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Weak solutions for the stationary Schrödinger equation and its application
Applied Mathematics Letters 63 (2017) 34–39
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Weak solutions for the stationary Schr¨odinger equation and its application✩ Lei Qiao School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China
article
info
Article history: Received 12 April 2016 Received in revised form 19 July 2016 Accepted 19 July 2016 Available online 27 July 2016
abstract In this paper, we prove Carleman’s formula for weak solutions of the stationary Schr¨ odinger equation in a cylinder. As an application of it, the integral representation of solutions of the stationary Schr¨ odinger equation is also obtained. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Carleman’s formula Stationary Schr¨ odinger equation Integral representation
1. Introduction and main results We introduce a system of spherical coordinates (r, Θ), Θ = (θ1 , θ2 , . . . , θn ), in Rn (n ≥ 2) which are related to Cartesian coordinates (x1 , x2 , . . . , θn−1 , y) by y = r cos θ1 . For positive functions h1 and h2 , we say that h1 . h2 if h1 ≤ ch2 for some constant c > 0. If h1 . h2 and h2 . h1 , then we say that h1 ≈ h2 . Let ∆n be the Laplace operator and Ω be a bounded domain in Rn−1 with smooth boundary ∂Ω . Consider the Dirichlet problem (∆n−1 + λ)ϕ = 0 on Ω and ϕ = 0 on ∂Ω (see [1, p. 41]). We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by ϕ, Ω ϕ2 (X)dΩ = 1, where X ∈ Ω and dΩ is the (n − 1)-dimensional volume element. The set Ω × R = {P = (X, y) ∈ Rn ; X ∈ Ω , y ∈ R} in Rn is simply denoted by Tn (Ω ). We call it a cylinder (see [2]). In the following, we denote the sets Ω × I and ∂Ω × I with an interval I on R by Tn (Ω ; I) and Sn (Ω ; I) respectively. Hence Sn (Ω ; R) denoted simply by Sn (Ω ) is ∂Tn (Ω ). Let AΩ denote the class of nonnegative radial potentials a(P ) (i.e. 0 ≤ a(P ) = a(r) for P = (X, y) = (r, Θ) ∈ Tn (Ω )) such that a ∈ Lbloc (Tn (Ω )) with some b > n/2 if n ≥ 4 and with b = 2 if n = 2 or n = 3. ✩
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301140, U1304102). E-mail address: [email protected].
http://dx.doi.org/10.1016/j.aml.2016.07.018 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
L. Qiao / Applied Mathematics Letters 63 (2017) 34–39
35
For the identical operator I and the potential a(P ) ∈ AΩ , we define the stationary Schr¨odinger operator by SSE a = −∆n + a(P )I, where SSE a can be extended in the usual way from the space C0∞ (Tn (Ω )) to an essentially self-adjoint operator on L2 (Tn (Ω )) (see [3, Ch. 11] for more details). Furthermore SSE a has a a a Green–Sch function GΩ (P, Q) (associated with the stationary Schr¨odinger operator SSE a ). Here GΩ (P, Q) a is positive on Tn (Ω ) and its inner normal derivative ∂GΩ (P, Q)/∂nQ is non-negative, where ∂/∂nQ denotes a the differentiation at Q along the inward normal into Tn (Ω ). We write 1/cn ∂GΩ (P, Q)/∂nQ by PI aΩ (P, Q), which is called the Poisson–Sch kernel (associated with the stationary Schr¨odinger operator SSE a ) on Tn (Ω ). Here, c2 = 2 and cn = (n − 2)wn when n ≥ 3, where wn is the surface area of the unit sphere in Rn . The Poisson–Sch integral PI aΩ [g](P ) of g (associated with the stationary Schr¨odinger operator SSE a ) on Tn (Ω ) is defined as follows PI aΩ [g](P ) = Sn (Ω) PI aΩ (P, Q)g(Q)dσQ , where g is a continuous function on Sn (Ω ) and dσQ is the surface area element on Sn (Ω ). Let BΩ be the class of the potential a(P ) ∈ AΩ (P = (X, y) = (r, Θ) ∈ Tn (Ω )) such that limr→∞ r2 a(r) = κ0 ∈ [0, ∞) and r−1 |r2 a(r) − κ0 | ∈ L(1, ∞). We denote by SpHa (Ω ) the class of all weak solutions of SSE a u(P ) ≥ 0 for any P ∈ Tn (Ω ), which are continuous when a ∈ BΩ (see e.g. [4]). In the rest of paper, we will always assume that a ∈ BΩ . If u(P ) ∈ SpHa (Ω ) and −u(P ) ∈ SpHa (Ω ), then u(P ) is the solution of SSE a u(P ) = 0 for any P ∈ Tn (Ω ). The class of all solutions of it is denoted by Ha (Ω ). An important role will be played by the solutions of the ordinary differential equation − Π ′′ (y) + (λ + a(|y|)) Π (y) = 0
(−∞ < y < +∞),
(1.1)
which has two specially linearly independent positive solutions V (y) and W (y) on R such that V (y) is nondecreasing and W (y) is nonincreasing as y → ±∞ (see [3, Ch. 11, Appendix C] for more details). √ √ The solutions V (y) and W (y) of Eq. (1.1) have the asymptotics V (y) ≈ exp( λy) and W (y) ≈ exp(− λy), as y → ±∞ (see [3,5]). We remark that both V (y)ϕ(X) ∈ Ha (Ω ) and W (y)ϕ(X) ∈ Ha (Ω ) vanish continuously on Sn (Ω ). Let u(P ) ∈ SpHa (Ω ), we use the stand notations u+ = max{u, 0} and u− = − min{u, 0}. The integral u(P )ϕ(X)dΩ of u(P ) is denoted by N (u)(y) when it exists, where P = (X, y). The finite or infinite limits Ω limy→+∞ N (u)(y)/V (y) and limy→−∞ N (u)(y)/W (y) are denoted by UV (u) and VW (u) respectively, when they exist. About Carleman’s formula for harmonic functions in a half space and a smooth cone, we refer the reader to the papers by Zhang et al. (see [6]) and Ronkin (see [7]) respectively. Our main aim in this paper is to prove Carleman’s formula for weak solutions of the stationary Schr¨odinger equation in a truncated cylinder. An application of it will be also given later. Theorem 1.1. Let 0 < r < R < +∞ and define Ψ (y) = V (y)
W (y) W (R) − V (y) V (R)
where r < |y| < R. If u(X, y) ∈ SpHa (Ω ), then we have T(X, y)SSE a u(X, y)dw Tn (Ω,(−R,−r))
=
χ′ (R) N (u)(−R) + V (R)
u(X ′ , y ′ )Ψ (−y ′ )
Sn (Ω,(−R,−r))
∂ϕ(X ′ ) W (R) dσQ + d1 (−r) + d2 (−r) ∂nX ′ V (R)
(1.2)
and T(X, y)SSE a u(X, y)dw Tn (Ω,(r,R))
=
χ′ (R) N (u)(R) + V (R)
Sn (Ω,(r,R))
u(X ′ , y ′ )Ψ (y ′ )
∂ϕ(X ′ ) W (R) dσQ + d1 (r) + d2 (r) ∂nX ′ V (R)
(1.3)
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on Tn (Ω , (−R, −r)) and Tn (Ω , (r, R)) respectively, where dw denotes the elements of the Euclidean volume in Rn , dσQ denotes the surface area element on Sn (Ω ), χ′ (R) = w(W, V )|r=R is the Wronskian of W and V at r = R, T(X, y) = Ψ (y)ϕ(X), ∂u(X, −r) d1 (−r) = ϕ(X)dΩ , W ′ (r)u(X, −r) − W (r) ∂n Ω ∂u(X, −r) d2 (−r) = V (r) − V ′ (r)u(X, −r) ϕ(X)dΩ , ∂n Ω ∂u(X, r) ′ d1 (r) = W (r)u(X, r) − W (r) ϕ(X)dΩ ∂n Ω and d2 (r) = Ω
∂u(X, r) ′ − V (r)u(X, r) ϕ(X)dΩ . V (r) ∂n
As an application of Theorem 1.1, we finally give the integral representation for any h(P ) ∈ Ha (Ω ). To do this, we denote CΩ the class of f (X, y)((X, y) ∈ Tn (Ω )) satisfying +∞ V −1 (|y|)N (|f |)(y)dy < +∞ (1.4) −∞
and DΩ the class of g(Q) (Q = (X , y ′ ) ∈ Sn (Ω )) such that ∂ϕ(X ′ ) ′ dσX ′ dy ′ < +∞ V (y ) |g(Q)| ∂nX ′ −∞ ∂Ω ′
(1.5)
and
+∞
W (y ′ )
|g(Q)| ∂Ω
∂ϕ(X ′ ) dσX ′ ∂nX ′
dy ′ < +∞.
