# The nonlinear Schrödinger equation with sprinkled nonlinearity

Benjamin Harrop-Griffiths  Department of Mathematics & Statistics, Georgetown
University
Washington, DC 20057, USA<EMAIL_ADDRESS>,  Rowan
Killip  Department of Mathematics, University of California, Los Angeles
CA 90095, USA<EMAIL_ADDRESS>and  Monica Vişan  Department of
Mathematics, University of California, Los Angeles
CA 90095, USA<EMAIL_ADDRESS>
###### Abstract.

We prove global well-posedness for the cubic nonlinear Schrödinger equation
with nonlinearity concentrated on a homogeneous Poisson process.

##  1\. Introduction

We consider the dynamics of a field $\psi:\mathbb{R}\to\mathbb{C}$ under the
Hamiltonian

(1.1) $\displaystyle
E[\psi]:=\int_{\mathbb{R}}\tfrac{1}{2}|\psi^{\prime}(x)|^{2}\,dx+\int_{\mathbb{R}}\tfrac{1}{2}|\psi(x)|^{4}\,d\mu(x),$

namely, the nonlinear Schrödinger equation

(NLSμ) $\displaystyle i\psi_{t}=-\psi_{xx}+2|\psi|^{2}\psi\,d\mu.$

This model is relevant in physical scenarios where the wave self-interaction
is mediated by the environment. In optics, for example, nonlinear effects
originate from the interaction of light with the matter through which it is
traveling. This interaction takes place at the atomic level, which is
considerably smaller than the wavelength of visible light. In this way, we are
lead to consider highly concentrated measures $\mu$.

As we will discuss more fully below, the case of a single point nonlinearity
has received a great deal of attention. In this paper, we seek to address the
following question: What happens if the sources of such strong self-
interaction were sprinkled throughout the environment like impurities in a
crystal? This corresponds to choosing $d\mu$ according to a homogeneous
Poisson process:

######  Definition 1.1.

The (homogeneous) Poisson process of unit intensity is the unique random
measure $\mu$ with Laplace functionals

(1.2)
$\operatorname{\mathbb{E}}\Bigl{\\{}\exp\Bigl{(}-\textstyle\int\phi(x)\,d\mu\Bigr{)}\Bigr{\\}}=\exp\Bigl{(}-\textstyle\int[1-e^{-\phi(x)}]\,dx\Bigr{)}\quad\text{for
all}\quad\phi\in C^{\infty}_{c}(\mathbb{R}).$

For a thorough account of Poisson processes, emphasizing the interpretation as
random measures, see [7]. One may obtain homogeneous processes of other
intensities by a simple rescaling. Homogeneity of the process is not at all
needed for our methods; we simply regard it as the most natural and most
interesting case. The main result of this paper is well-posedness for this
model:

######  Theorem 1.2.

Let $\mu$ be distributed according to the Poisson process (1.2). Then the
equation (NLSμ) is almost surely globally well-posed in $H^{1}(\mathbb{R})$.
Specifically,
(a) Given any initial data $\psi_{0}\in H^{1}$, almost surely there is a
unique $\psi\in C(\mathbb{R};H^{1})$ that satisfies

(1.3)
$\psi(t)=e^{it\partial_{x}^{2}}\psi_{0}-2i\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}|\psi(\tau)|^{2}\psi(\tau)d\mu\Bigr{]}\,d\tau.$

(b) The energy (1.1) of the solution is both finite and conserved; so too is
the mass

(1.4) $M[\psi]=\int_{\mathbb{R}}|\psi|^{2}\,dx.$

(c) For any sequence of initial data, $\psi_{n,0}\to\psi_{0}$ in $H^{1}$, the
corresponding solutions $\psi_{n}\in C(\mathbb{R};H^{1})$, which exist almost
surely, converge to $\psi$ in
$L^{p}\bigl{(}d\mathbb{P};C([-T,T];H^{1})\bigr{)}\mkern-5.0mu:$

(1.5) $\lim_{n\to\infty}\operatorname{\mathbb{E}}\Bigl{[}\,\sup_{|t|\leq
T}\|\psi_{n}(t)-\psi(t)\|_{H^{1}}^{p}\Bigr{]}=0,$

for any time $T>0$ and any $1\leq p<\infty$.

We say that a $\psi\in C(\mathbb{R};H^{1})$ satisfying (1.3) is a _strong
solution_ to (NLSμ) with initial data $\psi_{0}$. Note that RHS(1.3) will,
almost surely, constitute a tempered distribution for any $\psi\in
C(\mathbb{R};H^{1})$. The crux of this is the convergence of

(1.6)
$\int_{0}^{t}\int_{-\infty}^{\infty}\overline{e^{-i(t-\tau)\partial_{x}^{2}}\phi(x)}|\psi(\tau,x)|^{2}\psi(\tau,x)\,d\mu(x)\,d\tau$

for any $t\in\mathbb{R}$ and any $\phi\in\mathcal{S}(\mathbb{R})$; indeed,
$\psi(\tau,x)$ is uniformly bounded on the region of integration and
$e^{-i(t-\tau)\partial_{x}^{2}}\phi$ belongs to Schwartz space, which ensures
sufficient decay to integrate $d\mu$.

On the other hand, it is far from clear that RHS(1.3) belongs to
$H^{1}(\mathbb{R})$, as claimed in Theorem 1.2. There is, of course, the
question of regularity; this appears already in the case of a point
nonlinearity. The more poignant problem here is decay: For almost every $\mu$
there is a choice of $\phi\in H^{1}$ and $\psi\in C(\mathbb{R};H^{1}_{x})$ so
that (1.6) diverges for every $t>0$!

The origins of such divergence can be seen more easily at the level of the
energy (1.1). For almost every sample $\mu$ of the Poisson process, there is a
sequence of unit intervals $[x_{n},x_{n}+1]$ with the property that
$\mu([x_{n},x_{n}+1])\geq n!$. It is then elementary to choose a $\psi_{0}\in
H^{1}(\mathbb{R})$, comprised of bumps (centered on these intervals) with
slowly decaying amplitudes, so that $\psi_{0}$ has infinite energy. This
divergence shows that the (zero probability) set of exceptional measures $\mu$
that is excluded in Theorem 1.2 must depend on $\psi_{0}$.

Earlier, we described highly concentrated nonlinearities as representative of
the case where the characteristic scale of the nonlinearity is much much
narrower than that of the wave. This can be made rigorous by considering a
family of models with mollified nonlinearities and showing that their
solutions converge to those of (NLSμ). Just such a result was proved for
finite combinations of point interactions in [3, 9]. We wish to do the same
for our model. To this end, we define the absolutely continuous measure
$\mu^{\varepsilon}$ with density (Radon–Nikodym derivative)

(1.7)
$\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,(x)=\int_{\mathbb{R}}\rho^{\varepsilon}(x-y)\,d\mu(y),\quad\text{where}\quad\rho^{\varepsilon}(x)=\tfrac{1}{\varepsilon}\rho\left(\tfrac{x}{\varepsilon}\right)$

and $\rho\in C_{c}^{\infty}(-1,1)$ is non-negative, even, and satisfies
$\int_{\mathbb{R}}\rho(x)\,dx=1$.

######  Theorem 1.3.

Fix $\psi_{0}\in H^{1}(\mathbb{R})$ and let $\mu$ be distributed according to
the Poisson process (1.2). Then, almost surely, the solutions
$\psi_{\varepsilon}(t)$, associated to the mollified measure
$\mu^{\varepsilon}$ defined in (1.7), converge to the solution $\psi(t)$ to
(NLSμ); indeed,

(1.8) $\displaystyle\lim_{\varepsilon\to
0}\operatorname{\mathbb{E}}\Bigl{[}\,\sup_{|t|\leq
T}\|\psi_{\varepsilon}(t)-\psi(t)\|_{H^{1}}^{p}\Bigr{]}=0,$

for any time $T>0$ and any $1\leq p<\infty$.

If $\mu$ were a finite measure, then each $\mu^{\varepsilon}$ would have a
bounded density and correspondingly, the well-posedness of the regularized
model would be covered by the standard arguments described in [4]. For the
Poisson process, however, the mollification procedure does nothing to reduce
the impact of the sequences of ‘rich’ intervals such as the $[x_{n},x_{n}+1]$
described above. In this way, the model with mollified coupling measure
retains many of the original difficulties and this compels us to develop a
unified well-posedness theory. This theory will be deterministic and can be
applied to very general measures $\mu$.

As we have explained, our archenemy in this work is the inevitable, but
extremely erratic, appearance of rich clusters of points in samples of the
Poisson process. It is crucial to control the wave’s interaction with these
clusters. For this purpose, we introduce a carefully chosen weighted $L^{2}$
norm that conforms to the highs and lows of the process.

Given a positive measure $\mu$ on $\mathbb{R}$ and an integer $k\in\mathbb{Z}$
we define

(1.9)
$\mathcal{N}_{k}(\mu)^{2}=4+\sup_{\ell\in\mathbb{Z}}\Bigl{[}\mu(I_{\ell})^{2}-|k-\ell|\Bigr{]},\quad\text{where}\quad
I_{\ell}=[\ell-\tfrac{1}{2},\ell+\tfrac{1}{2}).$

This function provides a slowly varying approximation to $\mu(I_{k})$ in which
rich intervals have a correspondingly wide range of influence. We then define
an associated weight via linear interpolation:

(1.10)
$\omega(x;\mu)=(1+k-x)\mathcal{N}_{k}(\mu)^{2}+(x-k)\mathcal{N}_{k+1}(\mu)^{2}\quad\text{for
$x\in[k,k+1)$}.$

Finally, we define ${L_{\mu}^{2}}:=L^{2}(\mathbb{R};\omega(x;\mu)\,dx)$, that
is, the Hilbert space with norm

$\|f\|_{L_{\mu}^{2}}^{2}=\int_{\mathbb{R}}|f(x)|^{2}\,\omega(x;\mu)\,dx.$

Naturally, we will only consider measures $\mu$ for which $\omega(x;\mu)$ is
finite. As we will show in Lemma 2.1, this is guaranteed by the single
condition $\mathcal{N}_{0}(\mu)<\infty$. Lemma 4.1 then ensures that
$\mathcal{N}_{0}(\mu)$ is almost surely finite for samples from the Poisson
process.

The engine of our paper is the following entirely deterministic well-posedness
result:

######  Theorem 1.4.

Let $\mu$ be a positive Borel measure for which $\mathcal{N}_{0}(\mu)<\infty$.
Then (NLSμ) is globally well-posed in $H^{1}\cap{L_{\mu}^{2}}$.

Precisely, given any initial data $\psi_{0}\in H^{1}\cap{L_{\mu}^{2}}$ there
exists a unique global strong solution $\psi\in
C(\mathbb{R};H^{1}\cap{L_{\mu}^{2}})$ of (NLSμ) satisfying $\psi(0)=\psi_{0}$.
This solution conserves both the mass and the energy. Moreover, for any time
$T>0$ the solution map $H^{1}\cap{L_{\mu}^{2}}\ni\psi_{0}\mapsto\psi\in
C([-T,T];H^{1}\cap{L_{\mu}^{2}})$ is continuous.

This result is used to prove Theorem 1.2 in Section 4. In particular, Lemma
4.1 shows that any fixed $\psi_{0}\in H^{1}(\mathbb{R})$ belongs to
$L^{2}_{\mu}$ for almost every $\mu$. The arguments used to prove Theorem 1.4
also yield a deterministic analogue of Theorem 1.3; see Corollary 3.7. This is
the basis for the proof of Theorem 1.3 in Section 4.

### Outline of the proof

We first focus on the deterministic result Theorem 1.4. Note that the
restriction on $\mu$ allows for samples from the Poisson process and so we
must accept that there are unit intervals with arbitrarily large $\mu$
measure. On the other hand, the condition that $\psi_{0}\in L^{2}_{\mu}$
ensures that initially, at least, the waveform is not strongly concentrated on
such rich clusters. In particular, the energy is initially finite; see Lemma
2.3.

To proceed, we must confront the natural (and inevitable) spreading of
dispersive waves into such rich regions. Indeed, a concentrated nonlinearity
naturally creates high frequency waves and these, we fear, may carry the mass
to regions where $\mu$ is large. While we believe such transportation does
take place, we will ultimately be able to control it in a satisfactory manner
by proving $L^{2}_{\mu}$ bounds.

For nonlinear Schrödinger equations, the transportation of $L^{2}$ norm is
governed by the mass current (= momentum density), which is given by
$2\operatorname{Im}[\;\\!\overline{\psi}\psi^{\prime}\;\\!](x)$. This we will
control using the conservation of energy.

Energy conservation is, of course, a property of true solutions, which is
precisely what we are trying to construct! It is not a property of Picard
iterates, for example. In fact, construction of solutions via Picard iteration
is problematic: one needs $H^{1}$ bounds on the iterates to control transport
and one needs transport bounds to control $H^{1}$ growth. Moreover, there is
no easy way to exhibit the smallness needed to close a bootstrap argument. One
may make the time interval small, but how short it needs to be is delicately
entwined with the spreading of the Picard iterates into rich intervals.

We will construct solutions in the manner of an energy method. The first step
is to regularize the model, not solely in the sense of (1.7), but also by
limiting the existence of rich clusters. This we do with a spatial truncation:

(1.11)
$V^{\varepsilon}(x)=\varphi^{\varepsilon}(x)\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,(x)\in
C_{c}^{\infty}(\mathbb{R})\quad\text{where}\quad\varphi^{\varepsilon}(x)=\varphi(\varepsilon
x)$

and $\varphi\in C_{c}^{\infty}(-2,2)$ is a non-negative bump function that is
identically $1$ on $[-1,1]$. The resulting model is easily solved by
contraction mapping; see Proposition 3.2.

