\documentstyle[twoside]{article}
\setlength{\textwidth}{125mm}
\setlength{\textheight}{185mm}
\setlength{\parindent}{8mm}
\frenchspacing
\setlength{\oddsidemargin}{0pt}
\setlength{\evensidemargin}{0pt}
\begin{document}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\n}{\noindent}
\newcommand{\vs}{\vspace{0.5cm}}
\newcommand{\f}{\frac}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\r}{\ref}
\newcommand{\la}[1]{\label{#1}}
\setcounter{page}{0}
%\pagestyle{empty}
\begin{center}
\Large{\bf A Simple Model of Concentrated Nonlinearity}
\footnote{\n
To appear in the Proceedings of the Conference ``Mathematical
Results in Quantum Mechanics'', Prague, June 22-26, 1998.}
\end{center}
\vspace{2cm}
\begin{center}
{\bf Riccardo Adami}\\
{\em Dipartimento di Matematica, Universit\`a di Roma ``La Sapienza'', Italy}
\vspace{1cm}
{\bf Alessandro Teta}\\
{\em Dipartimento di Matematica, Universit\`a di Roma ``La Sapienza'',
Italy}
\vspace{4cm}
{\bf Abstract}
\end{center}
\n
We study the nonlinear Schr\"odinger equation in dimension one with
a nonlinearity concentrated in a point. We give results on local and
global existence of the solution and find conditions for the blow-up.
In the critical case an explicit blow-up solution is constructed.
\newpage
\setcounter{page}{1}
%\pagestyle{}
\n
Recent studies in Solid State Physics concerning resonant tunnelling,
interactions with impurities etc. have motivated the analysis of the
nonlinear
Schr\"odinger equation (NLSE) with a nonlinearity concentrated in a (small)
region of space. Such equation exhibits interesting qualitative
properties which have been discussed in the physical
literature ([J-LPS], [MA], [BKB]). Some rigorous mathematical results are given
in [N], where the existence of steady states in the case of an open
system with smooth short-range nonlinearities is proved.
\n
In this note we approach what we consider the simplest case of
concentrated nonlinearity, i.e. the one-dimensional Schr\"odinger
equation with nonlinearity concentrated in a fixed point.
\n
In spite of its simplicity, such model seems to be not completely
trivial (for instance it produces blow-up solutions).
\n
We consider this work as the first step in a line of research devoted
to the mathematical analysis of the Schr\"odinger equation in
dimension $d$ with a nonlinearity concentrated on a set of lower
dimension.
\n
The paper is organized as follows.
\n
First we formulate the nonlinear evolution problem and give a local
existence theorem.
\n
The second step is the analysis of the conservation laws and the proof
of a global existence theorem.
\n
Finally we give conditions for the blow-up of the solution and, in
the critical case, we explicitely construct a class of blow-up
solutions.
\n
We start with the formulation of the problem. For any fixed real
function $\alpha(t)$, we denote by $H_{\alpha(t)}$ the
Schr\"odinger operator with point interaction of strength $\alpha(t)$
placed at the origin. It is well known ([AGH-KH]) that such operator can be
defined for all t
as a self-adjoint operator in $L^{2}(I \! \! R)$.
In particular, its form domain
is $H^{1}(I \! \! R)$ and the operator domain consists of functions
$u \in H^2(I \! \! R \setminus \{ 0 \})$ satisfying the boundary condition
at the origin
$u'(0^+) - u' (0^-) = \alpha (t) u (0)$.
Moreover the linear Schr\"odinger equation with
time-dependent hamiltonian $H_{\alpha(t)}$ can also be correctly
defined and it admits a unique solution for $\alpha (t)$ smooth.
\n
We define the nonlinear evolution problem by replacing
$\alpha(t)$ with a function of the value at the origin of the
solution itself. More precisely, for $\gamma \in I \! \! R$,
$\sigma >0$, we consider
\be
i \f{\partial \psi_{t}}{\partial t} = H_{\alpha(t)} \psi_{t} , \;\;\;
\alpha(t) = \gamma |\psi_{t}(0)|^{2 \sigma}, \;\;\; \psi_{t=0}=\psi_{0}
\ee
\n
For the sake of
simplicity we have considered in (1) a power law nonlinearity;
more general nonlinearities can also be introduced.