(1.6)
We denote by EΩ the class of h(X, y) ∈ Ha (Ω ) ((X, y) ∈ Tn (Ω )) with h+ (X, y) ∈ CΩ ((X, y) ∈ Tn (Ω )) and h+ (Q) ∈ DΩ (Q = (X ′ , y ′ ) ∈ Sn (Ω )). Recently, Li & Wu (see [8]) proved the existence of a soliton type solution for the fractional Schr¨odinger equation by using a constrained minimization argument. In a recent paper, Qiao & Pan (see [9]) also obtained solutions of the Schr¨ odinger equation with a slowly growing boundary potentials in a cone by constructing a modified Poisson–Sch kernel (with respect to SSE a ). Inspired by above results, our main results are as follows. Theorem 1.2. Let h ∈ EΩ . Then h ∈ DΩ . Theorem 1.3. Let 0 ≤ h ∈ EΩ . Then the solution h admits the following representation h(P ) = PI aΩ [h](P )+ (UV (h)V (y) + VW (h)W (y)) ϕ(X) for any P = (X, y) ∈ Tn (Ω ). 2. Lemmas In our discussions, the following estimates for the Poisson–Sch kernel PI aΩ (P, Q) are fundamental, which follow from [3, Ch. 11]. ′
′
) ∂ϕ(X ) a ′ Lemma 2.1. PI aΩ (P, Q) ≈ V (y)W (y ′ )ϕ(X) ∂ϕ(X ∂nX ′ and PI Ω (P, Q) ≈ V (y )W (y)ϕ(X) ∂nX ′ for any P = (X, Y ) ∈ Tn (Ω ) and any Q = (X ′ , y ′ ) ∈ Sn (Ω ) satisfying y ′ > y + 1 and y ′ < y − 1 respectively.
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Given a continuous function g on Sn (Ω ), we say that h is a solution of the Dirichlet–Sch problem (associated with the stationary Schr¨ odinger operator SSE a ) on Tn (Ω ) with g, if h ∈ Ha (Ω ) and limP →Q,P ∈Tn (Ω) h(P ) = g(Q) for every Q ∈ Sn (Ω ). Lemma 2.2 (See Qiao & Deng [10] in the Conical Version). Let g(Q)(Q = (X ′ , y ′ )) be a continuous function on Sn (Ω ) satisfying (1.5) and (1.6). Then the Poisson–Sch integral PI aΩ [g](P ) is a solution of the Dirichlet–Sch problem on Tn (Ω ) with g and satisfies VW (PI aΩ [g]) = 0 and UV (PI aΩ [g]) = 0. Lemma 2.3. Let 0 ≤ h(P ) ∈ Ha (Ω ) vanish continuously on Sn (Ω ). Then the solution h(P ) admits the following representation h(P ) = (UV (h)V (y) + VW (h)W (y)) ϕ(X) for any P = (X, y) ∈ Tn (Ω ). Proof. First of all, we note that h(X, y) is twice continuously differentiable on Tn (Ω ) (see [5, Th. 5.15]). ∂2 Now by differentiating twice under the integral sign, we have ∂y 2 N (h)(y) = − Ω △n−1 h(X, y)ϕ(X)dΩ . Hence, if we observe from Green–Sch’s formula (see [3, Ch.11]) that Ω △n−1 h(X, y)ϕ(X)dΩ = ∂2 h(X, y) △n−1 ϕ(X)dΩ , we see that ∂y 2 N (h)(y) = λN (h)(y) for any y ∈ R. This gives N (h)(y) = Ω c1 V (y) + c2 W (y) for any y ∈ R, where c1 and c2 are constants. Since c1 = UV (h) and c2 = VW (h), the conclusion of Lemma 2.3 follows immediately. 