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A nonlinear Schrödinger equation with two symmetric point interactions in one dimension. (English) Zbl 1189.35310

Summary: We consider a time-dependent one-dimensional nonlinear Schrödinger equation with a symmetric double-well potential represented by two Dirac’s \(\delta \). Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B32 Bifurcations in context of PDEs
45H05 Integral equations with miscellaneous special kernels
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
35B40 Asymptotic behavior of solutions to PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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