We then show that the sequence of solutions to this regularized model converge
as $\varepsilon\to 0$, at least on short time intervals, and moreover, the
limit $\psi(t,x)$ is an $L^{\infty}_{t}H^{1}_{x}$ solution to (NLSμ); see
Proposition 3.4. At this moment in the argument, we know that
$t\mapsto\psi(t)$ is weakly $H^{1}$-continuous, but not we do not know that it
is norm continuous. Nevertheless, Proposition 3.1 guarantees uniqueness of
such a solution.

To prove well-posedness of (NLSμ), we must show that the limiting solution
$\psi$ belongs to $C(\mathbb{R};H^{1})$ and demonstrate continuity of the
data-to-solution map. We start by proving that $\psi$ conserves both mass and
energy. This is then used to demonstrate that $\psi$ is a global in time
solution and that it belongs to $C(\mathbb{R};H^{1})$; see Proposition 3.5.
Finally, the proof of Theorem 1.4 is completed with Proposition 3.6, which
shows that the data-to-solution map $\psi_{0}\mapsto\psi(t)$ is
$H^{1}\cap{L_{\mu}^{2}}$-continuous. In this way, the deterministic analysis
is completed in Section 3.

Section 4 is devoted to proving our main results about the stochastic model,
namely, Theorems 1.2 and 1.3. Lemma 4.1 shows that our deterministic Theorem
1.4 is almost surely applicable for each initial data $\psi_{0}\in H^{1}$.
This leaves us to verify the convergence claims (1.5) and (1.8). Our earlier
results yield almost sure convergence. We deduce $L^{p}(d\mathbb{P})$
convergence by exhibiting a suitable dominating function and applying the
dominated convergence theorem.

### Summary of existing literature

To the best of our knowledge, the model discussed in this paper has not been
considered previously. It synthesizes two facets of the theory that have been
considered before: the coupling constant is given by a stationary process and
is concentrated on a zero-dimensional set.

Nonlinear Schrödinger equations with an inhomogeneous coupling constant in
front of the nonlinearity arise very naturally in many physical contexts. This
has lead to a great many mathematical investigations of this case, too
numerous to recount here. However, the literature on this subject (that we
know of) has centered on the case of a single localized perturbation of the
coupling constant. In an experimental setting, this corresponds to a carefully
engineered apparatus. By comparison, the paper [10] stands out to us for
treating coupling constants given by certain stationary stochastic processes,
including periodic, quasiperiodic, and alloy-type models. Concretely, [10]
proves homogenization to the case of constant coupling constant in the mass-
critical two-dimensional setting.

Nonlinear Schrödinger equations with nonlinearity concentrated at a single
point (or at finitely many points) have generated considerable interest in
both mathematics and physics. For a thorough account, we recommend the recent
review [12]. The quantity of literature on these models is so great that we
will confine our discussion to the cubic model in one dimension and focus only
on well-posedness questions.

A cubic nonlinearity with a single delta-function as coupling constant

(NLSδ) $\displaystyle i\psi_{t}=-\psi_{xx}\pm 2\delta(x)|\psi|^{2}\psi$

is a mass-critical model. This is the observation that the scaling

$\psi(t,x)\mapsto\sqrt{\lambda}\psi(\lambda^{2}t,\lambda x)$

that preserves the class of solutions also preserves the (conserved) mass
(1.4). As we will describe, the known phenomenology of (NLSδ) mimics that of
the mass-critical homogenous equation.

Local existence for (NLSδ) with initial data in $L^{2}(\mathbb{R})$ is known
and such solutions were shown to be global in the case of small initial data;
see [5]. As observed in [6], the model (NLSδ) is pseudoconformally invariant.
This transformation yields explicit blow-up solutions in the focusing case.

Most researchers, however, have discussed this problem for initial data
$\psi_{0}\in H^{1}$. In the defocusing case, (NLSδ) is globally well-posed [2,
9]. In the focusing case, it is globally well-posed below an explicit mass
threshold and ill-posed above; see [6]. For either sign of the nonlinearity,
small solutions scatter; see [1, §2].

As noted earlier in the context of Theorem 1.3, the papers [3, 9] have already
demonstrated the validity of the $\delta$ interaction as describing the limit
of less singular nonlinearities.

### Acknowledgements

Rowan Killip was supported by NSF grant DMS-2154022. Monica Visan was
supported by NSF grant DMS-2054194.

##  2\. Preliminaries

Throughout the paper, $\lesssim$ and $\gtrsim$ indicate bounds that hold up to
a multiplicative constant whose exact value is of no consequence. We
abbreviate $A\lesssim B\lesssim A$ as $A\simeq B$. We write $C(\mathbb{R})$
for the space of bounded continuous functions on $\mathbb{R}$.

For all $s>\frac{1}{2}$ we have the Sobolev embedding $H^{s}\hookrightarrow
C(\mathbb{R})$, as demonstrated by the Gagliardo–Nirenberg inequality

(2.1)
$\|f\|_{L^{\infty}}\lesssim\|f\|_{L^{2}}^{1-\frac{1}{2s}}\|f\|_{H^{s}}^{\frac{1}{2s}}.$

As we consider finite-energy solutions, we will typically use the case $s=1$.
The principal properties of the space $H^{1}(\mathbb{R})$ that we employ are
the following:

(2.2) $\displaystyle\|fg\|_{H^{1}}$
$\displaystyle\lesssim\|f\|_{L^{\infty}}\|g\|_{H^{1}}+\|f\|_{H^{1}}\|g\|_{L^{\infty}}$
(2.3) $\displaystyle\|fg\|_{H^{1}}$
$\displaystyle\lesssim\|f\|_{H^{1}}\|g\|_{H^{1}}$ (2.4)
$\displaystyle\|fg\|_{H^{-1}}$
$\displaystyle\lesssim\|f\|_{H^{-1}}\|g\|_{H^{1}}.$

The first estimate is a direct consequence of the product rule. The bound
(2.3) expresses the algebra property of $H^{1}(\mathbb{R})$ and follows from
(2.2) and (2.1). The final estimate (2.4) is implied by (2.3) and the
realization of $H^{-1}(\mathbb{R})$ as the dual of $H^{1}(\mathbb{R})$ through
the usual $L^{2}$-paring.

We fix a non-negative function $\chi\in C_{c}^{\infty}(-1,1)$ so that
$\chi_{k}(x)=\chi(x-k)$ satisfies $\sum_{k\in\mathbb{Z}}\chi_{k}(x)=1$. One
may readily verify that

$\|f\|_{L^{2}}^{2}\simeq\sum_{k\in\mathbb{Z}}\|\chi_{k}f\|_{L^{2}}^{2}\quad\text{and}\quad\|f\|_{H^{1}}^{2}\simeq\sum_{k\in\mathbb{Z}}\|\chi_{k}f\|_{H^{1}}^{2}.$

Using complex interpolation, it follows that

(2.5) $\|f\|_{H^{s}}^{2}\simeq\sum_{k\in\mathbb{Z}}\|\chi_{k}f\|_{H^{s}}^{2}$

for all $0\leq s\leq 1$.

We now turn to the properties of the space ${L_{\mu}^{2}}$. Recalling the
definitions (1.9) and (1.10), we have the following:

######  Lemma 2.1.

If $\mu$ is a positive Borel measure on $\mathbb{R}$ for which
$\mathcal{N}_{0}(\mu)<\infty$, then $\mathcal{N}_{k}(\mu)<\infty$ for all
$k\in\mathbb{Z}$ and we have the estimates

(2.6) $\displaystyle|\mathcal{N}_{k}(\mu)^{2}-\mathcal{N}_{\ell}(\mu)^{2}|$
$\displaystyle\leq|k-\ell|$ $\displaystyle\quad\text{for all
$k,\ell\in\mathbb{Z}$},$ (2.7)
$\displaystyle\mu(I_{\ell})\leq\mathcal{N}_{\ell}(\mu)$
$\displaystyle\simeq\mathcal{N}_{k}(\mu)$ $\displaystyle\quad\text{whenever
$|k-\ell|\leq 3$}.$

The corresponding weight $\omega$ is a $1$-Lipschitz function on $\mathbb{R}$
satisfying the estimate

(2.8) $|\omega(x;\mu)|\leq\mathcal{N}_{0}(\mu)^{2}+|x|,$

and we have the equivalent norm

(2.9)
$\|f\|_{L_{\mu}^{2}}^{2}\simeq\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)^{2}\|\chi_{k}f\|_{L^{2}}^{2}.$

###### Proof.

From the estimate

$\Bigl{|}\bigl{[}\mu(I_{\ell})^{2}-|k-\ell|\bigr{]}-\bigl{[}\mu(I_{\ell})^{2}-|k+1-\ell|\bigr{]}\Bigr{|}\leq
1,$

we see that if one of $\mathcal{N}_{k}(\mu)$, $\mathcal{N}_{k+1}(\mu)$ is
finite, then the other is also finite and we may bound

$|\mathcal{N}_{k}(\mu)^{2}-\mathcal{N}_{k+1}(\mu)^{2}|\leq 1.$

By induction on $|k|$, if $\mathcal{N}_{0}(\mu)<\infty$ then we have
$\mathcal{N}_{k}(\mu)<\infty$ for all $k\in\mathbb{Z}$ and the estimate (2.6)
holds.

The inequality in (2.7) follows directly from the definition (1.9). For the
equivalence, we note that (2.6) and the fact that
$\mathcal{N}_{k}(\mu)^{2}\geq 4$ ensures that if $|k-\ell|\leq 3$ then

$|\mathcal{N}_{k}(\mu)^{2}-\mathcal{N}_{\ell}(\mu)^{2}|\leq
3\leq\tfrac{3}{4}\mathcal{N}_{k}(\mu)^{2}.$

Next, by (2.6) and the definition (1.10), we see that for all
$x,y\in\mathbb{R}$ we have

(2.10) $|\omega(x;\mu)-\omega(y;\mu)|\leq|x-y|,$

and hence $\omega$ is $1$-Lipschitz. The estimate (2.8) then follows from the
fact that $\omega(0;\mu)=\mathcal{N}_{0}(\mu)^{2}$. Moreover, if
$x\in\operatorname{supp}\chi_{k}\subseteq(k-1,k+1)$ then (2.10) yields

$|\omega(x;\mu)-\mathcal{N}_{k}(\mu)^{2}|=|\omega(x;\mu)-\omega(k;\mu)|\leq
1\leq\tfrac{1}{4}\mathcal{N}_{k}(\mu)^{2}.$

Using this, (2.9) follows from (2.5). ∎

Next we discuss the relationship between $\mathcal{N}_{0}(\mu)$ and the
mollified measure $\mu^{\varepsilon}$ defined in (1.7):

######  Lemma 2.2.

If $\mu$ is a positive Borel measure on $\mathbb{R}$ with
$\mathcal{N}_{0}(\mu)<\infty$, then the function
$\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\in C^{\infty}(\mathbb{R})$
and we have the estimate

(2.11)
$\displaystyle\bigl{\|}\chi_{k}\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\bigr{\|}_{L^{1}}$
$\displaystyle\lesssim\mathcal{N}_{k}(\mu),$

uniformly for $0<\varepsilon\leq 1$, as well as the estimate

(2.12)
$\|\chi_{k}(d\mu^{\eta}-d\mu^{\varepsilon})\|_{H^{-1}}\lesssim\sqrt{\eta}\,\mathcal{N}_{k}(\mu),$

uniformly for $0\leq\varepsilon<\eta\leq 1$.

###### Proof.

As $\mathcal{N}_{0}(\mu)<\infty$, $\mu$ is a locally finite Borel measure and
so $\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\in
C^{\infty}(\mathbb{R})$.

To prove the estimates (2.11) and (2.12), we argue by duality. If
$0<\varepsilon\leq 1$ and $\phi\in L^{\infty}$ then the function
$\rho^{\varepsilon}*(\chi_{k}\phi)\in C_{c}^{\infty}(k-2,k+2)$ satisfies

$\|\rho^{\varepsilon}*(\chi_{k}\phi)\|_{L^{\infty}}\lesssim\|\phi\|_{L^{\infty}}.$

We may then apply (2.7) to obtain

$|\langle\chi_{k}d\mu^{\varepsilon},\phi\rangle|=\left|\int_{k-2}^{k+2}\rho^{\varepsilon}*(\chi_{k}\phi)\,d\mu\right|\lesssim\mathcal{N}_{k}(\mu)\|\rho^{\varepsilon}*(\chi_{k}\phi)\|_{L^{\infty}}\lesssim\mathcal{N}_{k}(\mu)\|\phi\|_{L^{\infty}},$

which gives us (2.11).

For (2.12) we take $\phi\in H^{1}$ and apply Plancherel’s Theorem followed by
the Fundamental Theorem of Calculus to obtain

$\displaystyle\bigl{\langle}\chi_{k}\bigl{(}d\mu^{\eta}-d\mu^{\varepsilon}\bigr{)},\phi\bigr{\rangle}$
$\displaystyle=\int_{k-2}^{k+2}\int_{\mathbb{R}}\bigl{[}\rho^{\eta}(x-y)-\rho^{\varepsilon}(x-y)\bigr{]}\phi(x)\chi_{k}(x)\,dx\,d\mu(y)$
$\displaystyle=\int_{k-2}^{k+2}\int_{\mathbb{R}}\int_{\varepsilon}^{\eta}\overline{{\widehat{\rho}{\mkern
3.0mu}}^{\prime}(h\xi)}e^{iy\xi}\xi\,\widehat{(\phi\chi_{k})}(\xi)\,dh\,d\xi\,d\mu(y).$

We then apply the Cauchy–Schwarz inequality, (2.7), and then (2.3) to bound

$\bigl{|}\bigl{\langle}\chi_{k}\bigl{(}d\mu^{\eta}-d\mu^{\varepsilon}\bigr{)},\phi\bigr{\rangle}\bigr{|}\lesssim\mathcal{N}_{k}(\mu)\int_{\varepsilon}^{\eta}\|{\widehat{\rho}{\mkern
3.0mu}}^{\prime}(h\xi)\|_{L_{\xi}^{2}}\|\xi\widehat{(\phi\chi_{k})}\|_{L_{\xi}^{2}}\,dh\lesssim\sqrt{\eta}{\mkern
2.0mu}\mathcal{N}_{k}(\mu)\|\phi\|_{H^{1}}.\qed$

We conclude this section with several nonlinear estimates involving the doubly
mollified density $V^{\varepsilon}$ that was introduced in (1.11):

######  Lemma 2.3.

Let $\mu$ be a locally finite Borel measure on $\mathbb{R}$.

