Abstract
The aim of this paper is to investigate the existence of weak solutions for a two-dimensional Schrödinger equation with a singular potential in \(\mathbb{C}_{+}\). Under appropriate assumptions on the nonlinearity, we introduce a new type of quantum calculus via the Morse theory and variational methods. By applying Schrödinger type inequalities and the well-known Banach fixed point theorem in conjunction with the technique of measures of weak noncompactness, the new and more accurate estimations for boundary behaviors of them are also deduced.
Similar content being viewed by others
1 Introduction
In this paper, we study the following two-dimensional Schrödinger equation (see [1]):
in the upper half plane \(\mathbb{C}_{+} =\{ z=t+ix:x>0\}\), where the variables t and x are complex numbers in \(\mathbb{C}_{+}\) and \(C_{1}\) and \(C_{2}\) are real numbers. Our first aim is to construct the solution in terms of hypergeometric functions.
Let a be a real number, \(p>1\) and \(C_{2}=ap\). Then Eq. (1) arises naturally by linearizing the Klein–Gordon equation variable boundary (see [2]):
in \(\mathbb{C}_{+}\), which shows that
In general, the study of the solutions of the two-dimensional Schrödinger and their related properties are very complicated; especially if the roots of the characteristic polynomial are double and not analytic at the origin. The explicit difficulties in dealing with quadratic-type non-linearities in two dimensions are our inability to use the Strichartz inequalities.
However, many authors have showed that the solution of the two-dimensional Schrödinger equations can be expressed by using a variational inequality. In recent years, various extensions and generalizations of the classical variational inequality models and complementarity problems have emerged in quantum and fluid mechanics, nonlinear programming, physics, optimization and control, economics, transportation, finance, structural, elasticity and applied sciences (see [3–7] and the references therein for details).
The classical Schrödinger solution spaces \(\mathfrak{H}^{p}(\mathbb {C_{+}})\) (see [8]), are defined to consist of solutions of (1), holomorphic in \(\mathbb{C}_{+}\) with the property that \(\mathcal{M}_{p}(u,x)\) is uniformly bounded for \(x> 0\), where
Since \(|u|^{p}\) is the weak solution of (1) for \(u \in\mathfrak {H}^{p}(\mathbb{C_{+}})\) (see [9]), the solution \(\mathcal {M}_{p}(u,x)\) decreases in \((0,\infty)\),
Define
We remark that \(\phi^{(k)}(t)\in\mathcal{L}^{p}\) and \(\phi(t)\in\mathcal {C}^{\infty}\) if and only if \(\phi(t)\) belongs to the space \(\mathcal{D}_{\mathcal{L}^{p}}\) (see [6]). Let \(\mathcal{F}\) denote the space, which consists of infinitely differentiable weak solution of (1) in \(\mathbb {C}_{+}\). Let \(\mathcal{F}'_{\mathcal{L}^{p}}\) denote the dual of the space \(\mathcal{F}_{\mathcal{L}^{q}}\), that is, \(\mathcal{F}'_{\mathcal{L}^{p}}= (\mathcal{F}_{\mathcal{L}^{q}})^{\prime}\). We also denote \(q=\frac{p}{p-1}\) and by \(D'\) the dual of the space D. So we can get \(D\subseteq\mathcal{F}_{\mathcal{L}^{p}}\) and \(\mathcal {F}'_{\mathcal{L}^{p}}\subseteq D^{\prime}\).
Definition 1.1
(see [10])
If \(u \in\mathcal{F}^{\prime}\), then it has the following representation:
for any test function \(\phi\in\mathcal{F}\) and any function \(g(z)\) in \(\mathbb{C}_{+}\), where \(g(z)\) is analytic on the complement of the support of u.
Definition 1.2
(see [9])
Let Du be the Stokes operator defined by
on \(\mathcal{F}'_{\mathcal{L}^{p}}\) for all \(\phi\in\mathcal{F}_{\mathcal {L}^{q}}\).
It is obvious that
where \(u \in\mathcal{F}'_{\mathcal{L}^{p}}\).