3. Proof of theorems 3.1. Proof of Theorem 1.1 We only prove (1.2), because (1.3) can be proved similarly. From the definition of T(X, y), we know that T(X, y) ∈ Ha (Ω ) and T(X, y) vanishes continuously on Sn (Ω ). By applying the second Green–Sch’s formula (see [3, Ch. 11]) to u(X, y) and T(X, y) on Tn (Ω ; (−R, −r)), we see that ∂u(X, y) ∂T(X, y) − T(X, y) dw I(X, y) =: u(X, y) ∂n ∂n Tn (Ω;(−R,−r)) = T(X, y)SSE a u(X, y)dw, (3.1) Tn (Ω,(−R,−r))
where ∂/∂n denotes the differentiation along the inward normal into Tn (Ω ; (−R, −r)). Put I(X, y) = I1 (X, y) + I2 (X, y) + I3 (X, y),
(3.2)
where
∂T(X, y) ∂u(X, y) I1 (X, y) = u(X, −R) − T(X, −R) dΩ , ∂n y=−R ∂n y=−R Ω ∂T(X, y) ∂u(X, y) I2 (X, y) = u(X, −r) − T(X, −r) dΩ ∂n y=−r ∂n y=−r Ω and
I3 (X, y) = Sn (Ω;(−R,−r))
u(X ′ , y ′ )
∂T(X ′ , y ′ ) ∂u(X ′ , y ′ ) − T(X ′ , y ′ ) ∂n ∂n
dσQ .
It is easy to see that T(X, −R) = 0,
∂T(X, y) χ′ (R) = ϕ(X), ∂n y=−R V (R)
(3.3)
L. Qiao / Applied Mathematics Letters 63 (2017) 34–39
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∂T(X, y) W (R) ′ ′ = W (r) − V (r) ϕ(X), ∂n y=−r V (R)
(3.4)
∂T(X ′ , y ′ ) ∂ϕ(X ′ ) = Ψ (y ′ ) . ∂n ∂nX ′
(3.5)
T(X ′ , y ′ ) = 0
and
Thus (1.2) follows from (3.1)–(3.5). 3.2. Proof of Theorem 1.2 We apply (1.2) with R > r = 1 to h = h+ − h− on Tn (Ω ; (−R, −1)), and then obtain W (R) ∂ϕ(X ′ ) dσQ + d1 (−1) + m+ (−R) + h+ (Q)Ψ (−y ′ ) d2 (−1) ′ ∂n V (R) X Sn (Ω,(−R,−1)) ∂ϕ(X ′ ) = m− (−R) + h− (Q)Ψ (−y ′ ) dσQ , ∂nX ′ Sn (Ω,(−R,−1))
(3.6)
′
(R) where Q = (X ′ , y ′ ) and m± (−R) = χV (R) N (h± )(−R). Without loss of generality we can assume R > 2, we have from (3.6) −1 ∂ϕ(X ′ ) W (R) V R2 ′ − m− (−R) + 1 − V (y ) h (Q) dσX ′ dy ′ ′ V (R) W R2 ∂n X −R ∂Ω 2 ∂ϕ(X ′ ) dσQ . m− (−R) + h− (Q)Ψ (−y ′ ) ∂nX ′ Sn (Ω,(−R,−1)) −1 ∂ϕ(X ′ ) ′ + dσX ′ dy ′ + |d1 (−1)| + |d2 (−1)|. . m+ (−R) + (3.7) V (y ) h (Q) ∂nX −R ∂Ω −1 −1 +∞ m+ (−R) Since h ∈ EΩ , we obtain from (1.4) 1 (−y)N (h+ )(y)dy < +∞, which gives that χ′ (R) dR = −∞ V
lim m+ (−R) < +∞.