If $\frac{2}{3}\leq s\leq 1$ and $f_{j}\in H^{s}\cap{L_{\mu}^{2}}$, then we
have the estimate

(2.13)
$\displaystyle\int_{\mathbb{R}}\prod_{j=1}^{4}|f_{j}|\,d\mu+\int_{\mathbb{R}}\prod_{j=1}^{4}|f_{j}|\,d\mu^{\varepsilon}+\int_{\mathbb{R}}\prod_{j=1}^{4}|f_{j}|\,V^{\varepsilon}\,dx\lesssim\prod_{j=1}^{4}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{4}}\|f_{j}\|_{H^{s}}^{\frac{3}{4}},$

uniformly for $0<\varepsilon\leq 1$, as well as

(2.14) $\displaystyle\lim_{\varepsilon\to
0}\int_{\mathbb{R}}\prod_{j=1}^{4}f_{j}\,V^{\varepsilon}\,dx$
$\displaystyle=\int_{\mathbb{R}}\prod_{j=1}^{4}f_{j}\,d\mu=\lim_{\varepsilon\to
0}\int_{\mathbb{R}}\prod_{j=1}^{4}f_{j}\,d\mu^{\varepsilon}.$

If $f_{j}\in H^{1}\cap{L_{\mu}^{2}}$ and $g\in C(\mathbb{R})$, then

(2.15)
$\displaystyle\int_{\mathbb{R}}|f_{1}f_{2}\,g|\,d\mu+\bigl{\|}f_{1}f_{2}\,g\,\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\bigr{\|}_{L^{1}}$
$\displaystyle\lesssim\|g\|_{L^{\infty}}\prod_{j=1}^{2}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{2}}\|f_{j}\|_{H^{1}}^{\frac{1}{2}},$
(2.16) $\displaystyle\|f_{1}f_{2}\,g\,d\mu\|_{H^{-1}}$
$\displaystyle\lesssim\|g\|_{L^{\infty}}\prod_{j=1}^{2}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{2}}\|f_{j}\|_{H^{1}}^{\frac{1}{2}},$

uniformly for $0<\varepsilon\leq 1$.

Finally, if $f\in H^{1}\cap{L_{\mu}^{2}}$ and $0<\varepsilon\leq\eta\leq 1$,
then

(2.17)
$\||f|^{2}f(d\mu^{\eta}-d\mu^{\varepsilon})\|_{H^{-1}}\lesssim\sqrt{\eta}\|f\|_{L_{\mu}^{2}}\|f\|_{H^{1}}^{2}.$

###### Proof.

Given $s>\frac{1}{2}$ and $f\in H^{s}\cap{L_{\mu}^{2}}$, we apply the
Gagliardo–Nirenberg inequality (2.1) followed by Hölder’s inequality and then
the equivalent norms (2.5) and (2.9) to estimate

(2.18)
$\displaystyle\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)^{2-\frac{1}{s}}\|\chi_{k}f\|_{L^{\infty}}^{2}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)^{2-\frac{1}{s}}\|\chi_{k}f\|_{L^{2}}^{2-\frac{1}{s}}\|\chi_{k}f\|_{H^{s}}^{\frac{1}{s}}$
$\displaystyle\lesssim\|f\|_{L_{\mu}^{2}}^{2-\frac{1}{s}}\|f\|_{H^{s}}^{\frac{1}{s}}.$

Arguing similarly, if $\frac{2}{3}\leq s\leq 1$ we may bound

$\displaystyle\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)\|\chi_{k}f\|_{L^{\infty}}^{4}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)\|\chi_{k}f\|_{L^{2}}^{4-\frac{2}{s}}\|\chi_{k}f\|_{H^{s}}^{\frac{2}{s}}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)\|\chi_{k}f\|_{L^{2}}\|\chi_{k}f\|_{H^{s}}^{3}$
(2.19)
$\displaystyle\lesssim\left[\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}(\mu)^{2}\|\chi_{k}f\|_{L^{2}}^{2}\right]^{\frac{1}{2}}\left[\sum_{k\in\mathbb{Z}}\|\chi_{k}f\|_{H^{s}}^{6}\right]^{\frac{1}{2}}\lesssim\|f\|_{L_{\mu}^{2}}\|f\|_{H^{s}}^{3}.$

Turning to the estimate (2.13), we first note that by Sobolev embedding we
have $f_{j}\in C(\mathbb{R})$. In particular, for any $k_{j}\in\mathbb{Z}$ we
have $\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\in C_{c}(\mathbb{R})$ and the integral
$\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|\,d\mu$ is well-defined.
Using that $\operatorname{supp}\chi_{k_{j}}\subseteq(k_{j}-1,k_{j}+1)$, we may
then apply (2.7) followed by Hölder’s inequality and (2.19) to obtain

$\displaystyle\sum_{k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}}\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|\,d\mu$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{j}-k_{\ell}|\leq
1\end{subarray}}\prod_{j=1}^{4}\mathcal{N}_{k_{j}}(\mu)^{\frac{1}{4}}\|\chi_{k_{j}}f_{j}\|_{L^{\infty}}$
$\displaystyle\lesssim\prod_{j=1}^{4}\left[\sum_{k_{j}\in\mathbb{Z}}\mathcal{N}_{k_{j}}(\mu)\|\chi_{k_{j}}f_{j}\|_{L^{\infty}}^{4}\right]^{\frac{1}{4}}\\!\\!\lesssim\prod_{j=1}^{4}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{4}}\|f_{j}\|_{H^{s}}^{\frac{3}{4}}.$

Applying the Fubini–Tonelli Theorem, we may then bound

$\displaystyle\left|\int_{\mathbb{R}}\prod_{j=1}^{4}f_{j}\,d\mu\right|$
$\displaystyle=\left|\sum_{k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}}\int_{\mathbb{R}}\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\,d\mu\right|$
$\displaystyle\leq\sum_{k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}}\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|\,d\mu\lesssim\prod_{j=1}^{4}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{4}}\|f_{j}\|_{H^{s}}^{\frac{3}{4}},$

as claimed. For the second and third terms on LHS(2.13), we first use (2.11)
and then (2.7) to estimate

$\displaystyle\sum_{k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}}\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|\,d\mu^{\varepsilon}+\int_{\mathbb{R}}$
$\displaystyle\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|V^{\varepsilon}\,dx$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4},k_{5}\in\mathbb{Z}\\\
|k_{j}-k_{\ell}|\leq
1\end{subarray}}\bigl{\|}\chi_{k_{5}}\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\bigr{\|}_{L^{1}}\prod_{j=1}^{4}\|\chi_{k_{j}}f_{j}\|_{L^{\infty}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{j}-k_{\ell}|\leq
1\end{subarray}}\prod_{j=1}^{4}\mathcal{N}_{k_{j}}(\mu)^{\frac{1}{4}}\|\chi_{k_{j}}f_{j}\|_{L^{\infty}},$

and the proof is then completed as for the first term.

Next we prove the first equality in (2.14); the second follows by the same
argument. We take $K>1$ to be an integer and estimate

$\displaystyle\left|\int_{\mathbb{R}}\prod_{j=1}^{4}f_{j}\,V^{\varepsilon}\,dx-\int_{\mathbb{R}}\prod_{j=1}^{4}f_{j}\,d\mu\right|$
$\displaystyle\qquad\leq\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{1}|\leq K,\ |k_{j}-k_{\ell}|\leq
1\end{subarray}}\left|\int_{\mathbb{R}}\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\,V^{\varepsilon}\,dx-\int_{\mathbb{R}}\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\,d\mu\right|$
$\displaystyle\qquad\quad+\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{1}|>K,\ |k_{j}-k_{\ell}|\leq
1\end{subarray}}\left[\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}||V^{\varepsilon}|\,dx+\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|\,d\mu\right].$

As $\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\in C_{c}(\mathbb{R})$, for any fixed
$k_{1},\ldots,k_{4}\in\mathbb{Z}$ we have

$\lim_{\varepsilon\to
0}\int_{\mathbb{R}}\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\,V^{\varepsilon}\,dx=\int_{\mathbb{R}}\prod_{j=1}^{4}\chi_{k_{j}}f_{j}\,d\mu.$

Moreover, from (2.13) we have

$\displaystyle\sup_{0<\varepsilon\leq
1}\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{1}|>K,|k_{j}-k_{\ell}|\leq
1\end{subarray}}\left[\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}||V^{\varepsilon}|\,dx+\int_{\mathbb{R}}\prod_{j=1}^{4}|\chi_{k_{j}}f_{j}|\,d\mu\right]$
$\displaystyle\qquad\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{1}|>K,|k_{j}-k_{\ell}|\leq
1\end{subarray}}\prod_{j=1}^{4}\|\chi_{k_{j}}f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{4}}\|\chi_{k_{j}}f_{j}\|_{H^{s}}^{\frac{3}{4}}$
$\displaystyle\qquad\lesssim\prod_{j=1}^{4}\left[\sum_{|k_{j}|\geq
K}\|\chi_{k_{j}}f_{j}\|_{L_{\mu}^{2}}\|\chi_{k_{j}}f_{j}\|_{H^{s}}^{3}\right]^{\frac{1}{4}}$
$\displaystyle\qquad\lesssim\prod_{j=1}^{4}\|(1-\varphi^{\frac{2}{K-1}})f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{4}}\|(1-\varphi^{\frac{2}{K-1}})f_{j}\|_{H^{s}}^{\frac{3}{4}},$

where $\varphi^{\frac{2}{K-1}}$ is the smooth cutoff function discussed in
(1.11). Thus (2.14) follows from first taking $\varepsilon\to 0$ and then
$K\to\infty$.

The estimate (2.15) is proved similarly to (2.13). For the first term on
LHS(2.15) we apply (2.7) followed by the Cauchy–Schwarz inequality and (2.18)
with $s=1$ to bound

$\displaystyle\sum_{k_{1},k_{2}\in\mathbb{Z}}\int_{\mathbb{R}}|\chi_{k_{1}}f_{1}||\chi_{k_{2}}f_{2}||g|\,d\mu$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2}\in\mathbb{Z}\\\
|k_{1}-k_{2}|\leq
1\end{subarray}}\sqrt{\mathcal{N}_{k_{1}}\mathcal{N}_{k_{2}}}\|\chi_{k_{1}}f_{1}\|_{L^{\infty}}\|\chi_{k_{2}}f_{2}\|_{L^{\infty}}\|g\|_{L^{\infty}}$
$\displaystyle\lesssim\|g\|_{L^{\infty}}\prod_{j=1}^{2}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{2}}\|f_{j}\|_{H^{1}}^{\frac{1}{2}}.$

For the second term on LHS(2.15) we use (2.11) and (2.7) to estimate

$\displaystyle\sum_{k_{1},k_{2}}\int_{\mathbb{R}}|\chi_{k_{1}}f_{1}||\chi_{k_{2}}f_{2}||g|\bigl{|}\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\bigr{|}\,dx$
$\displaystyle\lesssim\\!\\!\\!\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3}\\\
|k_{j}-k_{\ell}|\leq
1\end{subarray}}\\!\\!\\!\bigl{\|}\chi_{k_{3}}\tfrac{d\mu^{\mkern-1.0mu\varepsilon}\\!\\!}{dx}\,\bigr{\|}_{L^{1}}\|\chi_{k_{1}}f_{1}\|_{L^{\infty}}\|\chi_{k_{2}}f_{2}\|_{L^{\infty}}\|g\|_{L^{\infty}}$
$\displaystyle\lesssim\\!\\!\\!\sum_{\begin{subarray}{c}k_{1},k_{2}\in\mathbb{Z}\\\
|k_{1}-k_{2}|\leq
1\end{subarray}}\\!\\!\\!\\!\sqrt{\mathcal{N}_{k_{1}}\mathcal{N}_{k_{2}}}\|\chi_{k_{1}}f_{1}\|_{L^{\infty}}\|\chi_{k_{2}}f_{2}\|_{L^{\infty}}\|g\|_{L^{\infty}},$

and then proceed as for the first term.

For (2.16) we take $\phi\in H^{1}$ and apply (2.15) followed by (2.1) to bound

$\displaystyle\bigl{|}\langle\phi,f_{1}f_{2}g\,d\mu\rangle\bigr{|}$
$\displaystyle\lesssim\|g\phi\|_{L^{\infty}}\prod_{j=1}^{2}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{2}}\|f_{j}\|_{H^{1}}^{\frac{1}{2}}$
$\displaystyle\lesssim\|g\|_{L^{\infty}}\|\phi\|_{H^{1}}\prod_{j=1}^{2}\|f_{j}\|_{L_{\mu}^{2}}^{\frac{1}{2}}\|f_{j}\|_{H^{1}}^{\frac{1}{2}}.$

The estimate then follows by duality.

