December  2021, 20(12): 4347-4377. doi: 10.3934/cpaa.2021163

Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry

Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama city, Kanagawa, JAPAN

Received  May 2021 Revised  August 2021 Published  December 2021 Early access  September 2021

We consider the Cauchy problems for Schrödinger equations with an inverse-square potential and a harmonic one. Since the Mehler type formulas are completed, the pseudo-conformal transforms can be constructed. Thus we can convert the problems into the nonautonomous Schrödinger equations without a harmonic oscillator.

Citation: Toshiyuki Suzuki. Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4347-4377. doi: 10.3934/cpaa.2021163
References:
[1]

K. AoudaN. KandaS. Naka and H. Toyoda, Ladder operators in repulsive harmonic oscillator with application to the Schwinger effect, Phys. Rev. D, 102 (2020), 025002.  doi: 10.1103/physrevd.102.025002.

[2]

N. BurqF. PlanchonJ. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680.  doi: 10.1512/iumj.2004.53.2541.

[3]

F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys., 10 (1969), 2191-2196. 

[4]

F. Calogero, Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 12 (1971), 419–436 ("Erratum", ibidem 37 (1996), 3646). doi: 10.1063/1.1665604.

[5]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.  doi: 10.1137/S0036141002416936.

[6]

N. Dunford and J. T. Schwartz, Linear operators. Part Ⅱ: Spectral theory, Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons New York-London, 1963.

[7]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.

[8]

G. Metafune and M. Sobajima, Spectral properties of non-selfadjoint extensions of the Calogero Hamiltonian, Funkcial. Ekvac., 59 (2016), 123-140.  doi: 10.1619/fesi.59.123.

[9]

J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16 (1975), 197-220.  doi: 10.1016/0001-8708(75)90151-6.

[10]

N. OkazawaT. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629.  doi: 10.1080/00036811.2011.631914.

[11]

N. OkazawaT. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.  doi: 10.3934/eect.2012.1.337.

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅱ, Academic Press, New York, 1975. 
[13]

T. Suzuki, Blowup of nonlinear Schrödinger equations with inverse-square potentials, Differ. Equ. Appl., 6 (2014), 309-333.  doi: 10.7153/dea-06-17.

[14]

T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.  doi: 10.1619/fesi.59.1.

[15]

T. Suzuki, Virial identities for nonlinear Schrödinger equations with an inverse-square potential of critical coefficient, Differ. Equ. Appl., 9 (2017), 327-352.  doi: 10.7153/dea-2017-09-24.

[16]

T. Suzuki, Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods, Evol. Equ. Control Theory, 8 (2019), 447-471.  doi: 10.3934/eect.2019022.

[17]

T. Suzuki, Semilinear Schrödinger equations with a potential of some critical inverse-square type, J. Differ. Equ., 268 (2020), 7629-7668.  doi: 10.1016/j.jde.2019.11.087.

[18]

S. Watanabe, The explicit solutions to the time-dependent Schrödinger equation with the singular potentials $k/(2x^{2})$ and $k/(2x^{2})+\omega^{2}x^{2}/2$, Commun. Partial Diff. Equ., 26 (2001), 571-593.  doi: 10.1081/PDE-100002238.

show all references

References:
[1]

K. AoudaN. KandaS. Naka and H. Toyoda, Ladder operators in repulsive harmonic oscillator with application to the Schwinger effect, Phys. Rev. D, 102 (2020), 025002.  doi: 10.1103/physrevd.102.025002.

[2]

N. BurqF. PlanchonJ. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680.  doi: 10.1512/iumj.2004.53.2541.

[3]

F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys., 10 (1969), 2191-2196. 

[4]

F. Calogero, Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 12 (1971), 419–436 ("Erratum", ibidem 37 (1996), 3646). doi: 10.1063/1.1665604.

[5]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.  doi: 10.1137/S0036141002416936.

[6]

N. Dunford and J. T. Schwartz, Linear operators. Part Ⅱ: Spectral theory, Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons New York-London, 1963.

[7]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.

[8]

G. Metafune and M. Sobajima, Spectral properties of non-selfadjoint extensions of the Calogero Hamiltonian, Funkcial. Ekvac., 59 (2016), 123-140.  doi: 10.1619/fesi.59.123.

[9]

J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16 (1975), 197-220.  doi: 10.1016/0001-8708(75)90151-6.

[10]

N. OkazawaT. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629.  doi: 10.1080/00036811.2011.631914.

[11]

N. OkazawaT. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.  doi: 10.3934/eect.2012.1.337.

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅱ, Academic Press, New York, 1975. 
[13]

T. Suzuki, Blowup of nonlinear Schrödinger equations with inverse-square potentials, Differ. Equ. Appl., 6 (2014), 309-333.  doi: 10.7153/dea-06-17.

[14]

T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.  doi: 10.1619/fesi.59.1.

[15]

T. Suzuki, Virial identities for nonlinear Schrödinger equations with an inverse-square potential of critical coefficient, Differ. Equ. Appl., 9 (2017), 327-352.  doi: 10.7153/dea-2017-09-24.

[16]

T. Suzuki, Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods, Evol. Equ. Control Theory, 8 (2019), 447-471.  doi: 10.3934/eect.2019022.

[17]

T. Suzuki, Semilinear Schrödinger equations with a potential of some critical inverse-square type, J. Differ. Equ., 268 (2020), 7629-7668.  doi: 10.1016/j.jde.2019.11.087.

[18]

S. Watanabe, The explicit solutions to the time-dependent Schrödinger equation with the singular potentials $k/(2x^{2})$ and $k/(2x^{2})+\omega^{2}x^{2}/2$, Commun. Partial Diff. Equ., 26 (2001), 571-593.  doi: 10.1081/PDE-100002238.

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