\n
Problem (1) corresponds to an evolution generated by the laplacian in
$I \! \! R$ with a nonlinear boundary condition at the origin, namely
\be
\psi_t^\prime (0^+) - \psi_t^\prime (0^-) = \gamma | \psi_t(0) |^{2 \sigma}
\psi_t(0) \ee
\n
It is convenient to reformulate problem (1) in integral form. One
obtains
\be
\psi_{t}(x) +i \gamma \int_{0}^{t} ds U \left( t-s,|x| \right)
|\psi_{s}(0)|^{2 \sigma} \psi_{s}(0) = \left( U(t) \psi_{0} \right) (x)
\la{due}
\ee
\n
where $U(t)$ denotes the free Schr\"odinger unitary group in dimension
one. From (\ref{due}) it is clear that the solution is enterely determined by
its value at the origin. Evaluating equation (\ref{due}) in $x=0$ one has a
closed equation for $\psi_{t}(0)$
\be
\psi_{t}(0) + i \gamma \int_{0}^{t} ds \f{1}{\sqrt{4 \pi i (t-s)}}
|\psi_{s}(0)|^{2 \sigma} \psi_{s}(0) = \left( U(t) \psi_{0} \right) (0)
\la{(3)}
\ee
\n
Equation (\ref{(3)}) is a nonlinear Abel integral equation, involving only the
time variable. The search for the solution of (\ref{due}) is then reduced to
the solution of (\ref{(3)}).
\n
As for the standard NLSE (see e.g. [GV]),
the natural space where we study the equation (\r{due}) is $H^{1}(I \! \! R)$,
i.e. the space of finite energy.
For technical reasons, in what follows
we choose an initial datum in $H^2 (I \! \! R)$ but we are convinced
that all the results remain valid under the more natural assumption
$\psi_0 \in H^1(I \! \! R)$.
\vs
\n
{\em Proposition 1}. For any $\psi_{0} \in H^{2}(I \! \! \!
\hspace{.06cm} R)$,
there
exists $\bar{t}>0$ s.t. problem (\ref{due}) has a unique solution
$\psi_{t} \in H^{1}(I \! \! R)$, $t \in [0,\bar{t})$.
\vs
\n
{\em Sketch of the proof}. The r.h.s. of (\ref{(3)}) is clearly continuous in
$t$. Then, using the contraction principle, equation (\ref{(3)}) can be
uniquely solved for small times in the space of continuous
functions ([M]). Substituting the solution $\psi_{t}(0)$ in (\ref{due}), one
easily sees that $\psi_{t} \in L^{2}(I \! \! R)$. Now,
exploiting the smoothing
properties of the Abel operator ([GoVe]) and the regularity of $\psi_{0}$, one
can prove that $\psi_{t}(0)$ is smooth enough to ensure $\psi_{t} \in
H^{1}(I \! \! R)$.
\vs
\n
One of the main tools for the analysis of nonlinear evolution problems
are conservation laws. In our case there are three conserved
quantities: $L^{2}$-norm (denoted by $\| \cdot \|$),
energy and parity.
\vs
\n
{\em Proposition 2.} For any $\psi_{0}\in H^{2}(I \! \! R)$ we have
\ba
& &\| \psi_{t}\| = \| \psi_{0} \| \label{norm} \\
& & E(\psi_{t}) \equiv \int_{I \! \! R}dx | \psi_{t}^{\prime}(x) |^{2} +
\f{\gamma}{\sigma +1} |\psi_{t}(0) |^{2 \sigma +2} = E(\psi_{0})\label{energy}
\\
& &\psi_{0} (x) = \pm \psi_{0}(-x) \;\; \mbox{implies} \;\;
\psi_{t}(x) = \pm \psi_{t}(-x) \la{parity}
\ea
\n
{\em Proof}. We give some details for (\r{energy}).
The proof of (\r{norm}) is similar
while (\r{parity}) is trivial.
\noindent
We define $q(t) \equiv | \psi_t (0) |^{2 \sigma} \psi_t (0)$. The hypotesis
$\psi_0 \in H^2 (I \! \! R)$ implies $q \in H^1((0, t))$ for all
$t < \bar t$.
\ba
\| \psi^{\prime}_t \|^2 & = & \int_{I \! \! R} dk \, k^2
\, | \psi_t (k) |^2 \nonumber \\
& = & \int_{I \! \! R} dk \, k^2
\, | \psi_0 (k) |^2 \, - \, \sqrt{\f 2 \pi} \, i
\, \gamma \, Re \int_{I \! \! R} dk \, k^2 \overline {\tilde
\psi_0 (k)} \int_0^t ds \,
e^{ik^2s} \, q(s)
\nonumber \\
& & + \, \f {\gamma^2} {2 \pi} \int_{I \! \! R}dk \,
k^2 \int_0^t ds \int_0^t ds' \, e^{ik^2(s'-s)} \overline {q(s)} q(s')
\label{energia}
\ea
where $\tilde \psi$ denotes the Fourier transform of $\psi$.
