V. B. Mandelzweig

Quasilinearization approach to computations with singular potentials

We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrodinger equation with singular potentials. The spiked harmonic oscillator r2 + λr−α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrodinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.

Quasilinearization method and WKB

Abstract Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers of ℏ, the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schrodinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schrodinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytic...

Quasilinearization approach to the resonance calculations: the quartic oscillator

We pioneered the application of the quasilinearization method (QLM) to resonance calculations. The quartic anharmonic oscillator (kx 2 /2) + x 4 with a negative coupling constant was chosen as the simplest example of the resonant potential. The QLM has been suggested recently for solving the bound state Schrodinger equation after conversion into Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. Comparison of our approximate analytic expressions for the resonance energies and wavefunctions obtained in the first QLM iteration with the exact numerical solutions demonstrate their high accuracy in the wide range of the negative coupling constant. The results enable accurate analytic estimates of the effects of the coupling constant variation on the positions and widths of the resonances.

Approximate analytic solutions of the Schrödinger equation for the generalized anharmonic oscillator

High precision approximate analytic expressions for the ground state energies and wavefunctions of the generalized anharmonic oscillator with the potential U(x)=g2x2/2+λ|x|p are obtained by first casting the Schrodinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The suitable initial guess for the first iteration ensures the high accuracy of the resulting expressions in a wide range of the parameters g≥0, λ≥0 and p≥2, the latter need not be an integer. It is found that the precision of the obtained wavefunctions is close to 0.1% that is more accurate than Wentzel–Kramers–Brillouin (WKB) solutions by two to three orders of magnitude. In addition, the SWKB excited state energies have been calculated using the obtained ground state wavefunctions. The latter results are in good agreement with the exact ones. All of the above mentioned enables us to make accurate analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the generalized anharmonic oscillators.

Boundary solutions of the two-electron Schrödinger equation at two-particle coalescences of the atomic systems

The limit relations for the partial derivatives of the two-electron atomic wave functions at the two-particle coalescence lines have been obtained numerically using accurate correlation function hyperspherical harmonic method wave functions. The asymptotic solutions of the proper two-electron Schroedinger equation have been derived for both electron-nucleus and electron-electron coalescence. It is shown that the solutions for the electron-nucleus coalescence correspond to the ground and singly excited bound states, including triplet ones. The proper solutions at small distances R from the triple coalescence point were presented as the second order expansion on R and ln R. The vanishing of the Focks logarithmic terms at the electron-nucleus coalescence line was revealed in the frame of this expansion, unlike the case of electron-electron coalescence. On the basis of the obtained boundary solutions the approximate wave function corresponding to both coalescence lines have been proposed in the two-exponential form with no variational parameters.

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.

Wavefunctions of Helium-Like Systems in Limiting Regions ¶

We find an approximate analytic form for the solution ψ(r1, r2, r12) of the Schrödinger equation for a system of two electrons bound to a nucleus in the spatial regions r1 = r2 = 0 and r12 = 0, which are of great importance for a number of physical processes. The forms are based on the well-known behavior of ψ(r1, r2, r12) near the singular triple coalescence point. The approximate functions are compared to the locally exact ones obtained earlier by the correlation function hyperspherical harmonic (CFHH) method for the helium atom, light helium-like ions, and the negative ion of hydrogen H−. The functions are shown to determine a natural basis for the expansion of CFHH functions in the considered spatial region. We demonstrate how these approximate functions simplify calculations of high-energy ionization processes.

Quasifree mechanism in ionization processes

We discuss a specific quasifree mechanism (QFM) of the ionization processes, which requires only small momentum transfer to the nucleus. The QFM mechanism is important in the kinematical regions where the two-electron system can absorb a photon without participation of the nucleus. The QFM leads to the complicated shape of the double photoionization spectrum curve, providing the breakdown of the nonrelativistic high energy asymptotic of the double-to single photoionization ratio. Calculation of the QFM contribution requires a consistent theoretical approach. Use of oversimplified approximations leads to misleading results. In certain kinematical regions the QFM provides the leading contribution, and thus the results obtained without inclusion of QFM are erroneous. Experimental detection of QFM is a complicated, but interesting problem.

Calculation of the photoionization with deexcitation cross sections of he and helium-like ions

We discuss the results of the calculation of the photoionization with deexcitation of excited He and helium-like ions Li+ and B3+ at high but nonrelativistic photon energies θ. Several lower 1S and 3S states are considered. We present and analyze the ratios Rd+* of the cross sections of photoionization with deexcitation, σ(d)+*(θ), and of the photoionization with excitation, σ+*(θ). The dependence of Rd+* on the excitation of the target object and the charge of its nucleus is presented. In addition to theoretical interest, the results obtained can be verified using long-lived excited states such as 23S of He.