Journals and Conference ProceedingsThe Journals and Conference Proceedings Collection includes publications reviewed and published by members of the university community and proceedings from conferences hosted by, or sponsored by, Texas State University.https://digital.library.txstate.edu/handle/10877/1362021-10-19T01:36:41Z2021-10-19T01:36:41ZBound states of the discrete Schrödinger equation with compactly supported potentialsAktosun, TuncayChoque-Rivero, Abdon E.Papanicolaou, Vassilishttps://digital.library.txstate.edu/handle/10877/146692021-10-18T18:25:09Z2019-02-11T00:00:00ZBound states of the discrete Schrödinger equation with compactly supported potentials
Aktosun, Tuncay; Choque-Rivero, Abdon E.; Papanicolaou, Vassilis
The discrete Schrödinger operator is considered on the half-line lattice n ∈ {1, 2, 3,...} with the Dirichlet boundary condition at n =0. It is assumed that the potential belongs to class A<sub>b</sub>, i.e. it is real valued, vanishes when n > b with b being a fixed positive integer, and is nonzero at n = b. The proof is provided to show that the corresponding number of bound states, N, must satisfy the inequalities 0 ≤ N ≤ b. It is shown that for each fixed nonnegative integer k in the set {0, 1, 2,..., b}, there exist infinitely many potentials in class A<sub>b</sub> for which the corresponding Schrödinger operator has exactly k bound states. Some auxiliary results are presented to relate the number of bound states to the number of real resonances associated with the corresponding Schrödinger operator. The theory presented is illustrated with some explicit examples.
2019-02-11T00:00:00ZUpper and lower solutions methods for impulsive Caputo-Hadamard fractional differential inclusionsBelhannache, FaridaHamani, SamiraHenderson, Johnnyhttps://digital.library.txstate.edu/handle/10877/146662021-10-18T14:35:50Z2019-02-06T00:00:00ZUpper and lower solutions methods for impulsive Caputo-Hadamard fractional differential inclusions
Belhannache, Farida; Hamani, Samira; Henderson, Johnny
In this article we use the method of lower and upper solutions combined with the fixed point theorem by Bohnnenblust-Karlin to show the existence of solutions for initial-value problems of impulsive Caputo-Hadamard fractional differential inclusions of order in (0,1).
2019-02-06T00:00:00ZExistence of global solutions to Cauchy problems for bipolar Navier-Stokes-Poisson systemsLiu, Jianhttps://digital.library.txstate.edu/handle/10877/146652021-10-18T14:18:12Z2019-01-29T00:00:00ZExistence of global solutions to Cauchy problems for bipolar Navier-Stokes-Poisson systems
Liu, Jian
In this article, we consider the Cauchy problem for one-dimensional compressible bipolar Navier-Stokes-Poisson system with density-dependent viscosities. Under certain assumptions on the initial data, we prove the existence and uniqueness of a global strong solution.
2019-01-29T00:00:00ZFundamental solutions and Cauchy problems for an odd-order partial differential equation with fractional derivativePskhu, Arsenhttps://digital.library.txstate.edu/handle/10877/146632021-10-15T16:53:47Z2019-02-04T00:00:00ZFundamental solutions and Cauchy problems for an odd-order partial differential equation with fractional derivative
Pskhu, Arsen
In this article, we construct a fundamental solution of a higher-order equation with time-fractional derivative, give a representation for a solution of the Cauchy problem, and prove the uniqueness theorem in the class of functions satisfying an analogue of Tychonoff's condition.
2019-02-04T00:00:00Z