R→+∞
(3.8)
Combining (3.7) and (3.8), we conclude that h− (Q) satisfies (1.5). If we apply (1.3) in place of (1.2) with R > r = 1 to h = h+ − h− on Tn (Ω ; (1, R)) h− (Q) satisfies (1.6). Hence Theorem 1.2 is proved from |h| = h+ + h− . 3.3. Proof of Theorem 1.3 For any P ∈ Tn (Ω ), define h′ (P ) = h(P ) − PI aΩ [h](P ). Then from Theorem 1.2 and Lemma 2.2 we know that UV (h′ ) = UV (h) and VW (h′ ) = VW (h). Hence, by applying Lemma 2.3 to h′ (P ), we obtain the conclusion of Theorem 1.3. Acknowledgments I would like to thank the anonymous referees for valuable comments and remarks. This paper was written while I was at the Laboratoire de Math´ematiques de Bretagne Atlantique of the Universit´e de Bretagne-Sud, Vannes, France, as a visiting scholar. I am grateful for the support and hospitality of this university, and also would like to express gratitude to Professor Quansheng Liu for his kind and cordial help. In particular, I am most grateful to the China Scholarship Council (Grant No. 201308410183) for giving me the financial support that allowed me to work in UBS for one year.
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References [1] G.V. Rozenblyum, M.Z. Solomyak, M.A. Shubin, Spectral theory of differential operators, in: Current Problems in Mathematics. Fundamental Directions, vol. 64, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform, Moscow, 1989, pp. 5–248. (in Russian). [2] H. Yoshida, Harmonic majorization of a subharmonic function on a cone or on a cylinder, Pacific J. Math. 148 (2) (1991) 369–395. [3] B. Levin, A. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schr¨ odinger operator, in: Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis, Science Press, Beijing, 2008, pp. 323–397. (Chapter 11). [4] B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. (NS) 7 (3) (1982) 447–526. [5] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, in: Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. [6] Y. Zhang, G. Deng, K. Kou, On the lower bound for a class of harmonic functions in the half space, Acta Math. Sci. Ser. B Engl. Ed. 32 (4) (2012) 1487–1494. [7] L. Ronkin, Functions of Completely Regular Growth, in: Mathematics and its Applications (Soviet Series), vol. 81, Kluwer Academic Publishers Group, Dordrecht, 1992, Translated from the Russian by A. Ronkin and I. Yedvabnik. [8] Q. Li, X. Wu, Soliton solutions for fractional Schr¨ odinger equations, Appl. Math. Lett. 53 (2016) 119–124. [9] L. Qiao, G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math. 17 (5) (2013) 1503–1521. [10] L. Qiao, G. Deng, Integral representation for the solution of the stationary Schr¨ odinger equation in a cone, Math. Nachr. 285 (16) (2012) 2029–2038.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Weak solutions for the stationary Schr¨odinger equation and its application✩ Lei Qiao School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China
article
info
Article history: Received 12 April 2016 Received in revised form 19 July 2016 Accepted 19 July 2016 Available online 27 July 2016
abstract In this paper, we prove Carleman’s formula for weak solutions of the stationary Schr¨ odinger equation in a cylinder. As an application of it, the integral representation of solutions of the stationary Schr¨ odinger equation is also obtained. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Carleman’s formula Stationary Schr¨ odinger equation Integral representation
1. Introduction and main results We introduce a system of spherical coordinates (r, Θ), Θ = (θ1 , θ2 , . . . , θn ), in Rn (n ≥ 2) which are related to Cartesian coordinates (x1 , x2 , . . . , θn−1 , y) by y = r cos θ1 . For positive functions h1 and h2 , we say that h1 . h2 if h1 ≤ ch2 for some constant c > 0. If h1 . h2 and h2 . h1 , then we say that h1 ≈ h2 . Let ∆n be the Laplace operator and Ω be a bounded domain in Rn−1 with smooth boundary ∂Ω . Consider the Dirichlet problem (∆n−1 + λ)ϕ = 0 on Ω and ϕ = 0 on ∂Ω (see [1, p. 41]). We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by ϕ, Ω ϕ2 (X)dΩ = 1, where X ∈ Ω and dΩ is the (n − 1)-dimensional volume element. The set Ω × R = {P = (X, y) ∈ Rn ; X ∈ Ω , y ∈ R} in Rn is simply denoted by Tn (Ω ). We call it a cylinder (see [2]). In the following, we denote the sets Ω × I and ∂Ω × I with an interval I on R by Tn (Ω ; I) and Sn (Ω ; I) respectively. Hence Sn (Ω ; R) denoted simply by Sn (Ω ) is ∂Tn (Ω ). Let AΩ denote the class of nonnegative radial potentials a(P ) (i.e. 0 ≤ a(P ) = a(r) for P = (X, y) = (r, Θ) ∈ Tn (Ω )) such that a ∈ Lbloc (Tn (Ω )) with some b > n/2 if n ≥ 4 and with b = 2 if n = 2 or n = 3. ✩
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301140, U1304102). E-mail address: [email protected].