Finally, we apply the product estimates (2.4) and (2.2), followed by (2.12),
(2.7), and finally (2.18) to estimate

$\displaystyle\||f|^{2}f(d\mu^{\eta}-d\mu^{\varepsilon})\|_{H^{-1}}$
$\displaystyle\qquad\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3},k_{4}\in\mathbb{Z}\\\
|k_{j}-k_{\ell}|\leq
1\end{subarray}}\|\chi_{k_{1}}f\|_{L^{\infty}}\|\chi_{k_{2}}f\|_{L^{\infty}}\|\chi_{k_{3}}f\|_{H^{1}}\|\chi_{k_{4}}(d\mu^{\eta}-d\mu^{\varepsilon})\|_{H^{-1}}$
$\displaystyle\qquad\lesssim\sum_{\begin{subarray}{c}k_{1},k_{2}\in\mathbb{Z}\\\
|k_{1}-k_{2}|\leq
1\end{subarray}}\sqrt{\eta}\sqrt{\mathcal{N}_{k_{1}}\mathcal{N}_{k_{2}}}\|\chi_{k_{1}}f\|_{L^{\infty}}\|\chi_{k_{2}}f\|_{L^{\infty}}\|f\|_{H^{1}}\lesssim\sqrt{\eta}\,\|f\|_{L_{\mu}^{2}}\|f\|_{H^{1}}^{2},$

which yields (2.17). ∎

##  3\. Deterministic well-posedness

In this section we prove Theorem 1.4. We assume throughout that $\mu$ is a
fixed Borel measure for which $\mathcal{N}_{0}(\mu)<\infty$. We denote the
space $H^{1}$ endowed with the weak topology by $H_{w}^{1}$.

We start by proving a Lipschitz estimate for the solution map of (NLSμ), which
yields uniqueness as an immediate corollary:

######  Proposition 3.1.

Suppose that for some $T>0$ there are two solutions $\psi,\phi\in
C([-T,T];H_{w}^{1}\cap{L_{\mu}^{2}})$ of the Duhamel formulation (1.3) of
(NLSμ). Then for all times $|t|\leq T$ we have the estimate

(3.1) $\|\psi(t)-\phi(t)\|_{H^{-1}}\leq Ce^{CK\langle
t\rangle}\|\psi(0)-\phi(0)\|_{H^{1}},$

where $C>0$ is some absolute constant and

(3.2) $K=\sup_{|t|\leq
T}\Bigl{[}\|\psi\|_{H^{1}}\|\psi\|_{{L_{\mu}^{2}}}+\|\phi\|_{H^{1}}\|\phi\|_{{L_{\mu}^{2}}}\Bigr{]}.$

###### Proof.

Using the Duhamel formula (1.3) we may write the difference as

$\psi(t)-\phi(t)=e^{it\partial_{x}^{2}}\bigl{[}\psi(0)-\phi(0)\bigr{]}-2i\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}\bigl{(}|\psi(\tau)|^{2}\psi(\tau)-|\phi(\tau)|^{2}\phi(\tau)\bigr{)}d\mu\Bigr{]}\,d\tau.$

For the first term, we apply (2.1) to bound

(3.3)
$\bigl{\|}e^{it\partial_{x}^{2}}\bigl{[}\psi(0)-\phi(0)\bigr{]}\bigr{\|}_{L^{\infty}}\lesssim\bigl{\|}e^{it\partial_{x}^{2}}\bigl{[}\psi(0)-\phi(0)\bigr{]}\bigr{\|}_{H^{1}}\lesssim\|\psi(0)-\phi(0)\|_{H^{1}}.$

For the second term, we take a test function $f\in L^{1}$ so that for
$\tau\neq t$ we have $e^{-i(t-\tau)\partial_{x}^{2}}f\in C(\mathbb{R})$ with
the dispersive estimate

(3.4)
$\|e^{-i(t-\tau)\partial_{x}^{2}}f\|_{L^{\infty}}\lesssim\tfrac{1}{\sqrt{|t-\tau|}}\|f\|_{L^{1}}.$

Applying (2.15), for $0\leq\tau<t\leq T$ we may then bound

$\displaystyle\left|\left\langle
f,e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}\bigl{(}|\psi(\tau)|^{2}\psi(\tau)-|\phi(\tau)|^{2}\phi(\tau)\bigr{)}d\mu\Bigr{]}\right\rangle\right|$
$\displaystyle\qquad=\left|\int_{\mathbb{R}}\overline{e^{-i(t-\tau)\partial_{x}^{2}}f}\bigl{(}|\psi(\tau)|^{2}\psi(\tau)-|\phi(\tau)|^{2}\phi(\tau)\bigr{)}\,d\mu\right|$
$\displaystyle\qquad\lesssim\Bigl{[}\|\psi(\tau)\|_{H^{1}}\|\psi(\tau)\|_{L_{\mu}^{2}}+\|\phi(\tau)\|_{H^{1}}\|\phi(\tau)\|_{L_{\mu}^{2}}\Bigr{]}\Bigl{\|}\bigl{(}\psi(\tau)-\phi(\tau)\bigr{)}\overline{e^{-i(t-\tau)\partial_{x}^{2}}f}\Bigr{\|}_{L^{\infty}}$
$\displaystyle\qquad\lesssim\tfrac{1}{\sqrt{t-\tau}}K\|\psi(\tau)-\phi(\tau)\|_{L^{\infty}}\|f\|_{L^{1}},$

where $K$ is defined as in (3.2). By duality, we then have

(3.5)
$\Bigl{\|}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}\bigl{(}|\psi(\tau)|^{2}\psi(\tau)-|\phi(\tau)|^{2}\phi(\tau)\bigr{)}d\mu\Bigr{]}\Bigr{\|}_{L^{\infty}}\lesssim\tfrac{1}{\sqrt{t-\tau}}K\|\psi(\tau)-\phi(\tau)\|_{L^{\infty}}.$

Combining (3.3) and (3.5), for $0<t\leq T$ we may bound

$\|\psi(t)-\phi(t)\|_{L^{\infty}}\lesssim\|\psi(0)-\phi(0)\|_{H^{1}}+K\int_{0}^{t}\tfrac{1}{\sqrt{t-\tau}}\|\psi(\tau)-\phi(\tau)\|_{L^{\infty}}\,d\tau.$

The integral form of Gronwall’s inequality then yields

$\|\psi(t)-\phi(t)\|_{L^{\infty}}\lesssim
e^{CK\sqrt{t}}\|\psi(0)-\phi(0)\|_{H^{1}},$

for some $C>0$.

Finally, we apply (2.16) to estimate

$\displaystyle\|\psi(t)-\phi(t)\|_{H^{-1}}$
$\displaystyle\lesssim\|\psi(0)-\phi(0)\|_{H^{-1}}+\\!\int_{0}^{t}\bigl{\|}\bigl{(}|\psi(\tau)|^{2}\psi(\tau)-|\phi(\tau)|^{2}\phi(\tau)\bigr{)}d\mu\bigr{\|}_{H^{-1}}d\tau$
$\displaystyle\lesssim\|\psi(0)-\phi(0)\|_{H^{1}}+K\\!\int_{0}^{t}\|\psi(\tau)-\phi(\tau)\|_{L^{\infty}}\,d\tau$
$\displaystyle\lesssim\Bigl{[}1+Kte^{CK\sqrt{t}}\Bigr{]}\|\psi(0)-\phi(0)\|_{H^{1}},$

which yields (3.1) for times $0<t\leq T$ (after possibly increasing the size
of the constant $C$). The corresponding estimate for times $-T\leq t<0$
follows from a parallel argument. ∎

Next, we turn to the problem of existence. Taking $V^{\varepsilon}\in
C_{c}^{\infty}(\mathbb{R})$ to be defined as in (1.11), we consider solutions
$\psi^{\varepsilon}\colon\mathbb{R}\times\mathbb{R}\to\mathbb{C}$ of the
regularized equation,

(NLSε) $\displaystyle
i\psi_{t}^{\varepsilon}=-\psi_{xx}^{\varepsilon}+2V^{\varepsilon}|\psi^{\varepsilon}|^{2}\psi^{\varepsilon}.$

The proof of the following proposition can be found in [4], with the exception
of the final statement that follows from the argument of [13]:

######  Proposition 3.2.

For all $\varepsilon>0$ the equation (NLSε) is globally well-posed in $H^{1}$.
More precisely, for all initial data $\psi^{\varepsilon}(0)\in H^{1}$ there
exists a unique solution $\psi^{\varepsilon}\in C(\mathbb{R};H^{1})$ of the
Duhamel formulation

$\psi^{\varepsilon}(t)=e^{it\partial_{x}^{2}}\psi^{\varepsilon}(0)-2i\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}V^{\varepsilon}|\psi^{\varepsilon}(\tau)|^{2}\psi^{\varepsilon}(\tau)\Bigr{]}\,d\tau$

of (NLSε). This solution conserves both the mass and the corresponding energy,

$E^{\varepsilon}[\psi^{\varepsilon}]=\int_{\mathbb{R}}\tfrac{1}{2}|\psi_{x}^{\varepsilon}|^{2}+\tfrac{1}{2}|\psi^{\varepsilon}|^{4}\,V^{\varepsilon}\,dx.$

Moreover, for all times $T>0$ the solution map
$H^{1}\ni\psi^{\varepsilon}(0)\mapsto\psi^{\varepsilon}\in C([-T,T];H^{1})$ is
locally Lipschitz. Finally, if $\psi^{\varepsilon}(0)\in\mathcal{S}$ then
$\psi^{\varepsilon}\in C^{\infty}(\mathbb{R};\mathcal{S})$.

We wish to prove that $\psi^{\varepsilon}(t)$ converges as $\varepsilon\to 0$.
A key first step is to obtain bounds that are uniform for
$\varepsilon\in(0,1]$:

######  Lemma 3.3.

If $\psi_{0}\in H^{1}\cap{L_{\mu}^{2}}$ and $\psi^{\varepsilon}\in
C(\mathbb{R};H^{1})$ is the solution of (NLSε) with initial data
$\psi^{\varepsilon}(0)=\psi_{0}$, then $\psi^{\varepsilon}\in
C(\mathbb{R};{L_{\mu}^{2}})$ and there exists a constant $C>0$ so that for all
$T>0$ we have the estimates

(3.6)
$\displaystyle\sup_{t\in\mathbb{R}}\|\psi^{\varepsilon}(t)\|_{H^{1}}^{2}$
$\displaystyle\leq
C\Bigl{[}1+\|\psi_{0}\|_{L_{\mu}^{2}}\|\psi_{0}\|_{H^{1}}\Bigr{]}\|\psi_{0}\|_{H^{1}}^{2},$
(3.7) $\displaystyle\sup_{|t|\leq
T}\|\psi^{\varepsilon}(t)\|_{{L_{\mu}^{2}}}^{2}$
$\displaystyle\leq\|\psi_{0}\|_{L_{\mu}^{2}}^{2}+CT\sup_{t\in\mathbb{R}}\|\psi^{\varepsilon}(t)\|_{H^{1}}^{2},$
(3.8) $\displaystyle\sup_{|t|\leq
T}\|(1-\varphi^{\lambda})\psi^{\varepsilon}(t)\|_{L^{2}}^{2}$
$\displaystyle\leq\|(1-\varphi^{\lambda})\psi_{0}\|_{L^{2}}^{2}+C\lambda
T\sup_{t\in\mathbb{R}}\|\psi^{\varepsilon}(t)\|_{H^{1}}^{2},$ (3.9)
$\displaystyle\sup_{|t|\leq
T}\|(1-\varphi^{\lambda})\psi^{\varepsilon}(t)\|_{{L_{\mu}^{2}}}^{2}$
$\displaystyle\leq\|(1-\varphi^{\lambda})\psi_{0}\|_{L_{\mu}^{2}}^{2}$
$\displaystyle\quad+CT\langle\mathcal{N}_{0}^{2}\lambda\rangle\sup_{|t|\leq
T}\Bigl{[}\|(1-\varphi^{\lambda})\psi^{\varepsilon}(t)\|_{L^{2}}\|\psi^{\varepsilon}(t)\|_{H^{1}}\Bigr{]},$

uniformly for $0<\varepsilon,\lambda\leq 1$.

###### Proof.

Combining an approximation argument with the well-posedness provided by
Proposition 3.2, we see that it suffices to prove these bounds for
$\psi_{0}\in\mathcal{S}$. As noted in Proposition 3.2, this ensures that that
the solution $\psi^{\varepsilon}\in C^{\infty}(\mathbb{R};\mathcal{S})$.

By conservation of mass and energy,

$\displaystyle\sup_{0<\varepsilon\leq
1}\sup_{t\in\mathbb{R}}\|\psi^{\varepsilon}(t)\|_{H^{1}}^{2}\lesssim\sup_{0<\varepsilon\leq
1}\sup_{t\in\mathbb{R}}\Bigl{[}M[\psi^{\varepsilon}(t)]+E^{\varepsilon}[\psi^{\varepsilon}(t)]\Bigr{]}$
$\displaystyle=M[\psi_{0}]+\sup_{0<\varepsilon\leq
1}E^{\varepsilon}[\psi_{0}].$

Thus by (2.13), we have

$\displaystyle\sup_{0<\varepsilon\leq
1}\sup_{t\in\mathbb{R}}\|\psi^{\varepsilon}(t)\|_{H^{1}}^{2}$
$\displaystyle\lesssim\Bigl{[}1+\|\psi_{0}\|_{L_{\mu}^{2}}\|\psi_{0}\|_{H^{1}}\Bigr{]}\|\psi_{0}\|_{H^{1}}^{2},$

which proves (3.6).

