Axel Schulze-Halberg
PhD (Mathematics), MSc (Theoretical Physics)

Professor of Mathematics
Adjunct Professor of Physics

Address
 Area of specialization
Department of Mathematics and Actuarial Science
Hawthorn Hall 443
Indiana University Northwest
3400 Broadway
Gary, IN 46408
USA

Email
axgeschu "at" iun.edu

Phone
++1 219 980 6780


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 Mathematical Physics

Some recent papers (see below for full publication list)
  • Closed-form bound states for two Dunkl-Lienard oscillator systems,
    Int. J. Mod. Phys. A (2024), to appear

  • Closed-form solutions of the Dunkl-Klein-Gordon equation for two inverse power-law interactions,
    Mod. Phys. Lett. A 38 (2023), 2350113

  • Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism,
    Acta Polytech. 63 (2023), 273

  • Darboux algorithms for two-dimensional Dirac equations with upper triangular potential matrix,
    J. Comput. Appl. Math. 427 (2023), 115143

Student papers in undergraduate research journals
  • A. Zuccolo, Wronskian representations of hypergeometric integrals,
    SIAM Undergraduate Research Online 11 (2018)

  • A. Buss, Construction of generalized integral formulas by means of Laplace transformations,
    Indiana University Journal of Undergraduate Research 2 (2016), 36-40


