To our knowledge, there are no previous results of this type. We will show that the schrodinger and dirac equations are spacetime versions of the averaged energy relations. A copublication of the ams and the courant institute. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or the kortewegde vries equation. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
We will indicate possible extensions and references. The derivations are based on the assumption that these wave equations are homogeneous and soluble via separation of variables. From the mathematical point of view, schrodingers equation is a delicate problem, possessing a mixture of the properties of parabolic and elliptic equations. This paper is concerned with the initial value problem ivp associated. Numerous and frequentlyupdated resource results are available from this search. For the sake of simplicity, we will mostly consider two model equations. The cauchy problem for the quasilinear schrodinger. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or kortewegde vries equation. The schrodinger map equation is also related to the ishimori equation see for some recent results on this equation in one dimension local well posedness is known for smooth data by the general theory of derivative nonlinear schrodinger equations, however this is not yet established in higher dimensions. Solutions of semilinear schrodinger equations in hs numdam. The method of multiple scales for nonlinear kleingordon. Suppose for example that our initial wave form is a highly peaked gaussian.
It should go without saying that this work is a major contribution to mathematics. Newtons laws, the schrodinger equation does not give the trajectory of a particle, but rather the wave function of the quantum system, which carries information about the wave nature of the particle, which allows us to only discuss the probability of nding the particle in. One can now substitute these expressions into the full 3d schrodinger equation and see that they solve it even at the points r where r 0. Then in the case of the linear advection equation all the fourier modes travel together at the same speed and. Several anomalies are highlighted and resolutions proposed. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. In principle, the methods presented apply to a large class of dispersive semilinear equations. Mathematical and computational methods for semiclassical schrodinger equations shi jin department of mathematics, university of wisconsin, madison, wi 53706. The description of nature is essentially probabilistic, with the probability of an. Find all the books, read about the author, and more. This paper presents the derivation of the schrodinger, kleingordon and dirac equations of particle physics, for free particles, using classical methods. Semilinear schrodinger equations courant lecture notes in mathematics by cazenave, thierry and a great selection of related books, art and collectibles available now at. Mathematical and computational methods for semiclassical.
The global wellposedness of the problem is proved in the sobolev spaceh shsr n of fractional ordersn2 under the following assumptions. Semilinear schrodinger equations courant lecture notes in. American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. Buy semilinear schrodinger equations courant lecture notes in mathematics on free shipping on qualified orders. Schrodinger schrodinger equation with a cubic nonlinearity. In particular, this means that semilinear equations are ones in which the coe. Picture below is from the 56th page of lyapunov stability of ground states of nonlinear dispersive evolution equations. A system is completely described by a wave function. Consider the cauchy problem for semilinear schrodinger equations in. The schrodingers schrodingers equation is the basic equation of quantum mechanics. Analytical and numerical results for some classes of nonlinear schr odinger equations xiao liu doctor of philosophy graduate department of mathematics university of toronto 20 this thesis is devoted to the study of nonlinear dispersive partial di erential equations of schr odinger type. The energy method 1 problems for lecture 1 10 lecture 2. An introduction to semilinear evolution equations revised edition thierry cazenave cnrsand university of paris vi, france and alain haraux cnrs and university of paris vi, france translated by yvan martel university of cergypontoise, france clarendon press oxford 1998. An introduction to nonlinear schrodinger equations thierry cazenave.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Nonlinear schrodinger equations part 1 conservation laws. For one dimensional quadratic schrodinger equations we can even prove firstorder convergence without any loss of. Part i 3 and then, by superposition, the solution to our wave equation will be ux. Semilinear schrodinger equations courant lecture notes in mathematics uk ed.
Semilinear schrodinger equations pdf free download epdf. An introduction to semilinear evolution equations, oxford lecture series in mathematics and its applications, oxford university press, oxford, 1998. Derivation of the nonlinear schrodinger equation from. This monograph contains all the material taht was treated in a course of lectures given by dr. A brief introduction to classical, statistical, and quantum mechanics.
The simple harmonic oscillator recall our rule for setting up the quantum mechanical problem. The main questions we investigate are longtime behavior. We consider the cubic nonlinear schrodinger equation on the spatial domain r. Modified scattering for the cubic schrodinger equation on product. Derivation of the nonlinear schrodinger equation from first principles theodore bodurov eugene, oregon, usa, email.
We study local well posedness in classical sobolev space hsof the associated initial value problem and periodic boundary value problem. Analytical and numerical results for some classes of. Schrodinger equation with a cubic nonlinearity eqworld. Therefore, the solution of the 3d schrodinger equation is obtained by multiplying the solutions of the three 1d schrodinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation. Semilinear schrodinger equations thierry cazenave download. Low regularity exponentialtype integrators for semilinear schr. It studies both problems of local nature local existence of solutions, uniqueness, regularity, smoothing effect and problems of global nature finitetime blowup, global existence, asymptotic behavior of solutions. Szeftel, almost global existence for hamiltonian semilinear. The cauchy problem for the quasilinear schrodinger equation following kenigponcevega 1 lecture 1. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to boseeinstein condensates confined to highly anisotropic cigarshaped traps, in the meanfield regime.
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