Since \(u\in\mathcal{F}^{\prime}_{\mathcal{L}^{p}}\), \(D\varphi\in D_{\mathcal{L}^{p^{\prime}}}\), Du defined as above is a functional on \(D_{\mathcal{L}^{p^{\prime}}}\). Linearity of Du is nontrivial. If \(\{\varphi_{v}\}\to\varphi\) in \(D_{\mathcal {L}^{p^{\prime}}}\), then it is easy to see that
2 Construction of the solutions
By virtue of the weak maximum principle of superposition, it is necessary to consider the following Riemann problems:
and
First we solve (2). Put \(x=\tau_{1}\) and \(4x-t^{2}=\tau_{2}\) in Eqs. (2) and (3). Let
where
Substituting \(\tau_{1}^{l}\vartheta(z) \) with ϑ, it follows that \(\mathfrak {R}\vartheta=0\), from which one concludes that
By a simple calculation, we know that
By replacing \(\frac{\tau_{1}}{\tau_{2}}\) by \(\frac{1}{4(1-x) }\), it is obvious that
which is equivalent to a hyperbolic–parabolic differential equation with
iff
It follows from the hypergeometric equation theory that the first and the second solutions for the hyperbolic–parabolic equation are
and
respectively.
Let \(z=\frac{t^{2}}{4x}\), where \(| z| <1\). It is easy to see that a complete solution of the hyperbolic–parabolic equation is
So \(\vartheta=x^{l}(1-x)^{\sigma}x\) is a solution of \(\mathfrak{R}\vartheta=0\). Notice that
which immediately shows that
Similarly, we can solve the problem (2) by letting
which shows that
3 Boundary behaviors
Theorem 3.1
If \(u\in\mathcal{F}'_{\mathcal{L}^{p}}\), then
is one of the representations of the solution u such that
and
where \(x\to\infty\) and there exists a function \(G_{k}(z)\in\mathfrak{H}^{p}(\mathbb{C}_{+})\) such that
and
Theorem 3.2
If g is defined in (10) and \(G_{k}\in\mathfrak{H}^{p}(\mathbb {C}_{+})\), then there exists a Schrödinger distributional solution \(u(t)\in\mathcal{F}^{\prime}_{\mathcal{L}^{p}}\) such that \(g(z)\) is one of the analytic representations of u.
Corollary 1
If \(u(t)\in\mathcal{F}^{\prime}_{\mathcal{L}^{p}}\), then
satisfies
and
as \(x\to\infty\) and there exists a function \(G_{k}\) in \(\mathfrak{H}^{p}(\mathbb{C}_{+})\) such that
4 Lemmas
The following lemmas are required in this section.
Lemma 4.1
(see [11, p. 69])
If \(u\in\mathcal {L}^{p}(\mathbb{R})\) and G is defined by
then
Lemma 4.2
(see [11, p. 77])
Let \(g(z)\) be any weak solution of Eq. (1) such that the following properties hold.
-
(I)
\(g(t+ix)\in\mathcal{L}^{p}\) for any fixed \(x>0\);
-
(II)
$$\lim_{x\to0^{+}}g(t+ix)=g^{+}(t) $$
in \(\mathcal{F}^{\prime}_{\mathcal{L}^{p}}\) (weakly),
as \(x\to\infty\) and
Then
where \(\operatorname{Im}z>0\).
5 Proofs of main results
5.1 Proof of Theorem 3.1
By virtue of the fixed point theorem with respect to the stationary Schrödinger operator in [8], we have
for any \(u\in\mathcal{F}^{\prime}_{\mathcal{L}^{p}}\), which shows that
where r is a nonnegative integer and \(u_{l}\in L_{p}\).
So
which shows that
from the Hölder inequality.
Put
So
which yields
where C is a positive constant.