List of publications

  • Closed-form bound states for two Dunkl-Lienard oscillator systems, Int. J. Mod. Phys. A (2024), to appear
  • A new Darboux transformation for mapping Schrodinger onto Klein-Gordon equations, Eur. Phys. J. Plus 138 (2023), 1117
  • Darboux transformations for a class of Duffin-Kemmer-Petiau equations governing spin-zero systems, Few Body Syst. 64 (2023), 64
  • Closed-form solutions of the Dunkl-Klein-Gordon equation for two inverse power-law interactions, Mod. Phys. Lett. A 38 (2023), 2350113
  • Darboux transformations and the decoupling process for Dunkl-Dirac equations with pseudoscalar potentials, Eur. Phys. J. Plus 138 (2023), 491
  • Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism, Acta Polytech. 63 (2023), 273
  • Generalized Darboux algorithms for Dunkl-Schrodinger equations with position-dependent mass at fixed energy, Int. J. Mod. Phys. A 38 (2023), 2350070
  • Darboux algorithms for two-dimensional Dirac equations with upper triangular potential matrix, J. Comput. Appl. Math. 427 (2023), 115143
  • (with W.S. Chung and H. Hassanabadi) Realization of the Wigner Heisenberg algebra through generalized momentum operators, Eur. Phys. J. Plus 138 (2023), 66
  • Arbitrary-order Darboux transformations for Dunkl-Schrödinger equations with position-dependent mass, Mod. Phys. Lett. A 37 (2022), 2250209
  • Bound states of the isotonic Mathews-Lakshmanan oscillator system within the Dunkl formalism, Mod. Phys. Lett. A 37 (2022), 2250178
  • Generalized Dunkl-Schrödinger equations: solvable cases, point transformations, and position-dependent mass systems, Phys. Scr. 97 (2022), 085213
  • Darboux transformations for Dirac equations in polar coordinates with vector potential and position-dependent mass, Eur. Phys. J. Plus 137 (2022), 832
  • Darboux transformations and reality conditions for stationary Dirac and Klein-Gordon equations in one dimension, Int. J. Mod. Phys. A 37 (2022), 2250080
  • A new type of Darboux transformations for the one-dimensional Burgers equation with forcing, Phys. Scr. 96 (2021), 125247
  • Darboux transformations for Dirac equations with arbitrary diagonal potential matrix in two dimensions, J. Phys. Conf. Ser. 2090 (2021), 012038
  • Characterization of Darboux transformations for quantum systems with quadratically energy-dependent potentials, J. Math. Phys. 62 (2021), 083504
  • First-order Darboux transformations for Dirac equations with arbitrary diagonal potential matrix in two dimensions, Eur. Phys. J. Plus 136 (2021), 790
  • (with P. Roy) Dirac systems with magnetic field and position-dependent mass: Darboux transformations and equivalence with generalized Dirac oscillators, Ann. Phys. 431 (2021), 168534
  • Closed-form representations of iterated Darboux transformations for the massless Dirac equation, Int. J. Mod. Phys. A 36 (2021), 2150064
  • (with A. Ishkhanyan) Inverse-root and inverse-root-exponential potentials: Darboux transformations and elementary Darboux partners, Phys. Scr. 96 (2021), 025206
  • Higher-order Darboux transformations for the Dirac equation with position-dependent mass at nonvanishing energy, Eur. Phys. J. Plus 135 (2020), 863
  • Generalized Schrödinger equations with quadratical energy-dependence in the potential: Darboux transformations and application to the Heun class, J. Math. Phys. 61 (2020), 083502
  • (with A. Ishkhanyan) Darboux partners of Heun-class potentials for the two-dimensional massless Dirac equation, Ann. Phys. 421 (2020), 168273
  • (with A. Ishkhanyan) Exactly-Solvable Quantum Systems in Terms of Lambert-W Functions, Few-Body Syst. 61 (2020), 12
  • Higher-order Darboux transformations and Wronskian representations for Schrödinger equations with quadratically energy-dependent potentials, J. Math. Phys. 61 (2020), 023503
  • Arbitrary-order Darboux transformations for two-dimensional Dirac equations with position-dependent mass, Eur. Phys. J. Plus 135 (2020), 332
  • (with M. Paskash) Wronskian representation of second-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials, Phys. Scr. 95 (2020), 015001
  • Higher-order Darboux transformations for the massless Dirac equation at zero energy, J. Math. Phys. 60 (2019), 073505
  • Darboux transformations for the massless Dirac equation with matrix potential: Radially symmetric zero-energy states, Eur. Phys. J. Plus 134 (2019), 295
