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Schrödinger-type identity to the existence and uniqueness of a solution to the stationary Schrödinger equation
- Delin Sun^{1}Email author
- Received: 25 December 2018
- Accepted: 13 March 2019
- Published: 21 March 2019
Abstract
In this article we study applications of the Schrödinger-type identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It is possible to derive a wide range of transmutation operators by this method. Classical Riesz transforms are involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. In this paper, we present a modified Schrödinger-type identity for solutions of a class of linear Schrödinger equations with mixed boundary conditions. The techniques used in our proofs are quite different, and most remarkably some of the proofs become simpler and more straightforward. As an application, we obtain the existence and uniqueness of a solution to the stationary Schrödinger equation in the sense of the Weyl law, which advances the recent results obtained in several articles even in a more general setting.
Keywords
- Schrödinger-type identity
- Neumann boundary condition
- Stationary Schrödinger equation
1 Introduction
The stationary Schrödinger equations are often used to express real-life problems. Naturally, we focus on finding solutions of linear Schrödinger equations. However, analytical solutions are frequently not possible to find and numerical solutions can be both theoretically and computationally complicated due to the complexity of a modified Schrödinger operator. Frequently, researchers investigate inequalities in weighted Banach spaces; see, for example, [1–6] and the references therein for details. Hence, it is appropriate to derive a generalized stationary Schrödinger inequality equation from modified Schrödinger differential equations (see [4, 5, 7]).
In recent years, under the frame of the auxiliary principle, some references, such as Ding [8], Noor [7], Chen and Zhang [9], Huang [10], and so on, introduced the related properties of weak solutions for generalized stationary Schrödinger equations of fluid dynamics and built the corresponding convergence theorems.
Due to the rapid advancement of computing resource, rigorous derivation of generalized stationary Schrödinger equations was done also for Schrödinger fluids. An asymptotic analysis of the generalized Schrödinger differential equation based on the asymptotic expansion was presented in [9, 10]. In this paper we investigate a new equation of generalized stationary Schrödinger inequalities and prove an existence and uniqueness theorem of solutions for this kind of equation.
The rest of the paper is organized as follows. In Sect. 2, we present some basic definitions, concepts, and some results that will be used later for our model problem. In Sect. 3, we prove the existence and uniqueness of weak solutions for generalized stationary Schrödinger equations. Finally, the paper is concluded in Sect. 4.
2 Preliminaries
- (C1)
\(a_{i}(\sigma _{i},\sigma _{i})\geq c_{i} \Vert \sigma _{i}\Vert _{i}^{2}\);
- (C2)
\(\vert a_{i}(\varrho _{i},\sigma _{i})\vert \leq d_{i} \Vert \varrho _{i} \Vert _{i}\cdot \Vert \sigma _{i}\Vert _{i}\)
- (C3)
\(b_{i}\) is a linear function for the first variable;
- (C4)
\(b_{i}\) is a convex function;
- (C5)There exists a positive constant \(\gamma _{i}\) satisfyingfor any \(\varrho _{i},\sigma _{i}\in H_{i}\).$$ \gamma _{i} \Vert \varrho _{i} \Vert _{i} \cdot \Vert \sigma _{i} \Vert _{i}\geq a_{i}( \varrho _{i},\sigma _{i}) $$
- (C6)for any \(\varrho _{i},\sigma _{i},w_{i}\in H_{i}\).$$ b_{i}(\varrho _{i},\sigma _{i}-w_{i}) \geq b_{i}(\varrho _{i},\sigma _{i})-b _{i}(\varrho _{i},w_{i}) $$
Remark 1
- (1)If \(A=B=I\), \(f_{i}=0\), and \(a_{i}(\varrho _{i},\sigma _{i})=0\), then (3) is equivalent tofor any \(\sigma _{1}\in H_{1} \) and$$ \bigl\langle F_{1}(x,y),\eta _{1}(\sigma _{1},x) \bigr\rangle _{1}+b_{1}(x,\sigma _{1})-b_{1}(x,x)\geq 0 $$for any \(\sigma _{2}\in H_{2}\).$$ \bigl\langle F_{2}(x,y),\eta _{2}(\sigma _{2},y) \bigr\rangle _{2}+b_{2}(y,\sigma _{2})-b_{2}(y,y)\geq 0 $$
- (2)If \(H_{1}=H_{2}=H\), \(f_{i}=f_{2}=f\), \(\eta _{1}=\eta _{2}= \eta \), \(a_{1}=a_{2}=a\), \(b_{1}=b_{2}=b\), then (3) is reduced tofor any \(v\in H\).$$ \bigl\langle F(Ax,Bx)-f,\eta (w,y) \bigr\rangle +a(y,w-y)+b(y,w)-b(y,y) \geq 0 $$
We then introduce the following definition (see [16, 17]), which will be useful for the proposed method.
