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[[File:Quantum particle probability density.png|thumb|upright=1.35|The probability density distribution of a quantum particle in three-dimensional space. The points in the image represent the probability of finding the particle at those locations, with darker colors indicating higher probabilities. To simplify and clarify the visualization, low-probability regions have been filtered out. In fact, the total probability 1 means that the particle exists everywhere in the entire space.]] {{Quantum mechanics}}
In mathematics, a '''normalized solution''' to an [[Ordinary differential equation|ordinary]] or [[partial differential equation]] is a solution with prescribed norm, that is, a solution which satisfies a condition like <math>\int_{\mathbb{R}^N} |u(x)|^2 \, dx = 1.</math> In this article, the normalized solution is introduced by using the [[nonlinear Schrödinger equation]]. The nonlinear [[Schrödinger equation]] (NLSE) is a fundamental equation in [[quantum mechanics]] and other various fields of physics, describing the evolution of complex [[wave functions]]. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.<ref>{{Cite journal |last1=Berestycki |first1=H. |last2=Lions |first2=P.-L. |date=1983 |title=Nonlinear scalar field equations. I. Existence of a ground state |journal=Arch. Rational Mech. Anal. |volume=82 |issue=4 |pages=313–345 |mr=0695535}}</ref>
==Definition and variational framework==
In order to illustrate this concept, consider the following nonlinear Schrödinger equation with prescribed norm:<ref name="Jeanjean1997">{{Cite journal |last=Jeanjean |first=L. |date=1997 |title=Existence of solutions with prescribed norm for semilinear elliptic equations |journal=Nonlinear Analysis: Theory, Methods & Applications |volume=28 |issue=10 |pages=
:<math> -\Delta u + \lambda u = f(u), \quad \int_{\mathbb{R}^N} |u|^2 \, dx = 1, </math>
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==History==
The exploration of normalized solutions for the nonlinear Schrödinger equation can be traced back to the study of standing wave solutions with prescribed <math>L^2</math>-norm. [[Jürgen Moser]]<ref>{{Cite journal |last=Moser |first=J. |date=1960 |title=A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations |journal=Communications on Pure and Applied Mathematics |volume=13 |pages=
For the variational problem, early foundational work in this area includes the concentration-compactness principle introduced by [[Pierre-Louis Lions]] in 1984, which provided essential techniques for solving these problems.<ref>{{Cite journal |last=Lions |first=P.-L. |date=1984 |title=The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 |journal=Annales de l'Institut Henri Poincaré C: Analyse Non Linéaire |volume=1 |issue=2 |pages=
For variational problems with prescribed mass, several methods commonly used to deal with unconstrained variational problems are no longer available. At the same time, a new critical exponent appeared, the <math> L^2</math>-critical exponent. From the [[Gagliardo-Nirenberg inequality]], we can find that the nonlinearity satisfying <math> L^2</math>-subcritical or critical or supercritical leads to a different geometry for functional. In the case the functional is bounded below, i.e., <math> L^2</math> subcritical case, the earliest result on this problem was obtained by Charles-Alexander Stuart<ref>{{Cite journal |last=Stuart |first=C.A. |date=1980 |title=Bifurcation for variational problems when the linearization has no eigenvalues |journal=Journal of Functional Analysis |volume=38 |pages=
In the case the functional is not bounded below, i.e., <math> L^2</math> supcritical case, some new difficulties arise. Firstly, since <math>\lambda</math> is unknown, it is impossible to construct the corresponding [[Nehari manifold]]. Secondly, it is not easy to obtain the boundedness of the Palais-Smale sequence. Furthermore, verifying the [[compactness]] of the Palais-Smale sequence is challenging because the embedding <math>H^1(\mathbb{R}^N) \hookrightarrow L^2(\mathbb{R}^N) </math> is not compact. In 1997, Louis Jeanjean using the following transform:
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:<math>P(u) := \partial_s \tilde{I}(u, s)|_{s=0} = \int_{\mathbb{R}^N} |\nabla u|^2 - N \int_{\mathbb{R}^N} \left( \frac{1}{2} f(u)u - F(u) \right) </math>
which corresponds exactly to the [[Pokhozhaev's identity]] of equation. Jeanjean used this additional condition to ensure the boundedness of the Palais-Smale sequence, thereby overcoming the difficulties mentioned earlier. As the first method to address the issue of normalized solutions in unbounded functional, Jeanjean's approach has become a common method for handling such problems and has been imitated and developed by subsequent researchers.<ref name="Jeanjean1997"
In the following decades, researchers expanded on these foundational results. Thomas Bartsch and Sébastien de Valeriola<ref>{{Cite journal |last1=Bartsch |first1=T. |last2=de Valeriola |first2=S. |date=2013 |title=Normalized solutions of nonlinear Schrödinger equations |journal=Archiv der Mathematik (Basel) |volume=100 |pages=
In [[bounded ___domain]], the situation is very different. Let's define <math>f(s)=|s|^{p-2}s </math> where <math> p \in (2, 2^*) </math>. Refer to Pokhozhaev's identity,
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:<math> p := 2+ \frac{4}{N}. </math>
From this, we can get different concepts about mass subcritical as well as mass supercritical. It is also useful to get whether the functional is bounded below or not.<ref name="Jeanjean1997"
===Palais-Smale sequence===
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==References==
{{Reflist}}
==Further reading==
* {{cite book | first=Lawrence C. | last=Evans |author-link=Lawrence C. Evans | title=Partial Differential Equations | publisher=American Mathematical Society | ___location=Providence, Rhode Island | year=1998 | isbn=0-8218-0772-2| url=https://math24.wordpress.com/wp-content/uploads/2013/02/partial-differential-equations-by-evans.pdf }}
* {{cite book | first=Michael | last=Struwe |author-link=Michael Struwe| title=Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-74013-1 | url=https://link.springer.com/book/10.1007/978-3-540-74013-1 }}
{{Quantum mechanics topics}}
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[[Category:Quantum mechanics]]
[[Category:Partial differential equations]] [[Category:Calculus of variations]] | |||