  • (with M. Ojel) Darboux transformations for the massless Dirac equation with matrix potential: construction of zero-energy states, Eur. Phys. J. Plus 134 (2019), 49
  • Regularity conditions for transformed potentials in the confluent supersymmetry algorithm, Int. J. Modern Phys. A 33 (2018), 1850214
  • (with . Yesiltas) Generalized Schrödinger equations with energy-dependent potentials: formalism and applications, J. Math. Phys. 59 (2018), 113503
  • The symmetrized square-root potential: exact solutions and application to the two-dimensional massless Dirac equation, Few Body Syst. 59 (2018), 115
  • (with . Yesiltas) The generalized confluent supersymmetry algorithm: representations and integral formulas, J. Math. Phys. 59 (2018), 043508
  • (with . Yesiltas) Rationally extended potentials and Xm-Jacobi type solutions of the two-dimensional massless Dirac equation, Eur. Phys. J. Plus 133 (2018), 26
  • (with B. Roy) Higher-order supersymmetric partners of generalized quantum nonlinear oscillators, Eur. Phys. J. Plus 133 (2018), 102
  • (with P. Roy) Bound states of the two-dimensional Dirac equation for an energy-dependent hyperbolic Scarf potential, J. Math. Phys. 58 (2017), 113507
  • (with P. Roy) Construction of zero-energy states in graphene through the supersymmetry formalism, J. Phys. A 50 (2017), 365205
  • (with P. Roy) Pseudo-hermitian and PT-symmetric quantum systems with energy-dependent potentials: Bound-state solutions and energy spectra, Ann. Phys. 380 (2017), 78-92
  • (with P. Roy) Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials, Ann. Phys. 378 (2017), 234-252
  • (with A. Contreras-Astorga) Recursive representation of Wronskians in confl uent supersymmetric quantum mechanics, J. Phys. A 50 (2017), 105301
  • (with B. Roy) Generalized quantum nonlinear oscillators: exact solutions and rational extensions, J. Math. Phys. 57 (2016), 102103
  • Trigonometric potentials arising from the spheroidal equation: supersymmetric partners and integral formulas, Eur. Phys. J. Plus 131 (2016), 202
  • Arbitrary-order Jordan chains associated with quantum-mechanical Hamiltonians: representations and integral formulas, J. Math. Phys. 57 (2016), 023521
  • Supersymmetric partners and confinement of a spiked inverted oscillator model, Eur. Phys. J. Plus 130 (2015), 228
  • (with J. Morris) Light wave propagation through a dilaton-Maxwell domain wall, Phys. Rev. D 92 (2015), 085026
  • (with J. Wang) Two-parameter double-oscillator model of Mathews-Lakshmanan type: series solutions and supersymmetric partners, J. Math. Phys. 56 (2015), 072106
  • Closed-form solutions and supersymmetric partners of the inverted Mathews-Lakshmanan oscillator, Eur. Phys. J. Plus 130 (2015), 134 (10pp)
  • (with A. Contreras-Astorga) On integral and differential representations of Jordan chains and the confluent supersymmetry algorithm, J. Phys. A 48 (2015), 315202
  • (with J. Wang) Confluent supersymmetric partners of quantum systems emerging from the spheroidal equation, Symmetry 7 (2015), 412-426
  • (with D. Dec) Index dependence of Bessel integral moments, Integr. Transf. Spec. F. 26 (2015), 460-469
  • (with A. Contreras-Astorga) The generalized zero-mode supersymmetry scheme and the confluent algorithm, Ann. Phys. (2015), 354-364
  • (with B. Roy) Time dependent potentials associated with exceptional orthogonal polynomials, J. Math. Phys. 55 (2014), 123506
  • (with A. Contreras-Astorga) The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials, J. Math. Phys. 55 (2014), 103506
  • (with J. Wang) Bound state solutions of the Klein-Gordon equation for the Mathews-Lakshmanan oscillator, Few-Body Syst. 55 (2014), 1223-1232
  • Burgers equation with time-dependent coefficients and nonlinear forcing term: linearization and exact solvability, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), 1068-1083
  • (with B. Roy) Darboux partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials, Ann. Phys. 349 (2014), 159 170
  • (with J. Garcia-Ravelo, A Menendez and J. Garcia-Martinez) Effective Kratzer and Coulomb potentials as limit cases of a multiparameter exponential-type potential, Phys. Lett. A 378 (2014), 2038 2042
  • (with B. Roy) Rational extension and Jacobi-type Xm solutions of a quantum nonlinear oscillator, J. Math. Phys. 54 (2013), 122104 (12pp)
  • (with J. Morris) An exactly-solvable three-dimensional nonlinear quantum oscillator, J. Math. Phys. 54 (2013), 112107 (17pp)
