Existence and quantum calculus of weak solutions for a class of two-dimensional Schrödinger equations in \(\mathbb{C}_{+}\)

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.

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

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

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

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.

1. (I)

   \(g(t+ix)\in\mathcal{L}^{p}\) for any fixed \(x>0\);

2. (II)
   $$\lim_{x\to0^{+}}g(t+ix)=g^{+}(t) $$

in \(\mathcal{F}^{\prime}_{\mathcal{L}^{p}}\) (weakly),

as \(x\to\infty\) and

Then

where \(\operatorname{Im}z>0\).

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

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.

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.

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.

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.

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Funding

This work was supported by National Natural Science Foundation of China (No. 61403356) and Zhejiang Provincial Natural Science Foundation of China (No. LY18F030012).

Affiliations

1. School of Mechanotronics and Vehicle Engineering, Qingyuan Polytechnic, Qingyuan, China

   • Yifan Xing

2. Shenzhen Quantum Wisdom Culture Development Company, Shenzhen, China

   • Yifan Xing

3. Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore

   • Yifan Xing

4. College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China

   • Jinhui Zhao

Contributions

The authors contributed exclusively in writing this paper. They read and approved the final manuscript.

Corresponding author

Correspondence to Jinhui Zhao.

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The authors declare that they have no competing interests.

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