\noindent
After some calculations one obtains
\vspace{.15cm}
\noindent
$ Re \int_{I \! \! R} dk \, k^2 \overline {\tilde
\psi_0 (k)} \int_0^t ds \,
e^{ik^2s} \, q(s) $
\ba
& ~~ = & \f {\sqrt \pi} {\sqrt 2 \, (\sigma + 1)} \left( | \psi_t
(0) |^{2 \sigma + 2} - | \psi_0 (0) |^{2 \sigma + 2} \right)
\nonumber \\
& ~~ & + \ \frac {\gamma} 2 \, \int_0^t \frac {ds}
{\sqrt {t-s}} \left[ Re (q(t) \overline
{q(s)}) + Im (q(t) \overline {q(s)}) \right]
\nonumber \\
& ~~ & - \ \frac {\gamma} 2 \, \int_{0 \leq s^{\prime}
\leq s \leq t} \, \frac {ds \, ds^{\prime} }
{\sqrt {s-s^{\prime}}} \left[ Re (\dot q(s) \, \overline{q(s^{\prime})}) +
Im (\dot q(s) \, \overline {q(s^{\prime})}) \right] \label{II}
\ea
and
\vspace{0.15cm}
\noindent
$\int_{I \! \! R}dk \,
k^2 \int_0^t ds \int_0^t ds' \, e^{ik^2(s'-s)} \overline {q(s)} q(s')$
\ba
& = & {\sqrt {2 \pi}} \int_0^t \frac {ds}
{\sqrt {t-s}} \left[ Re (q(t) \overline {q(s)}) + Im (q(t) \overline {q(s)}) \right]
\nonumber \\
& & - \ {\sqrt {2 \pi}} \int_{0 \leq s^{\prime}
\leq s \leq t} \frac {ds \, ds^{\prime} }
{\sqrt {s-s^{\prime}}} \left[ Re (\dot q(s) \, \overline {q(s^{\prime})}) +
Im (\dot q(s) \, \overline {q(s^{\prime})}) \right] \label{III}
\ea
Then, from (\ref{energia}), (\ref{II})
and (\ref{III}), conservation of energy is proved.
\vs
\n
{\em Remark 1.}
%For an odd initial datum the solution of (\ref{due}) is
%simply $U(t)\psi_{0}$. Then, without loss of generality, from now on
%we shall restrict to the case of an even initial datum.
Since $\psi_0^{odd}(0) = 0$, from (\ref{due}) one has
$\psi_t^{odd} = U(t)\psi_0^{odd}$
and $\psi_t^{even}$ is given by equation (\ref{due}), where $\psi_0$
is replaced by $\psi_0^{even}$ (here we have
denoted by $f^{even}$ and
$f^{odd}$ respectively the even and the odd part of the function $f$).
One can therefore restrict oneself to even initial data.
\vs
\n
Using the conservation laws and the Sobolev inequality we
easily obtain the following global existence theorem.
\vs
\n
{\em Proposition 3.} Assume $\gamma >0$ (case a)) or $\gamma
<0$, $\sigma <1$ (case b)). Then for any $\psi_{0} \in H^{2}(I \! \! R)$ and
any $T>0$ problem (\ref{due}) has a unique solution $\psi_{t} \in H^{1}(I \! \! R)$,
for $t \in [0,T]$.
\vs
\n
{\em Proof.} In case a) the conservation of the energy directly
provides an a priori estimate of $\| \psi_{t}^{\prime}\|$, which
implies that
the solution is global in time.
\n
In case b) we apply the Sobolev inequality ([A]) and the conservation
of the $L^{2}$-norm to estimate the ``potential'' term in the energy
\be
|\psi_{t}(0)|^{2 \sigma +2} \leq \|\psi_{0}\|^{\sigma +1}
\|\psi_{t}^{\prime}\|^{\sigma +1} \la{sobolev}
\ee
\n
Combining (\r{energy}) and (\r{sobolev}) we have
\be
E(\psi_{0}) \geq \| \psi_{t}^{\prime} \|^{2} \left( 1 -
\f{|\gamma|}{\sigma +1} \| \psi_{0} \|^{\sigma +1} \| \psi_{t}^{\prime}
\|^{\sigma -1} \right) \la{ineq}
\ee
\n
Inequality (\ref{ineq})
implies that $\| \psi_{t}^{\prime} \|$ cannot diverge in
finite time and this implies global existence.
