Schrodinger equations

Overview

The free Schrodinger equation

$i\partial _{t}u+\Delta u=0$

where u is a complex-valued function in $R\times R^{d}$, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions. This equation can be modified in many ways, notably by adding a potential or an obstacle, but we shall be interested in non-linear perturbations such as

$i\partial _{t}u+\Delta u=f(u,{\overline {u}},Du,D{\overline {u}})$

where $D$ denotes spatial differentiation. In such full generality, we refer to this equation as a derivative non-linear Schrodinger equation (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a semilinear Schrodinger equation (NLS). These equations (particularly the cubic NLS) arise as model equations from several areas of physics.

Some linear perturbations of the free Schrodinger equation are also of interest in the nonlinear theory (in part because one can view nonlinear equations as linear equations in which certain coefficients themselves depend on the solution). For instance, one can add a potential term $Vu$ to the right-hand side, yielding the Schrodinger equation with potential. Or one replace the Laplacian $\Delta =\partial _{k}\partial _{k}$ with a covariant Laplacian $(\partial _{k}+iA_{k})(\partial _{k}+iA_{k})$, leading to the magnetic Schrodinger equation. Finally, one can replace the underlying spatial domain $R^{d}$ with a Riemannian manifold $(M,g)$, and the Laplacian with the Laplace-Beltrami operator $\Delta _{g}$, yielding the Schrodinger equation on manifolds. One can also allow the manifolds to have boundaries (and assume appropriate boundary conditions), leading to the Schrodinger equation with obstacles.

One can combine these linear perturbations with a nonlinear one, leading for instance to the NLS with potential and the NLS on manifolds and obstacles.

One can generalize both the linear and nonlinear perturbations to these equations and consider the class of quasilinear Schrodinger equations or even fully nonlinear Schrodinger equations. Needless to say, these equations are significantly more difficult to analyse than the simpler model cases discussed above.

Unique continuation

A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if u is a solution to a nonlinear Schrodinger equation, and u(t_0) and u(t_1) is specified on a domain D at two different times t_0, t_1, does this uniquely specify the solution everywhere at all other intermediate times?

• For the 1D cubic NLS, with D equal to a half-line, and u assumed to vanish on D, this is in Zg1997.
• For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in Bo1997b.
• For D the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in KnPoVe2003
• For D in a half-plane, and allowing potentials in various Lebesgue spaces, this is in [IonKn-p]
• A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in reference:Isk1993 Isk1993

Specific Schrodinger Equations

Monomial semilinear Schrodinger equations are indexed by the degree of the nonlinearity and the spatial domain. A taxonomy of these and other specific Schrodinger equations appears on the specific equations page.

Quadratic NLS

Equations of the form

$i\partial _{t}u+\Delta u=Q(u,{\overline {u}})$

which $Q(u,{\overline {u}})$ a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.

Cubic NLS

The cubic nonlinear Schrodinger equation is of the form

$i\partial _{t}u+\Delta u=\pm |u|^{2}u$

Quartic NLS

A nonlinear Schrodinger equation with nonlinearity of degree 4 is a quartic nonlinear Schrodinger equation.

Quintic NLS

Equations of the form

$i\partial _{t}u+\Delta u=\pm |u|^{4}u$

are quintic nonlinear Schrodinger equations.

Septic NLS

Equations of the form

$i\partial _{t}u+\Delta u=\pm |u|^{6}u$

are septic nonlinear Schrodinger equations.

$L^{2}$-critical NLS

The nonlinear Schrodinger equation

$i\partial _{t}u+\Delta u=\pm |u|^{\frac {4}{d}}u$

posed for $x\in R^{d}$ is scaling invariant in $L_{x}^{2}$. This family of nonlinear Schrodinger equations is therefore called the mass critical nonlinear Schrodinger equation.

Higher order NLS

One can study higher-order NLS equations in which the Laplacian is replaced by a higher power.One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable cubic NLS on $R$. Another is the nonlinear Schrodinger-Airy equation.

Schrodinger maps

A geometric Schodinger equation that has been intensively studied is the Schrodinger map equation.

Cubic DNLS on $R$

The cubic DNLS on R deriviative cubic nonlinear Schrodinger equation has nonlinearity of the form $i\partial _{x}(|u|^{2}u).$

Hartree Equation

The Hartree equation has a nonlocal nonlinearity given by convolution.

Maxwell-Schrodinger system

A Schrodinger-wave system closely related to the [wave:Maxwell-Klein-Gordon|Maxwell-Klein-Gordon equation]] is the Maxwell-Schrodinger system.

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms $L_{t}^{q}L_{x}^{r}$ or $L_{x}^{r}L_{t}^{q}$, or in $X^{s,b}$ spaces, defined by

$\|u\|_{X^{s,b}}=\|u\|_{s,b}:=\|\langle \xi \rangle ^{s}\langle \tau -|\xi |^{2}\rangle ^{b}{\hat {u}}\|_{L_{\tau ,\xi }^{2}}.$

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities. See also [Bo1993b], [KlMa1993].

See Schrodinger estimates

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