As $\psi^{\varepsilon}(t)\in\mathcal{S}$, the estimate (2.8) ensures that
$\|\psi^{\varepsilon}(t)\|_{L_{\mu}^{2}}<\infty$ for all $t\in\mathbb{R}$. We
may then integrate by parts to obtain

(3.10)
$\tfrac{d}{dt}\|\psi^{\varepsilon}\|_{L_{\mu}^{2}}^{2}=-2\operatorname{Im}\int_{\mathbb{R}}\bigl{(}\overline{\psi^{\varepsilon}}\psi_{x}^{\varepsilon}\bigr{)}_{x}\,\omega\,dx=2\operatorname{Im}\int_{\mathbb{R}}\overline{\psi^{\varepsilon}}\psi_{x}^{\varepsilon}\,\omega_{x}\,dx,$

and so, by (2.10),

$\sup_{|t|\leq
T}\|\psi^{\varepsilon}(t)\|_{L_{\mu}^{2}}^{2}\leq\|\psi_{0}\|_{L_{\mu}^{2}}^{2}+2T\sup_{|t|\leq
T}\|\psi^{\varepsilon}(t)\|_{L^{2}}\|\psi^{\varepsilon}(t)\|_{H^{1}},$

which proves (3.7). The estimates (3.8) and (3.9) follow from a parallel
argument, with $\omega$ replaced by $(1-\varphi^{\lambda})^{2}$ and
$(1-\varphi^{\lambda})^{2}\omega$, respectively. In the second instance, we
also use that

$\bigl{|}[(1-\varphi^{\lambda})^{2}\omega]_{x}(x)\bigr{|}\lesssim(1-\varphi^{\lambda}(x))\bigl{[}1+\lambda|\varphi_{x}(\lambda
x)||\omega(x)|\bigr{]}\lesssim\langle\mathcal{N}_{0}^{2}\lambda\rangle(1-\varphi^{\lambda}(x)),$

which follows from (2.8) and (2.10).

Given initial data $\psi_{0}\in H^{1}\cap{L_{\mu}^{2}}$, the estimate (3.7)
ensures that the solution $\psi^{\varepsilon}\in C(\mathbb{R};H^{1})\cap
L^{\infty}([-T,T];{L_{\mu}^{2}})$. To complete the proof we have to verify
that $\psi\in C([-T,T];{L_{\mu}^{2}})$. To this end, we take
$t,\tau\in[-T,T]$, $0<\lambda\leq 1$, and use (2.8) to bound

$\displaystyle\|\psi^{\varepsilon}(t)-\psi^{\varepsilon}(\tau)\|_{L_{\mu}^{2}}$
$\displaystyle\leq\|\varphi^{\lambda}[\psi^{\varepsilon}(t)-\psi^{\varepsilon}(\tau)]\|_{L_{\mu}^{2}}+2\sup_{|t|\leq
T}\|(1-\varphi^{\lambda})\psi^{\varepsilon}(t)\|_{L_{\mu}^{2}}$
$\displaystyle\lesssim\bigl{(}\mathcal{N}_{0}+\tfrac{1}{\sqrt{\lambda}}\bigr{)}\|\psi^{\varepsilon}(t)-\psi^{\varepsilon}(\tau)\|_{L^{2}}+\sup_{|t|\leq
T}\|(1-\varphi^{\lambda})\psi^{\varepsilon}(t)\|_{L_{\mu}^{2}}.$

As the map $\mathbb{R}\ni t\mapsto\psi^{\varepsilon}(t)$ is
$L^{2}$-continuous, we deduce that

$\limsup_{t\to\tau}\|\psi^{\varepsilon}(t)-\psi^{\varepsilon}(\tau)\|_{L_{\mu}^{2}}\lesssim\sup_{|t|\leq
T}\|(1-\varphi^{\lambda})\psi^{\varepsilon}(t)\|_{L_{\mu}^{2}}.$

Employing (3.6) and (3.8) in (3.9), we send $\lambda\to 0$ and obtain

$\lim_{t\to\tau}\|\psi^{\varepsilon}(t)-\psi^{\varepsilon}(\tau)\|_{L_{\mu}^{2}}=0.\qed$

Using these uniform estimates for $\psi^{\varepsilon}$, we now prove existence
of local-in-time solutions to (NLSμ):

######  Proposition 3.4.

Given $\psi_{0}\in H^{1}\cap{L_{\mu}^{2}}$, there exists $T>0$ and a solution
$\psi\in C([-T,T];H_{w}^{1}\cap H^{s}\cap{L_{\mu}^{2}})$, for every $0\leq
s<1$, to the Duhamel formulation (1.3) of (NLSμ). The solution $\psi(t)$
conserves mass and satisfies the energy inequality

(3.11) $E\bigl{[}\psi(t)\bigr{]}\leq E\bigl{[}\psi_{0}\bigr{]},$

for all $|t|\leq T$. Moreover, if $\psi^{\varepsilon}$ is a family of
solutions to (NLSε) with $\psi^{\varepsilon}(0)\to\psi_{0}$ in
$H^{1}\cap{L_{\mu}^{2}}$, then

(3.12) $\lim_{\varepsilon\to 0}\ \sup_{|t|\leq T}\
\bigl{\|}\psi^{\varepsilon}(t)-\psi(t)\bigr{\|}_{H^{s}\cap{L_{\mu}^{2}}}=0\quad\text{for
each}\quad 0\leq s<1.$

###### Proof.

We will construct $\psi(t)$ as a limit of solutions $\psi^{\varepsilon}(t)$ to
(NLSε). For this purpose, it would suffice to consider solutions
$\psi^{\varepsilon}$ to (NLSε) with $\psi^{\varepsilon}(0)=\psi_{0}$, rather
than with a sequence of initial data $\psi^{\varepsilon}(0)\to\psi_{0}$ in
$H^{1}\cap{L_{\mu}^{2}}$. However, the extra generality we consider here will
be useful later. It is no loss of generality to demand that

(3.13) $\sup_{0<\varepsilon\leq
1}\|\psi^{\varepsilon}(0)\|_{H^{1}}\leq\|\psi_{0}\|_{H^{1}}+1\quad\text{and
}\quad\sup_{0<\varepsilon\leq
1}\|\psi^{\varepsilon}(0)\|_{{L_{\mu}^{2}}}\leq\|\psi_{0}\|_{{L_{\mu}^{2}}}+1.$

Given $T>0$, we define the control quantities

$A=\sup_{0<\varepsilon\leq 1}\sup_{|t|\leq
T}\|\psi^{\varepsilon}(t)\|_{H^{1}}\quad\text{and}\quad
B=\sup_{0<\varepsilon\leq 1}\sup_{|t|\leq
T}\|\psi^{\varepsilon}(t)\|_{L_{\mu}^{2}}.$

Recalling (3.6) and (3.13), we have

(3.14) $\displaystyle A^{2}$
$\displaystyle\lesssim\bigl{[}1+\|\psi_{0}\|_{L_{\mu}^{2}}\bigr{]}\bigl{[}1+\|\psi_{0}\|_{H^{1}}\bigr{]}^{3},$

and using (3.7) as well, we have

(3.15) $B^{2}\leq\sup_{0<\varepsilon\leq
1}\|\psi^{\varepsilon}(0)\|_{L_{\mu}^{2}}^{2}+CTA^{2}\lesssim[1+T]\bigl{[}1+\|\psi_{0}\|_{L_{\mu}^{2}}\bigr{]}^{2}\bigl{[}1+\|\psi_{0}\|_{H^{1}}\bigr{]}^{3}.$

Taking $0<\varepsilon\leq\eta\leq 1$, we use the Duhamel formula (1.3) to
write

$\displaystyle\psi^{\eta}(t)-\psi^{\varepsilon}(t)$
$\displaystyle=e^{it\partial_{x}^{2}}[\psi^{\eta}(0)-\psi^{\varepsilon}(0)]+I+I\\!I+I\\!I\\!I,$

where

$\displaystyle I$
$\displaystyle:=-2i\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}\bigl{(}\varphi^{\eta}-\varphi^{\varepsilon}\bigr{)}|\psi^{\eta}(\tau)|^{2}\psi^{\eta}(\tau)d\mu^{\eta}\Bigr{]}\,d\tau,$
$\displaystyle I\\!I$
$\displaystyle:=-2i\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}\varphi^{\varepsilon}|\psi^{\eta}(\tau)|^{2}\psi^{\eta}(\tau)\bigl{(}d\mu^{\eta}-d\mu^{\varepsilon}\bigr{)}\Bigr{]}\,d\tau,$
$\displaystyle I\\!I\\!I$
$\displaystyle:=-2i\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}\varphi^{\varepsilon}\bigl{[}|\psi^{\eta}(\tau)|^{2}\psi^{\eta}(\tau)-|\psi^{\varepsilon}(\tau)|^{2}\psi^{\varepsilon}(\tau)\bigr{]}d\mu^{\varepsilon}\Bigr{]}\,d\tau.$

Given dyadic $N\geq 1$ we define the Littlewood–Paley projection

$\widehat{P_{\leq
N}f}(\xi)=\varphi\bigl{(}\tfrac{\xi}{N}\bigr{)}\widehat{f}(\xi),$

where $\varphi$ is the same bump function $\varphi\in C_{c}^{\infty}(-2,2)$
employed in (1.11). We also use the notation $P_{>N}=1-P_{\leq N}$. We begin
by bounding the low frequency part of each of the terms $I,I\\!I,I\\!I\\!I$ in
both $L^{\infty}$ and $H^{s}$ for $0\leq s<1$.

Using the dispersive estimate (3.4) and then (2.15), we bound the first term
by

$\displaystyle\sup_{|t|\leq T}\|P_{\leq N}I\|_{L^{\infty}}$
$\displaystyle\lesssim\sup_{|t|\leq
T}\left|\int_{0}^{t}\tfrac{1}{\sqrt{|t-\tau|}}\bigl{\|}(\varphi^{\eta}-\varphi^{\varepsilon}\bigr{)}|\psi^{\eta}(\tau)|^{2}\psi^{\eta}(\tau)\tfrac{d\mu^{\mkern-1.0mu\eta}\\!\\!}{dx}\,\bigr{\|}_{L^{1}}\,d\tau\right|$
$\displaystyle\lesssim\sqrt{T}AB\sup_{|t|\leq
T}\|(\varphi^{\eta}-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{L^{\infty}}.$

Replacing the dispersive estimate by Bernstein’s inequality, we similarly
obtain

$\displaystyle\sup_{|t|\leq T}\|P_{\leq N}I\|_{H^{s}}$ $\displaystyle\lesssim
N^{s+\frac{1}{2}}T\sup_{|t|\leq
T}\bigl{\|}(\varphi^{\eta}-\varphi^{\varepsilon}\bigr{)}|\psi^{\eta}(t)|^{2}\psi^{\eta}(t)\tfrac{d\mu^{\mkern-1.0mu\eta}\\!\\!}{dx}\,\bigr{\|}_{L^{1}}$
$\displaystyle\lesssim N^{s+\frac{1}{2}}TAB\sup_{|t|\leq
T}\|(\varphi^{\eta}-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{L^{\infty}},$

for any $0\leq s<1$.

The estimates for the third term are essentially identical. Applying the
dispersive estimate (3.4) followed by (2.15) we get

$\displaystyle\sup_{|t|\leq T}\|P_{\leq
N}I\\!I\\!I\|_{L^{\infty}}\lesssim\sqrt{T}AB\sup_{|t|\leq
T}\|\psi^{\eta}(t)-\psi^{\varepsilon}(t)\|_{L^{\infty}},$

while replacing the dispersive estimate by Bernstein’s inequality yields

$\sup_{|t|\leq T}\|P_{\leq N}I\\!I\\!I\|_{H^{s}}\lesssim
N^{s+\frac{1}{2}}TAB\sup_{|t|\leq
T}\|\psi^{\eta}(t)-\psi^{\varepsilon}(t)\|_{L^{\infty}}.$

For the remaining term, we first apply (2.17) to bound

$\displaystyle\sup_{|t|\leq T}\|P_{\leq N}I\\!I\|_{H^{-1}}$
$\displaystyle\lesssim T\sup_{|t|\leq
T}\bigl{\|}|\psi^{\eta}(t)|^{2}\psi^{\eta}(t)\bigl{(}d\mu^{\eta}-d\mu^{\varepsilon}\bigr{)}\bigr{\|}_{H^{-1}}\lesssim
TA^{2}B\sqrt{\eta}.$

We may then use Bernstein’s inequality to get

$\displaystyle\sup_{|t|\leq T}\|P_{\leq N}I\\!I\|_{L^{\infty}}$
$\displaystyle\lesssim N^{\frac{3}{2}}TA^{2}B\sqrt{\eta},$
$\displaystyle\sup_{|t|\leq T}\|P_{\leq N}I\\!I\|_{H^{s}}$
$\displaystyle\lesssim N^{s+1}TA^{2}B\sqrt{\eta}.$

We now combine our estimates at low frequencies with Bernstein’s inequality at
high frequencies and (2.1) to bound

$\displaystyle\sup_{|t|\leq T}\|\psi^{\eta}(t)$
$\displaystyle-\psi^{\varepsilon}(t)\|_{L^{\infty}}$
$\displaystyle\quad\lesssim\|\psi^{\eta}(0)-\psi^{\varepsilon}(0)\|_{H^{1}}+\sup_{|t|\leq
T}\|P_{>N}\psi^{\eta}(t)\|_{L^{\infty}}+\sup_{|t|\leq
T}\|P_{>N}\psi^{\varepsilon}(t)\|_{L^{\infty}}$
$\displaystyle\quad\qquad+\sup_{|t|\leq T}\|P_{\leq
N}I\|_{L^{\infty}}+\sup_{|t|\leq T}\|P_{\leq
N}I\\!I\|_{L^{\infty}}+\sup_{|t|\leq T}\|P_{\leq N}I\\!I\\!I\|_{L^{\infty}}$
(3.16)
$\displaystyle\quad\lesssim\|\psi^{\eta}(0)-\psi^{\varepsilon}(0)\|_{H^{1}}+N^{-\frac{1}{2}}A+\sqrt{T}AB\sup_{|t|\leq
T}\|(\varphi^{\eta}-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{L^{\infty}}$
$\displaystyle\quad\qquad+N^{\frac{3}{2}}TA^{2}B\sqrt{\eta}+\sqrt{T}AB\sup_{|t|\leq
T}\|\psi^{\eta}(t)-\psi^{\varepsilon}(t)\|_{L^{\infty}}.$

We now choose $T>0$ sufficiently small (depending only on $\psi_{0}$) to
ensure that

$\sqrt{T}AB\ll 1,$

and hence the final term on RHS(3.16) can be absorbed into the left-hand side.
From (2.1) and (3.8), we may bound the third term on RHS(3.16) as follows:

$\displaystyle\sup_{|t|\leq
T}\|(\varphi^{\eta}-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{L^{\infty}}^{4}$
$\displaystyle\qquad\lesssim\sup_{|t|\leq
T}\biggl{\\{}\Bigl{[}\|(1-\varphi^{\eta})\psi^{\eta}(t)\|_{L^{2}}^{2}+\|(1-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{L^{2}}^{2}\Bigr{]}\|(\varphi^{\eta}-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{H^{1}}^{2}\biggr{\\}}$
$\displaystyle\qquad\lesssim\Bigl{[}\|(1-\varphi^{\eta})\psi_{0}\|_{L^{2}}^{2}+\|(1-\varphi^{\varepsilon})\psi_{0}\|_{L^{2}}^{2}+\eta
TA^{2}\Bigr{]}A^{2}.$

This converges to zero as $\eta,\varepsilon\to 0$. As a consequence, we may
pass to the limit in (3.16) to obtain

$\limsup_{\eta,\varepsilon\to 0}\sup_{|t|\leq
T}\|\psi^{\eta}(t)-\psi^{\varepsilon}(t)\|_{L^{\infty}}\lesssim
N^{-\frac{1}{2}}A.$

Taking the limit as $N\to\infty$, we see that $\psi^{\varepsilon}$ converges
in $C([-T,T];L^{\infty})$ as $\varepsilon\to 0$. Let $\psi$ denote this limit.