  • Wronskian representation for confluent supersymmetric transformation chains of arbitrary order, Eur. Phys. J. Plus 128 (2013), 68 (17pp)
  • (with J. Morris) Higher-dimensional realization of a nonlinear, one-parameter quantum oscillator, Eur. Phys. J. Plus 128 (2013), 54 (11pp)
  • Darboux operators for linear first-order multi-component equations in arbitrary dimensions, Cent. Eur. J. Phys. 11 (2013), 457-469
  • (with C. Gordon) Spectral properties of a confined nonlinear quantum oscillator in one and three dimensions, J. Math. Phys. 54 (2013), 042108 (15pp)
  • Comment on "Nonsingular parametric oscillators Darboux-related to the classical harmonic oscillator" by H.C. Rosu et al., Europhys. Lett. 101 (2013), 30001 (2pp)
  • Darboux transformations for energy-dependent potentials and the Klein-Gordon equation, Math. Physics, Analysis and Geometry (2013), http://dx.doi.org/10.1007/s11040-012-9126-6
  • (with J. Garcia-Ravelo, C. Pacheco-Garcia and J.J. Pena Gil) A position-dependent mass model for the Thomas-Fermi potential: exact solvability and relation to δ-doped semiconductors, Ann. Physics 333 (2013), 323 334
  • (with J. Garcia-Ravelo and A. Trujillo) Explicit formulas for generalized harmonic perturbations of the infinite quantum well with an application to Mathieu equations, J. Math. Phys. 53 (2012), 102101 (14pp)
  • Two-dimensional magnetic Schrödinger equations, Darboux transformations and solutions of associated auxiliary equations, J. Math. Phys. 53 (2012), 082108 (15pp)
  • Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix, J. Math. Phys. 53 (2012), 103519 (11pp)
  • (with J. Morris) Special function solutions of a spectral problem for a nonlinear quantum oscillator, J. Phys. A 45 (2012), 305301 (9pp)
  • Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations, Appl. Math. Lett. 25 (2012), 1520 1527
  • Combined Sundman-Darboux transformations and solutions of nonlinear ordinary differential equations of second order, Adv. Pure Appl. Math. 3 (2012), 165-180
  • Explicit higher-dimensional Darboux transformations for the time-dependent Schrödinger equation, Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, AMS Contemporary Mathematics series 563 (2012)
  • Two-dimensional generalized harmonic oscillators and their Darboux partners, J. Phys. A 44 (2011), 495304 (20pp)
  • Higher-order Darboux transformations in two dimensions, linearizable auxiliary equations and invariant potentials, J. Math. Phys. 52 (2011), 083505 (9pp)
  • (with H. Cobian), Time-dependent Schrödinger equations with effective mass in (2+1) dimensions: intertwining relations and Darboux operators, J. Phys. A 44 (2011), 285301 (14pp)
  • Modified Darboux transformations with foreign auxiliary equations, Phys. Lett. A 375 (2011), 2513-2518
  • Two-dimensional Darboux transformations for non-separable angular equations and solvable non-central potentials, J. Math. Chem. 49 (2011), 1302-1310
  • Quasi-exact solvability of a hyperbolic intermolecular potential induced by an effective mass step, Internat. J. Math. Math. Sci. 2011 (2011), article #358198 (9pp)
  • Propagators of generalized Schrödinger equations related by first-order supersymmetry, Acta Polytechnica 51 (2011), 63-69
  • (with E. Pozdeeva), Propagators of generalized Schrödinger equations related by higher-order supersymmetry, Internat. J. Modern Phys. A 26 (2011), 191-207
  • (with E. Pozdeeva), Darboux transformations for a generalized Dirac equation in two dimensions, J. Math. Phys. 51 (2010), 113501 (15pp)
  • (book chapter) Linear generalizations of the time-dependent Schrödinger equation: point transformations, Darboux transformations, and their properties, Quantum Mechanics, (Nova Science Publishers, New York, 2010)
  • Exact quantization formula for affine linearly energy-dependent potentials, Cent. Eur. J. Phys. 9 (2010), 57-64
  • Green's functions and trace formulas for generalized Sturm-Liouville problems related by Darboux transformations, J. Math. Phys. 51 (2010), 053501 (13pp)
  • Intertwining relations and Darboux transformations for Schrödinger equations in (n+1) dimensions, J. Math. Phys. 51 (2010), 033521 (11 pp)
  • (with J. Garcia-Ravelo, J. Pena and A. Queijeiro), Two-parameter ladder operators for spherical Bessel functions, Appl. Math. Lett. 23 (2010), 462-465
  • (with J. Carballo Jimenez), Darboux transformations for the time-dependent nonhomogeneous Burgers equation in (1+1) dimensions, Phys. Scripta 80 (2009), 065014 (8pp)