Definition 1
- (1)\(F_{1}\) is \(\alpha _{1}\)-strongly monotone if the following inequality holds:where \(\varrho _{1},\sigma _{1}\in H_{1}\), \(\varrho _{2}\in H_{2}\) and \(\alpha _{1}\) is a positive constant;$$ \bigl\langle F_{1}(A\varrho _{1},\varrho _{2})-F_{1}(A\sigma _{1},\varrho _{2}), \varrho _{1}-\sigma _{1}) \bigr\rangle _{1} \geq \alpha _{1} \Vert \varrho _{1}- \sigma _{1} \Vert _{1}^{2}, $$
- (2)If there exist two positive constants \(\beta _{1}\) and \(\xi _{1}\) such thatfor \(\varrho _{1},\sigma _{1}\in H_{1}\) and \(\varrho _{2},\sigma _{2} \in H_{2}\), then \(F_{1}\) is \((\beta _{1},\xi _{1})\)-Lipschitz continuous;$$ \bigl\Vert F_{1}(A\varrho _{1},B\varrho _{2})-F_{1}(A\sigma _{1},B\sigma _{2}) \bigr\Vert _{1}\leq \beta _{1} \Vert \varrho _{1}-\sigma _{1} \Vert _{1}+\xi _{1} \Vert \varrho _{2}-\sigma _{2} \Vert _{2} $$
- (3)Iffor \(\varrho _{1},\sigma _{1}\in H_{1}\), then \(\eta _{1}\) is \(\delta _{1}\)-Lipschitz continuous, where \(\delta _{1}\) is a positive constant.$$ \bigl\Vert \eta _{1}(\varrho _{1},\sigma _{1}) \bigr\Vert _{1}\leq \delta _{1} \Vert \varrho _{1}- \sigma _{1} \Vert _{1} $$
- (4)Iffor \(\varrho _{1},\sigma _{1}\in H_{1}\), then \(\eta _{1}\) is \(\sigma _{1}\)-strongly monotone, where \(\sigma _{1}\) is a positive constant.$$ \bigl\langle \varrho _{1}-\sigma _{1},\eta _{1}(\varrho _{1},\sigma _{1}) \bigr\rangle _{1} \geq \sigma _{1} \Vert \varrho _{1}-\sigma _{1} \Vert _{1}^{2} $$
3 Existence and uniqueness
In this section, we give an existence and uniqueness theorem of the solution of the auxiliary problem for (3). Under the frame of this theorem, we develop a new linearization iterative algorithm for the proposed model. First of all, let us present the following auxiliary problem for (3).
By applying a fixed point theorem, which is due to Yao et al. (see [19]), we have the following.
Theorem 1
Proof
Theorem 2
- (1)
\(a_{i}: H_{i}\times H_{i}\rightarrow \mathbb{R}\) satisfy (C1) and (C2), \(b_{i}: H_{i}\times H_{i}\rightarrow \mathbb{R}\) with (C3)–(C6);
- (2)
\(\eta _{i}\) is \(\sigma _{i}\)-strongly monotone;
- (3)
\(\langle F_{i}(Ax,By),\eta _{i}(\sigma _{i},\cdot )\rangle \) is concave and upper semicontinuous.
Proof
It follows that \(b_{i}\) is convex and lower semicontinuous, which means that \(\langle F_{i}(Ax,By), \eta _{i}(\sigma _{i},\cdot )\rangle \) is concave and upper semicontinuous. Hence, \(\phi _{i}(\sigma _{i},\cdot )\) and \(\{\sigma _{i}\in H_{i}: \psi _{i}(\sigma _{i},z_{i})\}\) are convex. That is to say, conditions (2) and (3) hold.
In fact, in the above formulation, the second inequality comes from that \(\eta _{i}\) is affine in the second variable, \(\eta _{i}(z_{i}^{*},z _{i}^{*})=0\) and \(b_{i}\) satisfies (C3)–(C6).
Therefore, \(z_{1}^{*}\in H_{1}\) and \(z_{2}^{*}\in H_{2}\) are the solutions of auxiliary problems (4) and (5), respectively.
4 Conclusions
In this work, we have studied applications of the Schrödinger-type identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It was possible to derive a wide range of transmutation operators by this method. Classical Riesz transforms were involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. In this paper, we presented a modified Schrödinger-type identity for solutions of a class of linear Schrödinger equations with mixed boundary conditions. The techniques used in our proofs were quite different, and most remarkably some of the proofs became simpler and more straightforward. As an application, we obtained the existence and uniqueness of a solution to the stationary Schrödinger equation in the sense of the Weyl law, which advanced the recent results obtained in several articles even in a more general setting.
Declarations
Acknowledgements
The author is very grateful to the editor and referees for their useful suggestions, which have improved the paper.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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.
Authors’ Affiliations
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