\vs
\n
Let us consider the problem of the existence of blow-up
solutions of (\ref{due}).
\n
Following the usual definition, we say that $\psi_{t}$ is a blow-up
solution if there exists $t_{0}< \infty$ s.t. $\|\psi_{t}^{\prime} \|
\rightarrow \infty$ for $t \rightarrow t_{0}$.
\n
Due to the conservation of the energy, it is evident that $\psi_t$ is a
blow-up solution if and only if $| \psi_t (0)| $ diverges for
$t \rightarrow t_0$. This means that, in our model, a blow-up
solution $\psi_t$ can be equivalently defined by the condition
$\| \psi_t \|_{L^\infty} \rightarrow \infty$ for $t \rightarrow t_0$.
\n
Using standard arguments, i.e. the computation of the second
time-derivative of the moment of inertia and the uncertainty
principle ([RR]), we can prove existence of blow-up solutions.
\vs
\n
{\em Proposition 4.} Assume $\gamma <0$ and $\sigma \geq 1$. Then for
any $\psi_{0} \in H^{2}(I \! \! R)$ s.t.
%$\psi_0 (x) = \psi_0 (-x)$ for any $x \in {I \! \! R}$,
%$\psi_0$ is an even function of $x$,
$\psi_0$ is an even function,
$\|x \psi_{0} \|Ê< \infty$ and
$E(\psi_{0}) < 0$ the solution $\psi_{t}$ is a blow-up solution.
\vs
\n
{\em Proof.} Let us define the moment of inertia (with respect to
the origin) of a solution of
(\ref{due})
\be
I(t) \equiv \int_{I \! \! R} dx x^{2} |\psi_{t}(x)|^{2}
\ee
\n
An explicit calculation yields
\ba
& &\dot{I}(t) = 4 \; Im \int_{I \! \! R} dx \overline{\psi_{t}(x)} x
\psi_{t}^{\prime}(x)\\
& &\ddot{I}(t) = 8 E(\psi_{0}) -4 |\gamma| \left( 1 - \f{2}{\sigma
+1} \right) |\psi_{t}(0) |^{2 \sigma +2}
\ea
\n
Since $\ddot{I}(t) <0$, there exists $t_{0} < \infty $ s.t. $I(t)
\rightarrow 0$ for $t \rightarrow t_{0}$. Applying the uncertainty
principle
\be
\| \psi_{0} \|^{2} \leq 2 \; \|Êx \psi_{t} \| \; \| \psi_{t}^{\prime}\|
\ee
\n
we conclude that $\psi_{t}$ is a blow-up solution.
\vs
\n
{\em Remark 2.}
%As we stated in remark 1,
The initial datum in the above
proposition is chosen to be even. For an arbitrary initial datum the
conclusion of the theorem still holds if the condition $E(\psi_{0}) < 0$
is replaced by $E(\psi_{0}^{even}) < 0$.
\vs
\n
>From propositions 3, 4 it is clear that the critical exponent of the
nonlinearity is $\sigma =1$.
\n
The critical case is of special interest
because of the existence of an additional
symmetry, the so-called pseudo-conformal invariance.
This symmetry can
be expressed as follows: if $u_{t}(x)$ is a solution then
\be
v_{\tau}(y) \equiv
\f{e^{-i \f{|y|^{2}}{4(t_{0} -\tau)}}}{\left[ a (
t_{0} - \tau ) \right]^{\f{1}{2}}}
\; u_{\f{1}{a^{2}(t_{0} - \tau)}}
\left( \f{y}{a (t_{0} -\tau )}
\right) \;\;\; a,t_{0}>0 \la{blowup}
\ee
\n
is again a solution. Moreover, a stationary solution of the problem, i.e.
a solution of
the form $Q(x) e^{i \lambda t}$, is easily computed
\be
u_{t}(x) \equiv e^{- \f{|x|}{2} + i \f{t}{4}} \la{stationary}
\ee
\n
Combining (\r{blowup}) and (\r{stationary}), we
explicitly construct the blow-up solution
\be
\psi_{t}(x) = \f{1}{\sqrt{a(t_{0} -t)}} e^{- \f{|x|}{2a(t_{0} - t)}
-i \f{|x|^{2}}{4(t_{0} -t)} +i \f{1}{4 a^{2} (t_{0} -t)}}
\la{quindici}
\ee
\n
An advantage to have the explicit solution (\r{quindici}) is that one can
directly check the properties of the solution near the blow-up
(non existence of strong $L^{2}$-limit,
$L^{2}$-concentration phenomenon, local behaviour, etc.).