  • (with J. Carballo Jimenez), Supersymmetry of generalized linear Schrödinger equations in (1+1) dimensions, Symmetry 1 (2009), 115-144
  • (with J. Garcia-Ravelo, J. Garcia Martinez and J. Pena), Exactly solvable energy dependent potentials, Phys. Lett. A 373 (2009), 3619-3623
  • (with A. Suzko), Darboux transformations and supersymmetry for generalized Schrödinger equations in (1+1) dimensions, J. Phys. A 42 (2009), 295203 (14pp)
  • (with E. Pozdeeva and A. Suzko), Explicit Darboux transformations of arbitrary order for generalized time-dependent Schrödinger equations, J. Phys. A 42 (2009), 115211 (12pp)
  • (with J. Garcia-Ravelo, J. Morales, J. Pena and P. Roy), Exact solutions of the Fokker-Planck equation from an n-th order supersymmetric quantum mechanics approach, Phys. Lett. A 373 (2009), 1610-1615
  • (with J. Garcia-Ravelo and J. Pena), An implicit spectral formula for generalized linear Schrödinger equations, Internat. J. Modern Phys. E 18 (2009), 1831-1844
  • (with A. Suzko and E. Velicheva), Supersymmetry and Darboux transformations for the generalized Schrödinger equation, Physics of Atomic Nuclei 72 (2009), 858-865
  • (with J. Pena and E. Zamora-Gallardo), Arbitrary-order Darboux transformations for position-dependent mass Schrödinger equations with noncentral potential, Int. J. Quant. Chem. 109 (2009), 1464-1471
  • (with A. Suzko), Intertwining operator method and supersymmetry for effective mass Schrödinger equations, Phys. Lett. A 372 (2008), 5865-5871
  • (with E. Pozdeeva), Trace formula for Dirac Green's functions related by Darboux transformations, J. Phys. A 41 (2008), 265201 (13pp)
  • Generalized linear Schrödinger equations and their Darboux transformations, Cent. Eur. J. Phys. 6 (2008), 654-661
  • (with E. Pozdeeva), Trace formula for Green's functions of effective mass Schrödinger equations and N-th order Darboux transformations, Internat. J. Modern Phys. A 23 (2008), 2635-2647
  • Darboux transformations for effective mass Schrödinger equations with energy-dependent potentials, Internat. J. Modern Phys. A 23 (2008), 537-546
  • Supersymmetry of time-dependent Schrödinger equations with effective mass, Cent. Eur. J. Phys. 6 (2008), 153-157
  • Reduction of time-dependent Schrödinger equations with effective mass to stationary Schrödinger equations, Research Letters in Physics (2008), DOI:10.1155/2008/589269
  • Darboux transformations for N-dimensional effective mass Schrödinger equations with hyperspherical symmetry, Internat. J. Modern Phys. A 22 (2007), 3293-3304
  • Fractional Darboux transformations are point canonical transformations, Il Nuovo Cimento B 121 (2007), 1025-1028
  • Form-preserving transformations for the time-dependent Schrödinger equation in (n+1) dimensions, Math. Physics, Analysis and Geometry 9 (2007), 371-388
  • Fractional Darboux transformations for the time-dependent Schrödinger equation, International Journal of Pure and Applied Mathematics 37 (2007), 489-506
  • New exact solutions of the non-homogeneous Burgers equation in (1+1) dimensions, Phys. Scr. 75 (2007), 531-536
  • Effective mass Hamiltonians with linear terms in the momentum: Darboux transformations and form-preserving transformations, International J. Modern Phys. A 22 (2007), 1735-1769
  • Form-preserving transformations of the time-dependent Klein-Gordon equation and the generation of solvable potentials, Phys. Scripta 74 (2006), 452-458
  • Exactly solvable combinations of scalar and vector potentials for the Dirac equation interrelated by Riccati equations, Chinese Phys. Lett. 23 (2006), 1365-1368
  • Reality condition for Darboux transformations of time-dependent Schrödinger equations with effective mass, Internat. J. Modern Phys. A 21 (2006), 4853-4868
  • A simple method for constructing non-liouvillian .rst integrals of autonomous planar systems, Czechoslovak Math. J. 56 (2006), 987-999
  • (with D. Stoffer and A. Aschwanden), Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof, Discrete Contin. Dyn. Syst. 14 (2006), 721-736
  • Darboux transformations for time-dependent Schrödinger equations with effective mass, Internat. J. Modern Phys. A 21 (2006), 1359-1377
  • Quantum systems with effective and time-dependent masses: form-preserving transformations and reality conditions, Cent. Eur. J. Phys. 3 (2005), 591-609
