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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 715252, 5 pages
http://dx.doi.org/10.1155/2013/715252
Research Article

Solving Integral Representations Problems for the Stationary Schrödinger Equation

Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China

Received 24 April 2013; Accepted 9 June 2013

Academic Editor: Pavel Kurasov

Copyright © 2013 Yudong Ren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

When solutions of the stationary Schrödinger equation in a half-space belong to the weighted Lebesgue classes, we give integral representations of them, which imply known representation theorems of classical harmonic functions in a half-space.

1. Introduction and Results

Let and be the sets of all real numbers and of all positive real numbers, respectively. Let denote the -dimensional Euclidean space with points , where and . The unit sphere and the upper half unit sphere in are denoted by and , respectively. The boundary and closure of an open set of are denoted by and , respectively. The upper half space is the set , whose boundary is .

We identify with and with , writing typical points as , where and putting

For and , let denote the open ball with center at and radius in . , where is the origin of . For a set , , we denote and by and , respectively. By we denote . We denote by the -dimensional volume elements induced by the Euclidean metric on .

Let denote the class of nonnegative radial potentials , that is, , , such that with some if and with if or .

This paper is devoted to the stationary Schrödinger equation: where , is the Laplace operator and . Note that solutions of (2) are (classical) harmonic functions in the case . Under these assumptions the operator can be extended in the usual way from the space to an essentially self-adjoint operator on (see [1]). We will denote it by as well. This last one has a Green function . Here is positive on and its inner normal derivative . We denote this derivative by , which is called the Poisson -kernel with respect to . If and are denoted by the Green function and Poisson kernel of the Laplace operator in , respectively, then where , and is the area of the unit sphere in .

Let be a continuous function on . We say that is a solution of the Dirichlet problem for the stationary Schrödinger operator on with , if Note that is a solution of the classical Dirichlet problem for the Laplace operator on with in the case .

Let be a Laplace-Beltrami operator (spherical part of the Laplace) on the unit sphere. It is known (see, e.g., [2, page 41]) that the eigenvalue problem has the eigenvalues , where . Corresponding eigenfunctions are denoted by , where is the multiplicity of . We norm the eigenfunctions in and .

Let and stand, respectively, for the increasing and nonincreasing, as , solutions of the equation: normalized under the condition .

We shall also consider the class , consisting of the potentials such that there exists a finite limit ; moreover, . If , then solutions of (2) are continuous (see [3]).

In the rest of paper, we assume that and we shall suppress this assumption for simplicity. Further, we use the standard notations , , and is the integer part of and , where is a positive real number.

Denote

Remark 1. in the case , where .

It is known (see [4]) that in the case under consideration the solutions to (6) have the asymptotics where and are some positive constants.

If , it is known that the following expansion for the Green function (see [5, Chapter 11], [6]): where and is its Wronskian. The series converges uniformly if either or , where .

For a nonnegative integer and two points , we define a modified Green function: where

If the generalized Poisson kernel with respect to is defined by then we have and coincides with ones in Finkelstein and Scheinberg [7], Siegel and Talvila [8], Deng [9], Qiao and Deng [10], and Qiao [11] (see [5, Chapter 11]).

Put where is a continuous function on .

For real numbers , we denote the class of all measurable functions satisfying the following inequality: and the class consists of all measurable functions satisfying

We will also consider the class of all continuous functions , which is the solution of (2), with and , is denoted by .

Remark 2. If and , then (14) and (15) are equivalent to respectively, from Remark 1 and (14), which yield that is equivalent to in the notation of [9].
Let us recall the classical case . If is harmonic in , continuous on , and , then there exists a constant such that (see [12]) where .

Deng (see [9]) has constructed a similar representation to (17) for , which is the integral with a modified Poisson kernel derived by subtracting some special harmonic polynomials from . By virtue of this modified Poisson kernel, Qiao (see [11, 13]) and Qiao and Deng (see [1418]) have constructed different integral representations for harmonic functions of finite order and infinite order.

Especially, Su (see [6, 19]) recently writes solutions to the half-space Dirichlet problem with respect to the stationary Schrödinger operator . Now we state our main results as follows.

Theorem 3. If , then .

Theorem 4. If , is an integer such that , and then the following properties hold.(I) If , then there exists a constant such that for .(II) If , then we have , where vanishing continuously on , and are constants.

2. Lemmas

The following Lemma plays an important role in our discussions, which is due to Levin and Kheyfits (see [5, page 356]).

Lemma 5. If and is a solution of (2) on a domain containing , then where

Lemma 6 (see [6, Corollary 1.6]). If is a continuous function on satisfying then is a solution of the Dirichlet problem for on with and

Lemma 7 (see [6, Lemma 2.1] or [20, Theorem 1]). If is a solution of (2) on satisfying then (19) holds.