Arguing as in (3.16) and choosing $T$ in the same way, but using our $H^{s}$
low frequency bounds in place of those in $L^{\infty}$, we get

$\displaystyle\sup_{|t|\leq
T}\|\psi^{\eta}(t)-\psi^{\varepsilon}(t)\|_{H^{s}}$
$\displaystyle\lesssim\|\psi^{\eta}(0)-\psi^{\varepsilon}(0)\|_{H^{s}}+N^{s-1}A$
$\displaystyle\quad+N^{s+\frac{1}{2}}\sqrt{T}\sup_{|t|\leq
T}\|(\varphi^{\eta}-\varphi^{\varepsilon})\psi^{\eta}(t)\|_{L^{\infty}}$
$\displaystyle\quad+N^{s+1}\sqrt{T}A\sqrt{\eta}+N^{s+\frac{1}{2}}\sqrt{T}\sup_{|t|\leq
T}\|\psi^{\eta}(t)-\psi^{\varepsilon}(t)\|_{L^{\infty}}.$

Passing to the limit as $\eta,\varepsilon\to 0$ and then $N\to\infty$ now
shows that $\psi^{\varepsilon}\to\psi$ in $C([-T,T];H^{s})$ for all $0\leq
s<1$.

As the solutions $\psi^{\varepsilon}$ conserve mass, we may pass to the limit
as $\varepsilon\to 0$ to obtain

$\|\psi(t)\|_{L^{2}}=\|\psi_{0}\|_{L^{2}}\quad\text{for all $|t|\leq T$}.$

In view of (2.8) and the $C([-T,T];L^{2})$ convergence already proved,

$\limsup_{\eta,\varepsilon\to 0}\sup_{|t|\leq
T}\bigl{\|}\varphi^{\lambda}(\psi^{\eta}(t)-\psi^{\varepsilon}(t))\bigr{\|}_{{L_{\mu}^{2}}}^{2}\lesssim\Bigl{(}\mathcal{N}_{0}^{2}+\tfrac{1}{\lambda}\Bigr{)}\limsup_{\eta,\varepsilon\to
0}\bigl{\|}\psi^{\eta}-\psi^{\varepsilon}\bigr{\|}_{C_{t}L^{2}}^{2}=0$

for any fixed $\lambda>0$. Thus, in order to demonstrate that
$\psi^{\varepsilon}\to\psi$ in $C([-T,T];{L_{\mu}^{2}})$, it suffices to show
that

(3.17) $\limsup_{\lambda\to 0}\ \sup_{0<\varepsilon\leq 1}\ \sup_{|t|\leq
T}\bigl{\|}[1-\varphi^{\lambda}]\psi^{\varepsilon}(t)\bigr{\|}_{{L_{\mu}^{2}}}^{2}=0.$

To this end, we first employ (3.8) to obtain

$\limsup_{\lambda\to 0}\ \sup_{0<\varepsilon\leq 1}\ \sup_{|t|\leq
T}\bigl{\|}[1-\varphi^{\lambda}]\psi^{\varepsilon}(t)\bigr{\|}_{L^{2}}^{2}\leq\limsup_{\lambda\to
0}\ \sup_{0<\varepsilon\leq
1}\bigl{\|}[1-\varphi^{\lambda}]\psi^{\varepsilon}(0)\bigr{\|}_{L^{2}}^{2}=0.$

Using this in (3.9), we may then conclude that

$\text{LHS\eqref{777}}\leq\limsup_{\lambda\to 0}\ \sup_{0\leq\varepsilon\leq
1}\bigl{\|}[1-\varphi^{\lambda}]\psi^{\varepsilon}(0)\bigr{\|}_{{L_{\mu}^{2}}}^{2}=0.$

Weak compactness ensures that for each time $t\in[-T,T]$ we have weak
convergence $\psi^{\varepsilon}(t)\rightharpoonup\psi(t)$ in $H^{1}$. As
$\psi^{\varepsilon}\in C([-T,T];H^{1})$, it follows that $\psi\in
C([-T,T];H_{w}^{1})$.

Fixing $\frac{2}{3}\leq s<1$, we use (2.13) to obtain

$\displaystyle\left|\int_{\mathbb{R}}|\psi^{\varepsilon}|^{4}\,V^{\varepsilon}\,dx-\int_{\mathbb{R}}|\psi|^{4}\,d\mu\right|$
$\displaystyle\qquad\leq\left|\int_{\mathbb{R}}\Bigl{[}|\psi^{\varepsilon}|^{4}-|\psi|^{4}\Bigr{]}\,V^{\varepsilon}\,dx\right|+\left|\int_{\mathbb{R}}|\psi|^{4}\,V^{\varepsilon}\,dx-\int_{\mathbb{R}}|\psi|^{4}\,d\mu\right|$
$\displaystyle\qquad\lesssim\Bigl{[}\|\psi^{\varepsilon}\|_{H^{s}\cap{L_{\mu}^{2}}}^{3}+\|\psi\|_{H^{s}\cap{L_{\mu}^{2}}}^{3}\Bigr{]}\|\psi^{\varepsilon}-\psi\|_{H^{s}\cap{L_{\mu}^{2}}}+\left|\int_{\mathbb{R}}|\psi|^{4}\,V^{\varepsilon}\,dx-\int_{\mathbb{R}}|\psi|^{4}\,d\mu\right|,$

which then converges to zero as $\varepsilon\to 0$ by (2.14). As a
consequence, for all $|t|\leq T$ we have

(3.18) $\displaystyle E\bigl{[}\psi(t)\bigr{]}$
$\displaystyle\leq\liminf_{\varepsilon\to
0}E^{\varepsilon}[\psi^{\varepsilon}(t)]=\liminf_{\varepsilon\to
0}E^{\varepsilon}\bigl{[}\psi_{0}\bigr{]}=E\bigl{[}\psi_{0}\bigr{]},$

giving us the energy inequality (3.11).

To complete the proof, we have one final task: verifying that $\psi$ is a
solution to (1.3). Given $\phi\in\mathcal{S}$, we may bound

$\|e^{-i(t-\tau)\partial_{x}^{2}}\phi\|_{H^{s}}=\|\phi\|_{H^{s}}\quad\text{and}\quad\||x|e^{-i(t-\tau)\partial_{x}^{2}}\phi\|_{L^{2}}\lesssim\||x|\phi\|_{L^{2}}+|t-\tau|\|\phi\|_{H^{1}},$

so (2.8) yields

$\|e^{-i(t-\tau)\partial_{x}^{2}}\phi\|_{H^{s}\cap{L_{\mu}^{2}}}\lesssim_{\phi}\mathcal{N}_{0}+\sqrt{|t-\tau|}.$

We may then argue as in the proof of (3.11), using (2.13) and (2.14) to show
that for all $|t|\leq T$ we have

$\displaystyle\lim_{\varepsilon\to
0}\left\langle\phi,\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}V^{\varepsilon}|\psi^{\varepsilon}(\tau)|^{2}\psi^{\varepsilon}(\tau)\Bigr{]}\,d\tau\right\rangle$
$\displaystyle\qquad=\lim_{\varepsilon\to
0}\int_{0}^{t}\int_{\mathbb{R}}\overline{e^{-i(t-\tau)\partial_{x}^{2}}\phi}\,V^{\varepsilon}|\psi^{\varepsilon}(\tau)|^{2}\psi^{\varepsilon}(\tau)\,dx\,d\tau$
$\displaystyle\qquad=\int_{0}^{t}\int_{\mathbb{R}}\overline{e^{-i(t-\tau)\partial_{x}^{2}}\phi}\,|\psi(\tau)|^{2}\psi(\tau)\,d\mu\,d\tau$
$\displaystyle\qquad=\left\langle\phi,\int_{0}^{t}e^{i(t-\tau)\partial_{x}^{2}}\Bigl{[}|\psi(\tau)|^{2}\psi(\tau)d\mu\Bigr{]}\,d\tau\right\rangle,$

so $\psi$ satisfies (1.3) in the sense of (tempered) distributions. ∎

To complete the proof of the existence of solutions to (NLSμ) as stated in
Theorem 1.4, we must extend the local-in-time solution constructed in
Proposition 3.4 to a global-in-time solution. We then combine the uniqueness
result from Proposition 3.1 with the energy inequality (3.11) to improve the
time continuity properties of our solution. Following this strategy, we will
prove the following:

######  Proposition 3.5.

Given $\psi_{0}\in H^{1}\cap{L_{\mu}^{2}}$ there exists a unique strong
solution $\psi\in C(\mathbb{R};H^{1}\cap{L_{\mu}^{2}})$ of (NLSμ). This
solution conserves both the mass and the energy. Moreover, if
$\psi^{\varepsilon}$ denote the solutions to (NLSε) with initial data
$\psi_{0}$, then

(3.19) $\lim_{\varepsilon\to 0}\ \sup_{|t|\leq T}\
\bigl{\|}\psi^{\varepsilon}(t)-\psi(t)\bigr{\|}_{H^{1}\cap{L_{\mu}^{2}}}=0\quad\text{for
each}\quad 0<T<\infty.$

###### Proof.

We first note that uniqueness follows from Proposition 3.1.

Turning to the problem of global existence, let $I\subseteq(0,\infty)$ be the
set of times $T>0$ for which there exists a solution $\psi\in
C([-T,T];H_{w}^{1}\cap H^{s}\cap{L_{\mu}^{2}})$ for all $0\leq s<1$ of (NLSμ)
with $\psi(0)=\psi_{0}$, which conserves mass and satisfies the energy
inequality (3.11).

By definition, $I$ is (relatively) closed. Proposition 3.4 shows that $I$ is
non-empty. Moreover, if $T\in I$ then we may apply Proposition 3.4 with
initial data $\psi(\pm T)\in H^{1}\cap{L_{\mu}^{2}}$ to show that $I$ is open.
As a consequence, $I=(0,\infty)$, yielding global existence.

Let us now suppose for a contradiction that there exists some time $T\neq 0$
for which

$E\bigl{[}\psi(T)\bigr{]}<E\bigl{[}\psi_{0}\bigr{]}.$

We may then define

$\phi(t,x)=\psi(T+t,x)$

to obtain a solution $\phi\in C(\mathbb{R};H_{w}^{1}\cap
H^{s}\cap{L_{\mu}^{2}})$ for all $0\leq s<1$ of (NLSμ) for which

$E\bigl{[}\phi(-T)\bigr{]}>E\bigl{[}\phi(0)\bigr{]},$

contradicting the fact that the unique solution of (NLSμ) with initial data
$\phi(0)$ satisfies the energy inequality (3.11). As a consequence, we must
have

$E\bigl{[}\psi(t)\bigr{]}=E\bigl{[}\psi_{0}\bigr{]}$

for all times $t\in\mathbb{R}$.

From (2.13) and the fact that $\psi\in C(\mathbb{R};H^{s}\cap{L_{\mu}^{2}})$
for $\frac{2}{3}\leq s<1$, the map

$t\mapsto\int_{\mathbb{R}}|\psi(t)|^{4}\,d\mu$

is continuous on $\mathbb{R}$. Combining this with the conservation of energy
we see that for all $t_{0}\in\mathbb{R}$ we have

$\lim_{t\to t_{0}}\|\psi(t)\|_{H^{1}}^{2}=\lim_{t\to
t_{0}}\left[M[\psi(t)]+2E[\psi(t)]-\int_{\mathbb{R}}|\psi(t)|^{4}\,d\mu\right]=\|\psi(t_{0})\|_{H^{1}}^{2}.$

Consequently, the map $\mathbb{R}\ni t\mapsto\|\psi(t)\|_{H^{1}}$ is
continuous. As $\psi\in C(\mathbb{R};H_{w}^{1})$, by the Radon–Riesz theorem
we then have $\psi\in C(\mathbb{R};H^{1})$.

It remains only to prove (3.19). From Proposition 3.4, we obtain convergence
in a short time interval $[-T,T]$, but only in the weaker
$C([-T,T];H^{s}\cap{L_{\mu}^{2}})$ norm. Nevertheless, this allows us to
deduce weak convergence
$\psi^{\varepsilon}(t_{\varepsilon})\rightharpoonup\psi(t)$ in $H^{1}$,
whenever $t_{\varepsilon}\to t\in[-T,T]$. We will demonstrate how to use
conservation of energy to deduce convergence in $C([-T,T];H^{1})$ . This
combination of Proposition 3.4 and upgraded convergence can then be iterated
to cover any compact time interval.