http://dx.doi.org/10.1016/j.aml.2016.07.018 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
L. Qiao / Applied Mathematics Letters 63 (2017) 34–39
35
For the identical operator I and the potential a(P ) ∈ AΩ , we define the stationary Schr¨odinger operator by SSE a = −∆n + a(P )I, where SSE a can be extended in the usual way from the space C0∞ (Tn (Ω )) to an essentially self-adjoint operator on L2 (Tn (Ω )) (see [3, Ch. 11] for more details). Furthermore SSE a has a a a Green–Sch function GΩ (P, Q) (associated with the stationary Schr¨odinger operator SSE a ). Here GΩ (P, Q) a is positive on Tn (Ω ) and its inner normal derivative ∂GΩ (P, Q)/∂nQ is non-negative, where ∂/∂nQ denotes a the differentiation at Q along the inward normal into Tn (Ω ). We write 1/cn ∂GΩ (P, Q)/∂nQ by PI aΩ (P, Q), which is called the Poisson–Sch kernel (associated with the stationary Schr¨odinger operator SSE a ) on Tn (Ω ). Here, c2 = 2 and cn = (n − 2)wn when n ≥ 3, where wn is the surface area of the unit sphere in Rn . The Poisson–Sch integral PI aΩ [g](P ) of g (associated with the stationary Schr¨odinger operator SSE a ) on Tn (Ω ) is defined as follows PI aΩ [g](P ) = Sn (Ω) PI aΩ (P, Q)g(Q)dσQ , where g is a continuous function on Sn (Ω ) and dσQ is the surface area element on Sn (Ω ). Let BΩ be the class of the potential a(P ) ∈ AΩ (P = (X, y) = (r, Θ) ∈ Tn (Ω )) such that limr→∞ r2 a(r) = κ0 ∈ [0, ∞) and r−1 |r2 a(r) − κ0 | ∈ L(1, ∞). We denote by SpHa (Ω ) the class of all weak solutions of SSE a u(P ) ≥ 0 for any P ∈ Tn (Ω ), which are continuous when a ∈ BΩ (see e.g. [4]). In the rest of paper, we will always assume that a ∈ BΩ . If u(P ) ∈ SpHa (Ω ) and −u(P ) ∈ SpHa (Ω ), then u(P ) is the solution of SSE a u(P ) = 0 for any P ∈ Tn (Ω ). The class of all solutions of it is denoted by Ha (Ω ). An important role will be played by the solutions of the ordinary differential equation − Π ′′ (y) + (λ + a(|y|)) Π (y) = 0
(−∞ < y < +∞),
(1.1)
which has two specially linearly independent positive solutions V (y) and W (y) on R such that V (y) is nondecreasing and W (y) is nonincreasing as y → ±∞ (see [3, Ch. 11, Appendix C] for more details). √ √ The solutions V (y) and W (y) of Eq. (1.1) have the asymptotics V (y) ≈ exp( λy) and W (y) ≈ exp(− λy), as y → ±∞ (see [3,5]). We remark that both V (y)ϕ(X) ∈ Ha (Ω ) and W (y)ϕ(X) ∈ Ha (Ω ) vanish continuously on Sn (Ω ). Let u(P ) ∈ SpHa (Ω ), we use the stand notations u+ = max{u, 0} and u− = − min{u, 0}. The integral u(P )ϕ(X)dΩ of u(P ) is denoted by N (u)(y) when it exists, where P = (X, y). The finite or infinite limits Ω limy→+∞ N (u)(y)/V (y) and limy→−∞ N (u)(y)/W (y) are denoted by UV (u) and VW (u) respectively, when they exist. About Carleman’s formula for harmonic functions in a half space and a smooth cone, we refer the reader to the papers by Zhang et al. (see [6]) and Ronkin (see [7]) respectively. Our main aim in this paper is to prove Carleman’s formula for weak solutions of the stationary Schr¨odinger equation in a truncated cylinder. An application of it will be also given later. Theorem 1.1. Let 0 < r < R < +∞ and define Ψ (y) = V (y)