\n
On the other hand, the standard
NLSE with critical power nonlinearity is also
invariant under the pseudo-conformal transformation (\ref{blowup}).
This fact suggests that the behaviour of
a solution that blows up at a single point can be well approximated
by our solution
(\r{quindici}) near the
blow-up time. In this sense our model would exhibit a universal
behaviour near the blow-up.
\n
The euristic motivation is the following.
In the standard NLSE (critical case) the ``nonlinear potential'' is
$|\psi_{t}(x)|^{4}$ and, near the blow-up, it behaves like the square
of a $\delta$-function (see e.g. [Me]).
\n
In our case the ``potential term'' is $|\psi_{t}(0)|^{2} \delta(x)$ and
it exhibits the same behaviour along the solution (\r{quindici}) for
$t\rightarrow t_{0}$, since the square of the modulus of
(\r{quindici}) is a
$\delta$-function approximation for $t \rightarrow t_{0}$.
\n
In conclusion we point out that the natural generalization of the
model discussed here is the case of NLSE with nonlinearities concentrated
in $n$ fixed points in dimensions one, two and three.
\n
The two and three dimensional cases require some care in their
formulation since the space of finite energy is strictly larger
than $H^1$.
This means that natural instruments like the Sobolev inequalities and
the uncertainty principle cannot be directly used.
Moreover, in the $n$ points case the analysis of the blow-up solutions
becomes more subtle because of the lack of conservation of parity (in
dimension
one) and angular momentum (in dimension two and three).
We plan to approach these questions in a further work.
\vs
\n
{\bf Acknowledgments.} We thank prof. G.F. Dell'Antonio and prof. R.
Figari for helpful discussions during the preparation of this work.
\vspace{.3cm}
\n
{\bf References}
\n
[A] Adams, R., {\em Sobolev Spaces}, Academic Press, New York, 1975.
\n
[AGH-KH] Albeverio, S., Gesztesy, F., H\"ogh-Krohn, R., Holden, H.,
{\em Solvable Models in Quantum Mechanics}, Springer-Verlag, New York,
1988.
\n
[BKB] Bulashenko, O.M., Kochelap, V.A., Bonilla L.L.,
{\em Coherent Patterns and Self-Induced Diffraction of Electrons on
a Thin Nonlinear Layer}, Phys. Rev. B, {\bf 54}, 3, 1996.
\n
[GV] Ginibre, J., Velo, G., {\em On a Class of Nonlinear Schr\"odinger
Equations. I. The Cauchy Problem, General Case}, J. Func. Anal., {\bf 32},
1-32, 1979.
\n
[GoVe] Gorenflo, R., Vessella, S., {\em Abel Integral Equations},
Springer-Verlag, Berlin Heidelberg, 1978.
\n
[J-LPS] Jona-Lasinio, G., Presilla, C., Sj\"ostrand J., {\em On
Schr\"odinger Equations with Concentrated Nonlinearities}, Ann. Phys.,
{\bf 240}, 1-21, 1995.
\n
[M] Miller, R. K., {\em Nonlinear Volterra Integral Equations}, W. A.
Benjamin Inc., 1971.
\n
[Me] Merle, F., {\em Construction of Solutions with Exactly k Blow-up
points for the Schr\"odinger Equation with Critical Nonlinearity},
Comm. Math. Phys., {\bf 129}, 223-240, 1990.
\n
[MA] Malomed, B., Azbel, M., {\em Modulational Instability of a
Wave Scattered by a Nonlinear Center}, Phys. Rev. B, {\bf 47},
16, 1993.
\n
[N] Nier, F., {\em The Dynamics of some Quantum Open Systems with
Short-Range Nonlinearities}, preprint Ecole Polytechnique, 1998.
\n
[RR] Rasmussen, J.J., Rypdal, K., {\em Blow-up in NLSE - A General
Review}, Physica Scripta, {\bf 33}, 481-497, 1986.
\end{document}
\vs
\n
[W] Weinstein, M.I., {\em NLSE and Sharp Interpolation Estimates},
C.M.P. 87, 567-576, 1983.
\end{document}