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Dispersive Estimates of Schrodinger and Schrodinger-Like Equations in One Dimension

Hill, Thomas
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1595850253465442

Abstract Details

2020, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
This dissertation will discuss one-dimensional dispersive estimates of the Schrodinger equation and a fourth-order Schrodinger-like equation. We prove dispersive estimates for the Schrodinger equation with Hamiltonians H =-Δ + V on R using a new approach inspired by work in the three-dimensional setting. Instead of constructing the resolvent of H out of scattering data for the potential, we identify its properties by working inside a Wiener algebra of operator-valued functions. The approach allows us to handle potentials satisfying (1+|x|)V ∈ L1 in all cases. This is the largest known class of potentials that supports an L1 to L∞ dispersive estimate in both the resonant and non-resonant case. We replicate both of those results and prove an additional weighted dispersive estimate with |t|-3/2 decay (in the non-resonant case) that previously required stronger decay conditions on V . We consider the L1 to L∞ dispersive estimates for the time evolution of Hamiltonians of the form H = -∂4x + V in dimension 1 with the bound |t|-1/2 . We require the potential, V = V (x), to have compact support centered around the origin and to be once-differentiable.
Michael Goldberg, Ph.D. (Committee Chair)
Leonid Slavin, Ph.D. (Committee Member)
Bingyu Zhang, Ph.D. (Committee Member)
125 p.
Mathematics
Schrodinger equation; dispersive estimates; fourth-order; Wiener algebra; one-dimensional

Recommended Citations

Citations

  • Hill, T. (2020). Dispersive Estimates of Schrodinger and Schrodinger-Like Equations in One Dimension [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1595850253465442

    APA Style (7th edition)

  • Hill, Thomas. Dispersive Estimates of Schrodinger and Schrodinger-Like Equations in One Dimension. 2020. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1595850253465442.

    MLA Style (8th edition)

  • Hill, Thomas. "Dispersive Estimates of Schrodinger and Schrodinger-Like Equations in One Dimension." Doctoral dissertation, University of Cincinnati, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1595850253465442

    Chicago Manual of Style (17th edition)

ucin1595850253465442
149
© 2020, some rights reserved.Creative Commons License Dispersive Estimates of Schrodinger and Schrodinger-Like Equations in One Dimension by Thomas Hill is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by University of Cincinnati and OhioLINK.