W (y) W (R) − V (y) V (R)
where r < |y| < R. If u(X, y) ∈ SpHa (Ω ), then we have T(X, y)SSE a u(X, y)dw Tn (Ω,(−R,−r))
=
χ′ (R) N (u)(−R) + V (R)
u(X ′ , y ′ )Ψ (−y ′ )
Sn (Ω,(−R,−r))
∂ϕ(X ′ ) W (R) dσQ + d1 (−r) + d2 (−r) ∂nX ′ V (R)
(1.2)
and T(X, y)SSE a u(X, y)dw Tn (Ω,(r,R))
=
χ′ (R) N (u)(R) + V (R)
Sn (Ω,(r,R))
u(X ′ , y ′ )Ψ (y ′ )
∂ϕ(X ′ ) W (R) dσQ + d1 (r) + d2 (r) ∂nX ′ V (R)
(1.3)
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on Tn (Ω , (−R, −r)) and Tn (Ω , (r, R)) respectively, where dw denotes the elements of the Euclidean volume in Rn , dσQ denotes the surface area element on Sn (Ω ), χ′ (R) = w(W, V )|r=R is the Wronskian of W and V at r = R, T(X, y) = Ψ (y)ϕ(X), ∂u(X, −r) d1 (−r) = ϕ(X)dΩ , W ′ (r)u(X, −r) − W (r) ∂n Ω ∂u(X, −r) d2 (−r) = V (r) − V ′ (r)u(X, −r) ϕ(X)dΩ , ∂n Ω ∂u(X, r) ′ d1 (r) = W (r)u(X, r) − W (r) ϕ(X)dΩ ∂n Ω and d2 (r) = Ω
∂u(X, r) ′ − V (r)u(X, r) ϕ(X)dΩ . V (r) ∂n
As an application of Theorem 1.1, we finally give the integral representation for any h(P ) ∈ Ha (Ω ). To do this, we denote CΩ the class of f (X, y)((X, y) ∈ Tn (Ω )) satisfying +∞ V −1 (|y|)N (|f |)(y)dy < +∞ (1.4) −∞
and DΩ the class of g(Q) (Q = (X , y ′ ) ∈ Sn (Ω )) such that ∂ϕ(X ′ ) ′ dσX ′ dy ′ < +∞ V (y ) |g(Q)| ∂nX ′ −∞ ∂Ω ′
(1.5)
and
+∞
W (y ′ )
|g(Q)| ∂Ω
∂ϕ(X ′ ) dσX ′ ∂nX ′
dy ′ < +∞.
(1.6)
We denote by EΩ the class of h(X, y) ∈ Ha (Ω ) ((X, y) ∈ Tn (Ω )) with h+ (X, y) ∈ CΩ ((X, y) ∈ Tn (Ω )) and h+ (Q) ∈ DΩ (Q = (X ′ , y ′ ) ∈ Sn (Ω )). Recently, Li & Wu (see [8]) proved the existence of a soliton type solution for the fractional Schr¨odinger equation by using a constrained minimization argument. In a recent paper, Qiao & Pan (see [9]) also obtained solutions of the Schr¨ odinger equation with a slowly growing boundary potentials in a cone by constructing a modified Poisson–Sch kernel (with respect to SSE a ). Inspired by above results, our main results are as follows. Theorem 1.2. Let h ∈ EΩ . Then h ∈ DΩ . Theorem 1.3. Let 0 ≤ h ∈ EΩ . Then the solution h admits the following representation h(P ) = PI aΩ [h](P )+ (UV (h)V (y) + VW (h)W (y)) ϕ(X) for any P = (X, y) ∈ Tn (Ω ). 2. Lemmas In our discussions, the following estimates for the Poisson–Sch kernel PI aΩ (P, Q) are fundamental, which follow from [3, Ch. 11]. ′
′
) ∂ϕ(X ) a ′ Lemma 2.1. PI aΩ (P, Q) ≈ V (y)W (y ′ )ϕ(X) ∂ϕ(X ∂nX ′ and PI Ω (P, Q) ≈ V (y )W (y)ϕ(X) ∂nX ′ for any P = (X, Y ) ∈ Tn (Ω ) and any Q = (X ′ , y ′ ) ∈ Sn (Ω ) satisfying y ′ > y + 1 and y ′ < y − 1 respectively.