Since \(u_{l} \in\mathcal{L}^{p}\), then
where M is a positive constant,
and
So
By virtue of the structure formula, we have
where
So we obtain \(G_{k}(z)\in\mathfrak{H}^{p}(C_{+})\) from Lemma 4.1, which shows that
5.2 Proof of Theorem 3.2
Since \(G_{k}(z)\in\mathfrak{H}^{p}(C_{+})\), where \(G_{k}(t+ix)\in\mathcal {L}^{p}\) for fixed x, there exists the solution \(u_{l}(t)\in\mathcal {L}^{p}\), where \(u_{l}\) is the nontangential limit of \(g(z)\).
Since \(D_{\mathcal{L}^{q}}\in\mathcal{L}^{q}\), we see that \(u_{l}(t)\in D^{\prime}_{\mathcal{L}^{p}}\) and
So
which shows that
where \(x>0\) and
We know that \(G_{k}(z)\) can be represented as follows:
from Lemma 4.2, which yields
Put
where \(u\in D^{\prime}_{\mathcal{L}^{p}}\), which shows that \(g(z)\) is one of the analytic representations of u.
5.3 Proof of Corollary 1
By virtue of the fixed point theorem with respect to the stationary Schrödinger operator in [8], we know that
The rest of the proof of the corollary is similar to the proof of Theorem 3.1. So we omit the details here for the sake of brevity.
The proof of Corollary 1 is complete.
6 Conclusions
In this paper, we investigated the existence of weak solutions for a two-dimensional Schrödinger equations with a singular potential in \(\mathbb{C}_{+}\). Under appropriate assumptions on the nonlinearity, we introduced a new type of quantum calculus via the Morse theory and variational methods. By applying the well-known Banach fixed point theorem in conjunction with the technique of measures of weak noncompactness, new and more accurate estimations for boundary behaviors of them were also deduced. We significantly extended and complemented some results from the current literature.
References
Gil’, A., Nogin, V.: Complex powers of a differential operator related to the Schrödinger operator. Vladikavkaz. Mat. Zh. 19(1), 18–25 (2017)
Nakao, M.L., Narazaki, T.: Existence and decay of solutions of some nonlinear wave equations in noncylindrical domains. Math. Rep. Coll. Gen. Educ. Kyushu Univ. 11(2), 117–125 (1978)
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions, 10th printing. National Bureau of Standards, Washington (1972)
Bardos, C., Chen, G.: Control and stabilization for the wave equation. III: domain with moving boundary. SIAM J. Control Optim. 19, 123–138 (1981)
Bresters, D.W.: On the equation of Euler–Poisson–Darboux. SIAM J. Math. Anal. 1, 31–41 (1973)
Ferreira, J.: Nonlinear hyperbolic–parabolic partial differential equation in noncylindrical domain. Rend. Circ. Mat. Palermo 44(1), 135–146 (1995)
Medeiros, L.A.: Nonlinear wave equations in domains with variable boundary. Arch. Ration. Mech. Anal. 47, 47–58 (1972)
Li, Z.: Boundary behaviors of modified Green’s function with respect to the stationary Schrödinger operator and its applications. Bound. Value Probl. 2015, Article ID 242 (2015)
Schwartz, L.: Théorie des Distributions. Hermann, Paris (1978)
Marion, O.: Hilbert transform, Plemelj relation, and Fourier transform of distributions. SIAM J. Math. Anal. 4(4), 656–670 (1973)
Pandey, J.: The Hilbert Transform of Schwartz Distributions and Applications. Wiley, New York (1996)
Acknowledgements
The authors are grateful to the anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.
Availability of data and materials
Not applicable.
Funding
This work was supported by National Natural Science Foundation of China (No. 61403356) and Zhejiang Provincial Natural Science Foundation of China (No. LY18F030012).
Author information
Authors and Affiliations
Contributions
The authors contributed exclusively in writing this paper. They read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Additional information
List of abbreviations
Not applicable.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Xing, Y., Zhao, J. Existence and quantum calculus of weak solutions for a class of two-dimensional Schrödinger equations in \(\mathbb{C}_{+}\). Bound Value Probl 2018, 59 (2018). https://doi.org/10.1186/s13661-018-0975-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-018-0975-1