If convergence of $\psi^{\varepsilon}$ to $\psi$ in $C([-T,T];H^{1})$ were to
fail, then (passing to a subsequence) we could find $t_{\varepsilon}\to
t\in[-T,T]$ such that

(3.20)
$\displaystyle\inf_{\varepsilon}\bigl{\|}\psi^{\varepsilon}(t_{\varepsilon})-\psi(t)\bigr{\|}_{H^{1}}>0.$

It is important here that we have already showed that $\psi\in
C(\mathbb{R};H^{1})$.

By energy conservation and (2.14), we have

$E[\psi(t)]=E[\psi_{0}]=\lim_{\varepsilon\to
0}E^{\varepsilon}[\psi_{0}]=\lim_{\varepsilon\to
0}E^{\varepsilon}[\psi^{\varepsilon}(t_{\varepsilon})].$

Combining this with the convergence of the potential energies (see the proof
of (3.18)), we deduce that the $H^{1}$ norms of
$\psi^{\varepsilon}(t_{\varepsilon})$ converge to that of $\psi(t)$. By the
Radon–Riesz theorem and the weak convergence noted above, this implies norm
convergence, contradicting (3.20). ∎

To complete the proof of Theorem 1.4, it remains to prove that the solution
map is continuous.

######  Proposition 3.6.

If $\psi_{n,0}\to\psi_{0}$ is a convergent sequence of
$H^{1}\cap{L_{\mu}^{2}}$ initial data with corresponding solutions
$\psi_{n},\psi\in C(\mathbb{R};H^{1}\cap{L_{\mu}^{2}})$, then for any time
$T>0$ we have

$\lim_{n\to\infty}\sup_{|t|\leq
T}\|\psi_{n}(t)-\psi(t)\|_{H^{1}\cap{L_{\mu}^{2}}}=0.$

###### Proof.

Our starting point is Proposition 3.1, which yields

$\lim_{n\to\infty}\sup_{|t|\leq T}\|\psi_{n}(t)-\psi(t)\|_{H^{-1}}=0.$

Next, we upgrade this to convergence in $C([-T,T];H^{\frac{2}{3}})$. From the
conservation of mass and energy followed by (2.13), we may bound

$M[\psi_{n}(t)]+E\bigl{[}\psi_{n}(t)\bigr{]}=M[\psi_{n,0}]+E\bigl{[}\psi_{n,0}\bigr{]}\lesssim\Bigl{[}1+\|\psi_{n,0}\|_{L_{\mu}^{2}}\|\psi_{n,0}\|_{L^{2}}\Bigr{]}\|\psi_{n,0}\|_{H^{1}}^{2},$

from which we obtain the uniform bound

(3.21) $\sup_{n\geq 1}\sup_{|t|\leq
T}\|\psi_{n}(t)\|_{H^{1}}^{2}\leq\sup_{n\geq 1}\sup_{|t|\leq
T}\Bigl{[}M[\psi_{n}(t)]+2E[\psi_{n}(t)]\Bigr{]}<\infty.$

As a consequence, for dyadic $N\geq 1$ we may bound

$\displaystyle\sup_{|t|\leq T}\|\psi_{n}(t)-\psi(t)\|_{H^{\frac{2}{3}}}$
$\displaystyle\leq\sup_{|t|\leq T}\|P_{\leq
N}(\psi_{n}(t)-\psi(t))\|_{H^{\frac{2}{3}}}+\sup_{|t|\leq
T}\|P_{>N}\psi_{n}(t)\|_{H^{\frac{2}{3}}}$ $\displaystyle\quad+\sup_{|t|\leq
T}\|P_{>N}\psi(t)\|_{H^{\frac{2}{3}}}$ $\displaystyle\lesssim
N^{\frac{5}{3}}\sup_{|t|\leq
T}\|\psi_{n}(t)-\psi(t)\|_{H^{-1}}+N^{-\frac{1}{3}}\sup_{n\geq 1}\sup_{|t|\leq
T}\|\psi_{n}(t)\|_{H^{1}}$ $\displaystyle\quad+N^{-\frac{1}{3}}\sup_{|t|\leq
T}\|\psi(t)\|_{H^{1}}.$

By first taking $n\to\infty$ and then $N\to\infty$, we see that
$\psi_{n}\to\psi$ in $C([-T,T];H^{\frac{2}{3}})$.

We recall that a compact subset $Q\subseteq H^{-1}$ is bounded and
equicontinuous, and consequently

$\lim_{N\to\infty}\sup_{\phi\in Q}\|P_{>N}\phi\|_{H^{-1}}=0;$

see, e.g., [8, 11]. Mimicking our argument for convergence in
$C([-T,T];H^{\frac{2}{3}})$, we see that for any compact subset $Q\subseteq
H^{-1}$ we have

$\displaystyle\sup_{|t|\leq T}\sup_{\phi\in
Q}|\langle\phi,\psi_{n}(t)-\psi(t)\rangle|$
$\displaystyle\qquad\leq\sup_{|t|\leq T}\sup_{\phi\in Q}|\langle\phi,P_{\leq
N}(\psi_{n}(t)-\psi(t))\rangle|+\sup_{|t|\leq T}\sup_{\phi\in
Q}|\langle\phi,P_{>N}\psi_{n}(t)\rangle|$
$\displaystyle\qquad\quad+\sup_{|t|\leq T}\sup_{\phi\in
Q}|\langle\phi,P_{>N}\psi(t)\rangle|$ $\displaystyle\qquad\lesssim
N^{2}\sup_{\phi\in Q}\|\phi\|_{H^{-1}}\sup_{|t|\leq
T}\|\psi_{n}(t)-\psi(t)\|_{H^{-1}}$ $\displaystyle\qquad\quad+\sup_{\phi\in
Q}\|P_{>N}\phi\|_{H^{-1}}\sup_{n\geq 1}\sup_{|t|\leq
T}\|\psi_{n}(t)\|_{H^{1}}+\sup_{\phi\in Q}\|P_{>N}\phi\|_{H^{-1}}\sup_{|t|\leq
T}\|\psi(t)\|_{H^{1}},$

so once again taking $n\to\infty$ and then $N\to\infty$ gives us

(3.22) $\lim_{n\to\infty}\;\sup_{|t|\leq T}\;\sup_{\phi\in
Q}\;|\langle\phi,\psi_{n}(t)-\psi(t)\rangle|=0.$

We will return to this result later when proving $C([-T,T];H^{1})$
convergence.

Turning to convergence in $C([-T,T];{L_{\mu}^{2}})$, we first note that as
$\psi_{n,0}\to\psi_{0}$ in ${L_{\mu}^{2}}$ we have

(3.23) $\lim_{\lambda\to 0}\;\sup_{n\geq
1}\;\|(1-\varphi^{\lambda})\psi_{n,0}\|_{L_{\mu}^{2}}=0\quad\text{and}\quad\lim_{\lambda\to
0}\;\sup_{n\geq 1}\;\|(1-\varphi^{\lambda})\psi_{n,0}\|_{L^{2}}=0.$

In view of the convergence (3.19), using (3.8), (3.9), (3.21), and (3.23), we
find that

$\lim_{\lambda\to 0}\ \sup_{|t|\leq T}\
\Bigl{[}\|(1-\varphi^{\lambda})\psi(t)\|_{L^{2}}^{2}+\sup_{n}\|(1-\varphi^{\lambda})\psi_{n}(t)\|_{L^{2}}^{2}\Bigr{]}=0$

and

(3.24) $\lim_{\lambda\to 0}\ \sup_{|t|\leq T}\
\Bigl{[}\|(1-\varphi^{\lambda})\psi(t)\|_{L_{\mu}^{2}}^{2}+\sup_{n}\|(1-\varphi^{\lambda})\psi_{n}(t)\|_{{L_{\mu}^{2}}}^{2}\Bigr{]}=0.$

We are now ready to address convergence in $C([-T,T];{L_{\mu}^{2}})$. Clearly

$\displaystyle\|\psi_{n}(t)-\psi(t)\|_{L_{\mu}^{2}}\leq\|\varphi^{\lambda}[\psi_{n}(t)-\psi(t)]\|_{L_{\mu}^{2}}+\|(1-\varphi^{\lambda})\psi_{n}(t)\|_{L_{\mu}^{2}}+\|(1-\varphi^{\lambda})\psi(t)\|_{L_{\mu}^{2}}$

holds for all $\lambda>0$ and $t\in[-T,T]$. Using (2.8) we may bound

$\displaystyle\sup_{|t|\leq
T}\|\varphi^{\lambda}[\psi_{n}(t)-\psi(t)]\|_{L_{\mu}^{2}}$
$\displaystyle\lesssim\bigl{(}\mathcal{N}_{0}+\tfrac{1}{\sqrt{\lambda}}\bigr{)}\sup_{|t|\leq
T}\|\psi_{n}(t)-\psi(t)\|_{L^{2}}.$

Recalling (3.24) and that we have already proved convergence in
$C([-T,T];H^{\frac{2}{3}})$, we may send $n\to\infty$ and then $\lambda\to 0$,
we deduce that $\psi_{n}\to\psi$ in $C([-T,T];{L_{\mu}^{2}})$.

As we have now shown that $\psi_{n}\to\psi$ in
$C([-T,T];H^{\frac{2}{3}}\cap{L_{\mu}^{2}})$, (2.13) gives us

$\lim_{n\to\infty}\sup_{|t|\leq
T}\left|\int_{\mathbb{R}}|\psi_{n}(t)|^{4}\,d\mu-\int_{\mathbb{R}}|\psi(t)|^{4}\,d\mu\right|=0.$

Conservation of mass and energy then yields

$\displaystyle\sup_{|t|\leq
T}\Bigl{|}\|\psi_{n}(t)\|_{H^{1}}^{2}-\|\psi(t)\|_{H^{1}}^{2}\Bigr{|}$
$\displaystyle\leq\Bigl{|}M\bigl{[}\psi_{n,0}\bigr{]}-M[\psi_{0}]\Bigr{]}+2\Bigl{|}E\bigl{[}\psi_{n,0}\bigr{]}-E\bigl{[}\psi_{0}\bigr{]}\Bigr{|}$
$\displaystyle\quad+\sup_{|t|\leq
T}\left|\int_{\mathbb{R}}|\psi_{n}(t)|^{4}\,d\mu-\int_{\mathbb{R}}|\psi(t)|^{4}\,d\mu\right|,$

which converges to zero as $n\to\infty$.

Finally, as $\psi\in C(\mathbb{R};H^{1})$, the set

$Q=\bigl{\\{}(1-\partial_{x}^{2})\psi(t):|t|\leq T\bigr{\\}}\subseteq H^{-1}$

is compact. We may then use (3.22) with the convergence of norms to see that

$\displaystyle\sup_{|t|\leq T}\|\psi_{n}(t)-\psi(t)\|_{H^{1}}^{2}$
$\displaystyle\leq\sup_{|t|\leq
T}\Bigl{|}\|\psi_{n}(t)\|_{H^{1}}^{2}-\|\psi(t)\|_{H^{1}}^{2}\Bigr{|}$
$\displaystyle\quad+2\sup_{|t|\leq T}\sup_{\phi\in
Q}\Bigl{|}\langle\phi,\psi_{n}(t)-\psi(t)\rangle\Bigr{|}$

converges to zero as $n\to\infty$. ∎

We now turn our attention to a deterministic version of Theorem 1.3. This
theorem concerns solutions $\psi_{\varepsilon}$ to
$(\text{NLS}_{\mu^{\varepsilon}})$ with $\mu^{\varepsilon}$ defined by (1.7).
Specifically, we prove an analogue of (3.19) for the solutions
$\psi_{\varepsilon}$ to

(3.25) $\displaystyle
i\partial_{t}\psi_{\varepsilon}=-\partial_{x}^{2}\psi_{\varepsilon}+2|\psi_{\varepsilon}|^{2}\psi_{\varepsilon}\,d\mu^{\varepsilon}\quad\text{with}\quad\psi_{\varepsilon}(0)=\psi_{0}\in{L_{\mu}^{2}}\cap
H^{1}.$

######  Corollary 3.7.

Suppose $\mathcal{N}_{0}(\mu)<\infty$ and $\psi_{0}\in{L_{\mu}^{2}}\cap
H^{1}$. For each $\varepsilon\in(0,1]$, there is a unique global solution
$\psi_{\varepsilon}$ to (3.25). Moreover,

(3.26) $\lim_{\varepsilon\to 0}\ \sup_{|t|\leq T}\
\bigl{\|}\psi_{\varepsilon}(t)-\psi(t)\bigr{\|}_{H^{1}\cap{L_{\mu}^{2}}}=0\quad\text{for
each}\quad 0<T<\infty.$

###### Proof.

For $\varepsilon\in(0,1]$, the definition (1.7) of $\mu^{\varepsilon}$
guarantees that

$\mu^{\varepsilon}(I_{k})\leq\mu(I_{k-1})+\mu(I_{k})+\mu(I_{k+1})\quad\text{and}\quad\mu(I_{k})\leq\mu^{\varepsilon}(I_{k-1})+\mu^{\varepsilon}(I_{k})+\mu^{\varepsilon}(I_{k+1})$

and consequently,

(3.27)
$\displaystyle\mathcal{N}_{k}(\mu)\simeq\mathcal{N}_{k}(\mu^{\varepsilon})\quad\text{and}\quad\omega(x;\mu)\simeq\omega(x;\mu^{\varepsilon})\quad\text{uniformly
for $\varepsilon\in(0,1]$.}$

This demonstrates that the norms on ${L_{\mu}^{2}}$ and
${L_{\mu^{\varepsilon}}^{2}}$ are equivalent, uniformly for
$\varepsilon\in(0,1]$. In particular, $\psi_{0}\in{L_{\mu}^{2}}$ implies that
$\psi_{0}\in L^{2}_{\mu^{\varepsilon}}$. Thus, Theorem 1.4 guarantees that
there is a unique global solution $\psi_{\varepsilon}$ to (3.26); moreover,
this solution is $H^{1}\cap{L_{\mu}^{2}}$-continuous and conserves both the
mass and the energy functional

$E_{\varepsilon}[\psi]:=\int\tfrac{1}{2}|\psi^{\prime}(x)|^{2}\,dx+\int\tfrac{1}{2}|\psi(x)|^{4}\,d\mu^{\varepsilon}(x).$

Using mass and energy conservation, as well as (3.19) and (3.7), we also see
that

$\displaystyle\sup_{0<\varepsilon\leq 1}\sup_{|t|\leq
T}\|\psi_{\varepsilon}(t)\|_{H^{1}\cap{L_{\mu}^{2}}}<\infty\quad\text{for
any}\quad 0<T<\infty.$

The existence of the solutions $\psi_{\varepsilon}$ and the fact that they
obey such bounds allow us to repeat the proof of Proposition 3.4 replacing
$\varphi^{\eta}\equiv\varphi^{\varepsilon}\equiv 1$ (but retaining
$\varphi^{\lambda}$). In this way, we find that $\psi_{\varepsilon}(t)$ are
Cauchy in $C([-T,T];{L_{\mu}^{2}}\cap H^{2/3})$ and that the limit is the
unique solution $\psi(t)$ to (NLSμ). Although this convergence is local in
time, once we demonstrate $C([-T,T];H^{1})$ convergence, this argument can be
iterated as in the proof of Proposition 3.5.