  • Comment on: D.-Y. Song and J.R. Klauder, "Generalization of the Darboux transformation and generalized harmonic oscillators", J. Phys. A 36 (2003), no. 32, 8673-8684, J. Phys. A 38 (2005), 5831-5836
  • Form-preserving transformations for Hamiltonians with linear terms in the momentum, Found. Phys. Lett. 18 (2005), 535-548.
  • On quasi-exact solvability of the Schrödinger equation for a free particle on the surface of a spindle torus, Found. Phys. Lett. 18 (2005), 291-300
  • Form-preserving transformations of the time-dependent Schrödinger equation with time and position-dependent mass, Commun. Theor. Phys. 43 (2005), 657-665
  • Orbitally asymptotically stable periodic solutions of delay equations with positive feedback, Mitt. Math. Sem. Giessen 252 (2003), 1-106 and Mitt. Math. Sem. Giessen 254 (2004), 48-90
  • Exact wavefunctions and energies of a non-relativistic free quantum particle on the surface of a degenerate torus, Modern Phys. Lett. A 19 (2004), 1759-1766
  • Non-existence of liouvillian solutions for a free quantum particle on a torus surface, part 1: polar states, Found. Phys. Lett. 17 (2004), 677-690
  • N-multiplets of the radial Dirac equation, Chinese J. Phys. 42 (2004), 234-242
  • Quasi-exactly solvable potentials of the radial Dirac equation with two known levels, Chinese J. Phys. 42 (2004), 117-126
  • A new formula to obtain exact Green's functions of the time-dependent Schrödinger equation, Commun. Theor. Phys. 41 (2004), 723-725
  • Exact solutions of the time-dependent Schrödinger equation with time-dependent mass generated by functional and coordinate transformations, Hadronic J. 27 (2004), 485-498
  • Functional transformation of the time-dependent Schrödinger equation with time- and spatially dependent mass, Hadronic J. 26 (2003), 707-714
  • Equivalence of time-dependent Schrödinger equations with constant and time-dependent mass, Modern Phys. Lett. A 18 (2003), 2829-2835
  • Hidden symmetry in quasi-exactly solvable fractional power potentials, Progr. Theoret. Phys. 110 (2003), 1235-1240
  • (with S.K. Bose), Improved SUSY inspired variational method to obtain bound state energies of the screened Coulomb potential, Modern Phys. Lett. A 18 (2003), 853-857
  • A method of generating exact solutions of the time-dependent Schrödinger equation with time-dependent mass, Modern Phys. Lett. A 17 (2002), 1567-1573
  • Closed-form solutions of the Schrödinger equation for a particle on the torus, Found. Phys. Lett. 15 (2002), 585-589
  • General solution of a system of di.erential equations modeling a class of exactly-solvable potentials, part 2: extended results, Proc. Estonian Acad. Sci. Phys. Math. 51 (2002), 179-193
  • Quasi-exactly solvable singular fractional power potentials emerging from the triconfluent Heun equation, Phys. Scripta 65 (2002), 373-376
  • A class of exact solutions of the time-dependent Schrödinger equation with time-dependent mass, Modern Phys. Lett. A 16 (2002), 2557-2566
  • The central symmetric screened Coulomb potential in N dimensions, Hadronic J. 24 (2001), 519-530
  • (with S.K. Bose and M. Singh), New exact solutions of the Dirac equation, Phys. Lett. A 287 (2001), 321-324
  • Comment on Phys. Scripta 60, 195 (1999), Phys. Scripta 64 (2001), 96
  • General solution of a system of di.erential equations modeling a class of exactly-solvable potentials, Proc. Estonian Acad. Sci. Phys. Math. 50 (2001), 42-48
  • On normalizability of zero energy solutions of the Schrödinger equation, Hadronic J. 23 (2000), 683-687
  • (with S.K. Bose), Exact solution of the Dirac equation for a singular central power potential, Modern Phys. Lett. A 15 (2000), 1583-1588
  • Exact solutions of the Schrödinger equation with irregular singularity, Internat. J. Theoret. Phys. 39 (2000), 2305-2325
  • (with S.K. Bose), Properties of an algebraic transformation for 1D Schrödinger potentials, Phys. Scripta 62 (2000), 241-247
  • (with S.K. Bose), An interesting and useful transformation for the nonrelativistic Schrödinger equation, Hadronic J. 23 (2000), 249-257
  • On finite normal series solutions of relativistic wave equations, Found. Phys. Lett. 13 (2000), 265-277
  • On finite normal series solutions of the Schrödinger equation with irregular singularity, Found. Phys. Lett. 13 (2000), 11-27
  • Exact bound states for the symmetric screened Coulomb potential, Phys. Scripta 61 (2000), 641-645
  • Exact algebraic bound states for the potential V(r) = k2⁄r2 + k3⁄r3+ k4⁄r4 , Hadronic J. 22 (1999), 343-350