 1-6.
    
    strong repulsion: M. Znojil,
    An analytic estimate of the number of bound states in the Lennard-Jones potentials.
    Phys. Lett. A 188 (1994) 113-6.
    
    Coulombic shape: M. Znojil,
    Non-numerical determination of the number of bound states in some screened Coulomb potentials.
    Phys. Rev. A. 51 (1995) 128 - 35.
    
    
    

  • MERE PARTIAL SOLVABILITY

    semi-relativistic model: M. Znojil,
    Screened Coulomb potential V(r) = (a+br)/(c+dr) in a semi-relativistic Pauli-Schroedinger equation
    J. Phys. A 29 (1996) 6443 - 53.
    
    complex-pole case: M. Znojil and Rajkumar Roychoudhury,
    Spiked and screened oscillators V(r) = A r^2 + B/r^2 + C/r^4 + D/r^6 + F/(1 + g r^2) and their elementary bound states.
    Czechosl. J. Phys. 48 (1998) 1 - 8.
    
    model in more dimensions: M. Znojil,
    Quantum exotic: A repulsive and bottomless confining potential.
    J. Phys. A 31 (1998) 3349 - 55.
    
    
    

• CONSEQUENCES

  
  

  • OCCURRENCE OF OLD NAMES IN NEW CONTEXTS

    discretized Schroedinger equation: M. Znojil,
    Minimal relativity and Hulthen potentials.
    Phys. Lett. A. 203 (1995) 1-4.
    
    shifted boundary conditions: M. Znojil,
    Jacobi polynomials and bound states.
    J. Math. Chem. 19 (1996) 205 - 13
    
    
    

  • MORE PHYSICS

    relativity: M. Znojil,
    Elementary doublets of bound states of the radial Dirac equation.
    Mod. Phys. Lett. A 14 (1999) 863 - 8.
    
    the presence of a core: M. Znojil,
    Singular potentials with quasi-exact Dirac bound states.
    Phys. Lett. A 255 (1999) 1 - 6.
    
    genuine many-body character: M. Znojil and M. Tater,
    Exactly solvable three-body Calogero-type model with translucent two-body barriers
    (nucl-th/0103015), Phys. Lett. A 284 (2001) 225 - 230.
    
    
    

4. NEW FORMS OF PERTURBATION THEORY

• DISPENSING WITH THE BASIS

  
  

  • PARTITIONED SHOOTING TECHNIQUE

    formulated in two dimensions: M. Znojil,
    A quick perturbative method for Schroedinger equations
    J. Phys. A 30 (1997) 8771 - 83.
    
    
  
  

  • PT-SYMMETRIC PERTURBATION THEORY

    re-arrangements and numerical tests:
    M. F. Fernández, R. Guardiola, J. Ros and M. Znojil,
    Strong-coupling expansions for the PT-symmetric oscillators V(x) = a x + b (ix)^2 + c (ix)^3.
    J. Phys. A 31 (1998) 10105 - 12.
    
    
  
  

• APPLICATION-FRIENDLY RECIPES

  
  

  • MODIFIED BASES

    double-well adapted bases: M. Znojil,
    r^D oscillators with arbitrary D > 0 and perturbation expansions with Sturmians.
    J. Math. Phys. 38 (1997) 5087 - 97.
    
    partitioned series: M. Znojil,
    Short-range oscillators in power-series picture
    (quant-ph/9909068), J. Phys. A: Math. Gen. 33 (2000) 1647 - 1659.
    
    
  
  

  • COMPLICATED ZERO-ORDER POTENTIALS

    multiple-well, Padé-approximated: M. Znojil,
    Perturbation method with triangular propagators and anharmonicities of intermediate strength
    (quant-ph/9910114), J. Math. Chem. 28 (2000) 139 - 167.
    
    
  
  
  
  

• ADAPTED-NORMALIZATION TECHNIQUES

  
  

  • HUECKEL PERTURBATION THEORY

    large-D alternative expansions using integer arithmetics: M. Znojil,
    Bound states in the Kratzer plus polynomial potentials and the new form of perturbation theory
    (quant-ph/9907054) J. Math. Chem. 26 (1999) 157 - 72.
    
    
  
  

  • MULTIPLE-SQUARE-WELL-FRIENDLY SHOOTING

    discontinuities-admitting normalization: M. Znojil,
    New perturbation method with the matching of wave functions
    (quant-ph/9812027), Intern. J. Quant. Chem. 79 (2000) 235 - 242.
    
    
  
  

  • THE METHOD WITH ASYMPTOTIC NORMALIZATION

    short-range version: M. Znojil,
    Perturbed Poeschl-Teller oscillators
    (math-ph/0001022), Phys. Lett. A 266 (2000) 254 - 259.