The proof of Proposition 3.5 also contains the key idea for upgrading
convergence, namely, energy conservation and the Radon–Riesz argument. Writing

$\displaystyle\biggl{|}\int|\psi|^{4}\,d\mu-\int|\psi_{\varepsilon}|^{4}\,d\mu^{\varepsilon}\biggr{|}$
$\displaystyle\leq\biggr{|}\int|\psi|^{4}\,[d\mu-d\mu^{\varepsilon}]\biggr{|}+\int\bigl{|}|\psi|^{4}-|\psi_{\varepsilon}|^{4}\bigr{|}\,d\mu^{\varepsilon}$

and then applying (2.14) and (2.15), we deduce that

$\displaystyle\lim_{\varepsilon\to
0}\int|\psi_{\varepsilon}(t)|^{4}\,d\mu^{\varepsilon}=\int|\psi(t)|^{4}\,d\mu\quad\text{for
all}\quad t\in[-T,T].$

In this way, energy conservation shows that
$\|\psi_{\varepsilon}(t)\|_{H^{1}}\to\|\psi(t)\|_{H^{1}}$ and the Radon–Riesz
theorem allows us to upgrade convergence in the weak topology to norm
convergence. ∎

##  4\. Proof of Theorem 1.2

To apply Theorem 1.4 to prove Theorem 1.2, we require one additional lemma:

######  Lemma 4.1.

Let $\mu$ be distributed according to the Poisson process (1.2). Then

(4.1) $\operatorname{\mathbb{E}}\,\mathcal{N}_{0}^{2}\lesssim 1,$

and $\mathcal{N}_{k}<\infty$ for all $k\in\mathbb{Z}$ almost surely. Moreover,
if $f\in L^{2}$ then

(4.2)
$\operatorname{\mathbb{E}}\|f\|_{L_{\mu}^{2}}^{2p}\lesssim\|f\|_{L^{2}}^{2p}$

for any $1\leq p<\infty$; in particular, $f\in{L_{\mu}^{2}}$ almost surely.

###### Proof.

Fixing a choice of $r>4$, we apply Young’s inequality to bound

$\mu(I_{\ell})^{2}|k-\ell|^{\frac{r}{2}-1}\leq\tfrac{2}{r}\bigl{(}\tfrac{r-2}{r}\bigr{)}^{\frac{r-2}{2}}\mu(I_{\ell})^{r}+|k-\ell|^{\frac{r}{2}};$

in particular, if $k\neq\ell$ then

$\mu(I_{\ell})^{2}-|k-\ell|\lesssim\frac{\mu(I_{\ell})^{r}}{\langle
k-\ell\rangle^{\frac{r}{2}-1}}$

and so, for any $1\leq p<\infty$,

$\mathcal{N}_{k}^{2p}\lesssim
1+\sum_{\ell\in\mathbb{Z}}\frac{\mu(I_{\ell})^{pr}}{\langle
k-\ell\rangle^{p(\frac{r}{2}-1)}}.$

Taking the expectation we obtain

(4.3) $\operatorname{\mathbb{E}}\bigl{(}\mathcal{N}_{k}^{2p}\bigr{)}\lesssim
1+\sum_{\ell\in\mathbb{Z}}\frac{\operatorname{\mathbb{E}}\mu(I_{\ell})^{pr}}{\langle
k-\ell\rangle^{p(\frac{r}{2}-1)}}\lesssim 1,$

which gives us (4.1). Consequently, $\mathcal{N}_{0}^{2}<\infty$ almost surely
and the estimate (2.6) then ensures that almost surely we have
$\mathcal{N}_{k}^{2}\leq\mathcal{N}_{0}^{2}+|k|<\infty$ for all
$k\in\mathbb{Z}$.

Similarly, if $f\in L^{2}$ then we may apply (2.9) and (4.3) to bound

$\displaystyle\Bigl{\\{}\operatorname{\mathbb{E}}\|f\|_{L_{\mu}^{2}}^{2p}\Bigr{\\}}^{\frac{1}{p}}\simeq\Bigl{\\{}\operatorname{\mathbb{E}}\Bigl{[}\sum_{k\in\mathbb{Z}}\mathcal{N}_{k}^{2}\|\chi_{k}f\|_{L^{2}}^{2}\Bigr{]}^{p}\Bigr{\\}}^{\frac{1}{p}}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\Bigl{\\{}\operatorname{\mathbb{E}}\Bigl{[}\mathcal{N}_{k}^{2p}\|\chi_{k}f\|_{L^{2}}^{2p}\Bigr{]}\Bigr{\\}}^{\frac{1}{p}}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\|\chi_{k}f\|_{L^{2}}^{2}\simeq\|f\|_{L^{2}}^{2},$

from which we obtain (4.2). ∎

We are now in a position to complete the proofs of our main theorems:

###### Proof of Theorem 1.2.

Given $\psi_{0}\in H^{1}$, Lemma 4.1 ensures that we almost surely have
$\mathcal{N}_{0}<\infty$ and $\psi_{0}\in{L_{\mu}^{2}}$. We may then apply
Proposition 3.5 to deduce that almost surely there exists a unique strong
solution $\psi\in C(\mathbb{R};H^{1}\cap{L_{\mu}^{2}})$ of (NLSμ) that
conserves mass and energy.

It remains to prove our continuity statement (1.5). For this, we will combine
Proposition 3.6 with the dominated convergence theorem. Our main task is
verifying the existence of a suitable dominating function. It is no loss of
generality to fix $p\in[2,\infty)$ and to assume that the sequence of initial
data $\psi_{n,0}\to\psi_{0}$ in $H^{1}$ satisfies

(4.4)
$\sup_{n}\|\psi_{n,0}\|_{H^{1}}\lesssim\|\psi_{0}\|_{H^{1}}\quad\text{and}\quad\sum_{n}\|\psi_{n,0}-\psi_{0}\|_{H^{1}}\lesssim\|\psi_{0}\|_{H^{1}}.$

Indeed, every subsequence of $\psi_{n,0}\to\psi_{0}$ has such a
subsubsequence.

For the dominating function, we choose

(4.5) $F_{T}(\mu):=\sup_{n}\sup_{|t|\leq
T}\Bigl{[}\|\psi(t)\|_{H^{1}}^{2}+\|\psi_{n}(t)\|_{H^{1}}^{2}\Bigr{]}^{\frac{p}{2}},$

where $\psi(t)$ and $\psi_{n}(t)$ denote the solutions to (NLSμ) with initial
data $\psi_{0}$ and $\psi_{n,0}$, respectively. In light of the conservation
of mass and energy,

$\displaystyle F_{T}(\mu)$
$\displaystyle\leq\sup_{n}\biggl{[}\|\psi_{0}\|_{H^{1}}^{2}+\|\psi_{n,0}\|_{H^{1}}^{2}+\int|\psi_{0}|^{4}+|\psi_{n,0}|^{4}\,d\mu\biggr{]}^{\frac{p}{2}}$
$\displaystyle\lesssim_{p}\|\psi_{0}\|_{H^{1}}^{p}+\biggl{[}\int|\psi_{0}|^{4}\,d\mu\biggr{]}^{\frac{p}{2}}\\!+\sum_{n}\biggl{\\{}\|\psi_{n,0}-\psi_{0}\|_{H^{1}}^{p}+\biggl{[}\int|\psi_{n,0}-\psi_{0}|^{4}\,d\mu\biggr{]}^{\frac{p}{2}}\biggr{\\}}.$

Employing (2.13), then (4.2), and finally (4.4), we find that

(4.6)
$\operatorname{\mathbb{E}}\\{F_{T}(\mu)\\}\lesssim_{p}\|\psi_{0}\|_{H^{1}}^{p}+\|\psi_{0}\|_{H^{1}}^{2p}.$

This shows that our dominating function has finite expectation.

By Jensen’s inequality, followed by (4.2) and (4.4), we also have

$\operatorname{\mathbb{E}}\Bigl{\\{}\sum_{n}\|\psi_{n,0}-\psi_{0}\|_{L_{\mu}^{2}}\Bigr{\\}}\leq\sum_{n}\sqrt{\operatorname{\mathbb{E}}\|\psi_{n,0}-\psi_{0}\smash[b]{\|_{L_{\mu}^{2}}^{2}}}<\infty,$

which shows that $\psi_{n,0}\to\psi_{0}$ in ${L_{\mu}^{2}}$ almost surely. By
hypothesis, we know that $\psi_{n,0}\to\psi_{0}$ in $H^{1}$. These two
observations allow us to apply Proposition 3.6 and so deduce that

(4.7) $\lim_{n\to\infty}\sup_{|t|\leq
T}\|\psi_{n}(t)-\psi(t)\|_{H^{1}}=0\quad\text{almost surely.}$

Thus, we see that (1.5) follows from the dominated convergence theorem. ∎

###### Proof of Theorem 1.3.

Lemma 4.1 shows that both $\mathcal{N}_{0}<\infty$ and
$\psi_{0}\in{L_{\mu}^{2}}$ almost surely. Thus, Corollary 3.7 shows that
(3.26) holds almost surely. To deduce $L^{p}(d\mathbb{P})$ convergence, we
employ the dominated convergence theorem as in the proof of Theorem 1.2. As a
dominating function, we choose

$F_{T}(\mu):=\sup_{|t|\leq
T}\;\Bigl{[}\|\psi(t)\|_{H^{1}}^{2}+\sup_{0<\varepsilon\leq
1}\,\|\psi_{\varepsilon}(t)\|_{H^{1}}^{2}\Bigr{]}^{\frac{p}{2}}.$

Using the conservation of mass and energy, as well as (2.13) and (3.27), we
have

$F_{T}(\mu)\lesssim\Bigl{[}M[\psi_{0}]+E[\psi_{0}]+\sup_{0<\varepsilon\leq
1}E_{\varepsilon}[\psi_{0}]\Bigr{]}^{\frac{p}{2}}\lesssim\|\psi_{0}\|_{H^{1}}^{p}+\|\psi_{0}\|_{H^{1}}^{3p/2}\|\psi_{0}\|_{{L_{\mu}^{2}}}^{p/2}.$

Thus, using (4.2), we see that $\operatorname{\mathbb{E}}\\{F_{T}\\}<\infty$
and the proof is complete. ∎

## References

  * [1] R. Adami, R. Fukuizumi, and J. Holmer.  Scattering for the $L^{2}$ supercritical point NLS.  Trans. Amer. Math. Soc., 374(1):35–60, 2021.
  * [2] R. Adami and A. Teta.  A class of nonlinear Schrödinger equations with concentrated nonlinearity.  J. Funct. Anal., 180(1):148–175, 2001.
  * [3] C. Cacciapuoti, D. Finco, D. Noja, and A. Teta.  The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit.  Lett. Math. Phys., 104(12):1557–1570, 2014.
  * [4] T. Cazenave.  Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics.  New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
  * [5] N. Hayashi, C. Li, T. Ogawa, and T. Sato.  Critical exponent for global existence of solutions to the Schrödinger equation with a nonlinear boundary condition.  Nonlinear Anal., 230:Paper No. 113229, 17, 2023.
  * [6] J. Holmer and C. Liu.  Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory.  J. Math. Anal. Appl., 483(1):123522, 20, 2020.
  * [7] O. Kallenberg.  Random measures.  Akademie-Verlag, Berlin; Academic Press, Inc., London, fourth edition, 1986.
  * [8] R. Killip and M. Vişan.  KdV is well-posed in $H^{-1}$.  Ann. of Math. (2), 190(1):249–305, 2019.
  * [9] A. I. Komech and A. A. Komech.  Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator.  Russ. J. Math. Phys., 14(2):164–173, 2007.
  * [10] M. Ntekoume.  Homogenization for the cubic nonlinear Schrödinger equation on $\mathbb{R}^{2}$.  Comm. Partial Differential Equations, 45(11):1561–1588, 2020.
  * [11] R. L. Pego.  Compactness in $L^{2}$ and the Fourier transform.  Proc. Amer. Math. Soc., 95(2):252–254, 1985.
  * [12] L. Tentarelli.  A general review on the NLS equation with point-concentrated nonlinearity.  Commun. Appl. Ind. Math., 14(1):62–84, 2023.
  * [13] M. Tsutsumi.  Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations.  J. Differential Equations, 42(2):260–281, 1981.