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Section: Scientific Foundations

Geometric schemes for the Schrödinger equation

Participants : François Castella, Philippe Chartier, Guillaume Dujardin, Erwan Faou.

Given the Hamiltonian structure of the Schrödinger equation, we are led to consider the question of energy preservation for time-discretization schemes.

At a higher level, the Schrödinger equation is a partial differential equation which may exhibit Hamiltonian structures. This is the case of the time-dependent Schrödinger equation, which we may write as

Im28 ${i\#949 \mfrac {\#8706 \#968 }{\#8706 t}=H\#968 ,}$(8)

where $ \psi$ = $ \psi$(x, t) is the wave function depending on the spatial variables Im29 ${x=(x_1,\#8943 ,x_N)}$ with Im30 ${x_k\#8712 \#8477 ^d}$ (e.g., with d = 1 or 3 in the partition) and the time Im31 ${t\#8712 \#8477 }$ . Here, $ \varepsilon$ is a (small) positive number representing the scaled Planck constant and i is the complex imaginary unit. The Hamiltonian operator H is written

H = T + V

with the kinetic and potential energy operators

Im32 ${T=-\munderover \#8721 {k=1}N\mfrac \#949 ^2{2m_k}\#916 _x_k~\mtext and~V=V{(x)},}$

where mk>0 is a particle mass and $ \upper_delta$xk the Laplacian in the variable Im30 ${x_k\#8712 \#8477 ^d}$ , and where the real-valued potential V acts as a multiplication operator on $ \psi$ .

The multiplication by i in (8 ) plays the role of the multiplication by J in classical mechanics, and the energy Im33 ${\#9001 \#968 |H|\#968 \#9002 }$ is conserved along the solution of (8 ), using the physicists' notations Im34 ${\#9001 u|A|u\#9002 =\#9001 u,Au\#9002 }$ where Im35 ${\#9001 ~~,~\#9002 }$ denotes the Hermitian L2 -product over the phase space. In quantum mechanics, the number N of particles is very large making the direct approximation of (8 ) very difficult.

The numerical approximation of (8 ) can be obtained using projections onto submanifolds of the phase space, leading to various PDEs or ODEs: see [50] , [49] for reviews. However the long-time behavior of these approximated solutions is well understood only in this latter case, where the dynamics turns out to be finite dimensional. In the general case, it is very difficult to prove the preservation of qualitative properties of (8 ) such as energy conservation or growth in time of Sobolev norms. The reason for this is that backward error analysis is not directly applicable for PDEs. Overwhelming these difficulties is thus a very interesting challenge.

A particularly interesting case of study is given by symmetric splitting methods, such as the Strang splitting:

$ \psi$1 = exp(-i($ \delta$t)V/2)exp(i($ \delta$t)$ \upper_delta$)exp(-i($ \delta$t)V/2)$ \psi$0(9)

where $ \delta$t is the time increment (we have set all the parameters to 1 in the equation). As the Laplace operator is unbounded, we cannot apply the standard methods used in ODEs to derive long-time properties of these schemes. However, its projection onto finite dimensional submanifolds (such as Gaussian wave packets space or FEM finite dimensional space of functions in x ) may exhibit Hamiltonian or Poisson structure, whose long-time properties turn out to be more tractable.


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