3. Proof of Theorem 3

We apply the formula (20) to in : where

Since , we obtain by (8)

From (15), we conclude that

Combining (25), (27), and (28), we obtain

Set where .

By the L’hospital’s rule, we have which yields that there exists a positive constant such that, for any ,

Then which shows that from . Then Theorem 3 is proved.

4. Proof of Theorem 4

To prove (II), notice that and condition (15) is stronger than condition (22) from Theorem 3.

Consider the function . Then it follows from Lemma 6 and Theorem 3 that this is a solution of (2) in and vanishes continuously on .

Then for any . Further, (8) gives that which together with (34) and Lemmas 6 and 7 give the result of (II).

If , then for each and there exists a constant such that for all . So if we take we see that (18) holds for all .

Thus we complete the proof of Theorem 4.

Acknowledgments

The author wishes to expresseses his appreciation to Professor Jingben Yin for very useful conversations related to this problem. He is grateful to the referee for her or his careful reading and helpful suggestions which led to the improvement of the original paper. This work is supported by the National Natural Science Foundation of China under Grant 11226093.

References

  1. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 3, Academic Press, London, UK, 1970.
  2. G. Rosenblum, M. Solomyak, and M. Shubin, Spectral Theory of Differential Operators, VINITI, Moscow, Russia, 1989.
  3. B. Simon, “Schrödinger semigroups,” Bulletin of the American Mathematical Society, vol. 7, no. 3, pp. 447–526, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. P. Hartman, Ordinary Differential Equations, John Wiley & Sons Inc., New York, NY, USA, 1964. View at MathSciNet
  5. B. Y. Levin and A. I. Kheyfits, “Asymptotic behavior of subfunctions of time-independent Schrödinger operators,” in Some Topics on Value Distribution and Differentiability in Complex and P-Adic Analysis, vol. 11, pp. 323–397, Science Press, Beijing, China, 2008. View at Zentralblatt MATH · View at MathSciNet
  6. B. Y. Su, “Dirichlet Problem for the Schrödinger operator in a half space,” Abstract and Applied Analysis, vol. 2012, Article ID 578197, 14 pages, 2012. View at Publisher · View at Google Scholar
  7. M. Finkelstein and S. Scheinberg, “Kernels for solving problems of Dirichlet type in a half-plane,” Advances in Mathematics, vol. 18, no. 1, pp. 108–113, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. Siegel and E. Talvila, “Sharp growth estimates for modified Poisson integrals in a half space,” Potential Analysis, vol. 15, no. 4, pp. 333–360, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. T. Deng, “Integral representations of harmonic functions in half spaces,” Bulletin des Sciences Mathématiques, vol. 131, no. 1, pp. 53–59, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Qiao and G. T. Deng, “Growth estimates for modified Green potentials in the upper-half space,” Bulletin des Sciences Mathématiques, vol. 135, no. 3, pp. 279–290, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. Qiao, “Integral representations for harmonic functions of infinite order in a cone,” Results in Mathematics, vol. 61, no. 1-2, pp. 63–74, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970. View at MathSciNet
  13. L. Qiao, “Modified Poisson integral and Green potential on a half-space,” Abstract and Applied Analysis, vol. 2012, Article ID 765965, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. L. Qiao and G. T. Deng, “The Riesz decomposition theorem for superharmonic functions in a cone and its application,” Scientia Sinica Mathematica, vol. 42, no. 8, pp. 763–774, 2012 (Chinese).
  15. L. Qiao and G. T. Deng, “Integral representations of harmonic functions in a cone,” Scientia Sinica Mathematica, vol. 41, no. 6, pp. 535–546, 2011 (Chinese).
  16. L. Qiao and G. T. Deng, “Growth estimate for a class of harmonic functions in a cone,” Acta Mathematica Sinica, vol. 54, no. 6, pp. 1021–1028, 2011 (Chinese). View at Zentralblatt MATH · View at MathSciNet
  17. L. Qiao and G. T. Deng, “Growth property and integral representation of harmonic functions in a cone,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 2, pp. 511–523, 2013. View at MathSciNet
  18. L. Qiao and G. T. Deng, “Integral representation for the solution of the stationary Schrödinger equation in a cone,” Mathematische Nachrichten, vol. 285, no. 16, pp. 2029–2038, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Y. Su, “Growth properties of harmonic functions in the upper half space,” Acta Mathematica Sinica, vol. 55, no. 6, pp. 1095–1100, 2012 (Chinese). View at MathSciNet
  20. A. I. Kheyfits, “Liouville theorems for generalized harmonic functions,” Potential Analysis, vol. 16, no. 1, pp. 93–101, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet