Exact and approximate solutions to Schrödinger’s equation with decatic potentials

Brandon, David; Saad, Nasser

doi:10.2478/s11534-013-0179-3

Cited by 10 publications

(14 citation statements)

References 61 publications

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“….} and unknown coefficients q i ∈ R. Note that the case of polynomial-type potential is not uncommon in physical realizations, especially in quantum mechanics, see for instance [9], where the use of a decatic potential is justified in the Schrödinger equation. Consider now a given δ ∈ (0, L] that ill determine the observation interval [0, δ].…”

Section: Problem Statement and Requirementsmentioning

confidence: 99%

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Estimation of the potential in a 1D wave equation via exponential observers

Kitsos

Bajodek

Baudouin

2021

2021 60th IEEE Conference on Decision and Control (CDC)

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The problem of estimation of the unknown potential in a 1-dimensional wave equation via state observers is considered in this work. The potential is supposed to depend on the space variable only and be polynomial. The main observation information is the value of the solution of the wave equation in a subinterval of the domain, including also some of its higher-order spatial derivatives. The method we propose to estimate the potential includes turning it into a new state as in finite-dimensional parameter estimation approaches. However, in this infinite dimensions setting, this requires an indirect approach that is introduced, including an infinite-dimensional state transformation. Sufficient conditions allow the design of an internal semilinear observer for the resulting cascade system, corresponding to the observed subinterval, which estimates the potential in an exponentially fast manner.

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Section: Problem Statement and Requirementsmentioning

confidence: 99%

“…These boundary conditions involve mappings that have been considered as measurements and are, thus, known. Now, for system (9), written in such an appropriate form, we are in a position to propose the following observer:…”

Section: Problem Statement and Requirementsmentioning

confidence: 99%

Estimation of the potential in a 1D wave equation via exponential observers

Kitsos

Bajodek

Baudouin

2021

2021 60th IEEE Conference on Decision and Control (CDC)

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“…Various analytical and computational approaches have been developed to calculate the energies of the anharmonic oscillators. Some of the approaches include an algebraic method based on the ladder operator [9], analytic quasilinearization method [10], Lie algebra [11,12], the Poincare-Linstedt method [13], multiple-scale perturbation theory [14], Wick's normal ordering technique [15], examination of polynomial solution [16], quantum Monte Carlo method [17], and pertur-bation theory [18]. Many other approaches have also been developed to calculate the energies of the systems.…”

Section: Introductionmentioning

confidence: 99%

Quantum Anharmonic Oscillators: A Truncated Matrix Approach

et al. 2021

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This study aims at implementing a truncated matrix approach based on harmonic oscillator eigenfunctions to calculate energy eigenvalues of anharmonic oscillators containing quadratic, quartic, sextic, octic, and decic anharmonicities. The accuracy of the matrix method is also tested. Using this method, the wave functions of the anharmonic oscillators were written as a linear combination of some finite number of harmonic oscillator basis states. Results showed that calculation with 100 basis states generated accurate energies of oscillators with relatively small coupling constants, with computation time less than 1 minute. Including more basis states could result in more correct digits. For instance, using 300 harmonic oscillator basis states in a simple Mathematica code in about 8 minutes, highly accurate energies of the oscillators were obtained for relatively small coupling constants, with up to 15 correct digits. Reasonable accuracy was also found for much larger coupling constants with at least three correct digits for some low lying energies of the oscillators reported in this study. Some of our results contained more correct digits than other results reported in the literature.

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“…Since its inception in 1980 [4,5], the QES models have received great attention in the field of quantum mechanics due to its wide use in modeling many physical phenomena ( [2] and references therein). Increasing interest has been noticed for getting analytical solutions [6][7][8][9][10] as well as highly accurate approximations [11][12][13][14][15][16] of various models involving QES potentials. While several existing methods provide excellent results for specific cases, an efficient scheme for obtaining energy eigenvalues and eigenfunctions simultaneously for any QES model is of great demand.…”

Section: Introductionmentioning

confidence: 99%

Efficient interpolating wavelet collocation scheme for quantum mechanical models in $$\mathbb {R}$$

et al. 2021

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This work deals with the development of an efficient interpolating wavelet collocation scheme for obtaining highly accurate eigenstates of Sturm-Liouville problem with a view to divulge some hidden bound state spectrum of quasi-exactly solvable models in non-relativistic quantum mechanics in R. The properties of scale functions in the interpolating wavelet basis generated by scale function and wavelets of symlets in Daubechies family have been judiciously used to provide an estimate of a posteriori error in the approximation of eigen functions. The efficiency of the scheme has been examined on a variety of exactly solvable models (e.g., harmonic oscillator problem having an infinite number of bound states, Schrödinger equation with Pöschl-Teller potential having a finite number of bound states, the quantum version of Mathews-Lakshmanan oscillator with bound states decaying algebraically in the asymptotic region and some quasi-exactly solvable models) and found highly efficient. The scheme has been subsequently applied to reveal some hidden bound states of a number of quasi-exactly solvable models with polynomial and non-polynomial potentials. From the careful analysis of results of the examples exercised here, it is observed that the proposed scheme seems to be quite useful and efficient compared to other methods [e.g., double exponential sinc collocation method of Gaudreau et al. (Ann Phys 360:520-538, 2015)] available in the literature to expose highly accurate approximation of bound state energies and wave functions of quasi-exactly solvable or non-solvable non-relativistic quantum mechanical models in R.

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Exact and approximate solutions to Schrödinger’s equation with decatic potentials

Cited by 10 publications

References 61 publications

Estimation of the potential in a 1D wave equation via exponential observers

Estimation of the potential in a 1D wave equation via exponential observers

Quantum Anharmonic Oscillators: A Truncated Matrix Approach

Efficient interpolating wavelet collocation scheme for quantum mechanical models in $$\mathbb {R}$$

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