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Given a continuous function g on Sn (Ω ), we say that h is a solution of the Dirichlet–Sch problem (associated with the stationary Schr¨ odinger operator SSE a ) on Tn (Ω ) with g, if h ∈ Ha (Ω ) and limP →Q,P ∈Tn (Ω) h(P ) = g(Q) for every Q ∈ Sn (Ω ). Lemma 2.2 (See Qiao & Deng [10] in the Conical Version). Let g(Q)(Q = (X ′ , y ′ )) be a continuous function on Sn (Ω ) satisfying (1.5) and (1.6). Then the Poisson–Sch integral PI aΩ [g](P ) is a solution of the Dirichlet–Sch problem on Tn (Ω ) with g and satisfies VW (PI aΩ [g]) = 0 and UV (PI aΩ [g]) = 0. Lemma 2.3. Let 0 ≤ h(P ) ∈ Ha (Ω ) vanish continuously on Sn (Ω ). Then the solution h(P ) admits the following representation h(P ) = (UV (h)V (y) + VW (h)W (y)) ϕ(X) for any P = (X, y) ∈ Tn (Ω ). Proof. First of all, we note that h(X, y) is twice continuously differentiable on Tn (Ω ) (see [5, Th. 5.15]). ∂2 Now by differentiating twice under the integral sign, we have ∂y 2 N (h)(y) = − Ω △n−1 h(X, y)ϕ(X)dΩ . Hence, if we observe from Green–Sch’s formula (see [3, Ch.11]) that Ω △n−1 h(X, y)ϕ(X)dΩ = ∂2 h(X, y) △n−1 ϕ(X)dΩ , we see that ∂y 2 N (h)(y) = λN (h)(y) for any y ∈ R. This gives N (h)(y) = Ω c1 V (y) + c2 W (y) for any y ∈ R, where c1 and c2 are constants. Since c1 = UV (h) and c2 = VW (h), the conclusion of Lemma 2.3 follows immediately. 3. Proof of theorems 3.1. Proof of Theorem 1.1 We only prove (1.2), because (1.3) can be proved similarly. From the definition of T(X, y), we know that T(X, y) ∈ Ha (Ω ) and T(X, y) vanishes continuously on Sn (Ω ). By applying the second Green–Sch’s formula (see [3, Ch. 11]) to u(X, y) and T(X, y) on Tn (Ω ; (−R, −r)), we see that ∂u(X, y) ∂T(X, y) − T(X, y) dw I(X, y) =: u(X, y) ∂n ∂n Tn (Ω;(−R,−r)) = T(X, y)SSE a u(X, y)dw, (3.1) Tn (Ω,(−R,−r))
where ∂/∂n denotes the differentiation along the inward normal into Tn (Ω ; (−R, −r)). Put I(X, y) = I1 (X, y) + I2 (X, y) + I3 (X, y),
(3.2)
where
∂T(X, y) ∂u(X, y) I1 (X, y) = u(X, −R) − T(X, −R) dΩ , ∂n y=−R ∂n y=−R Ω ∂T(X, y) ∂u(X, y) I2 (X, y) = u(X, −r) − T(X, −r) dΩ ∂n y=−r ∂n y=−r Ω and
I3 (X, y) = Sn (Ω;(−R,−r))
u(X ′ , y ′ )
∂T(X ′ , y ′ ) ∂u(X ′ , y ′ ) − T(X ′ , y ′ ) ∂n ∂n
dσQ .
It is easy to see that T(X, −R) = 0,
∂T(X, y) χ′ (R) = ϕ(X), ∂n y=−R V (R)
(3.3)
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∂T(X, y) W (R) ′ ′ = W (r) − V (r) ϕ(X), ∂n y=−r V (R)
(3.4)
∂T(X ′ , y ′ ) ∂ϕ(X ′ ) = Ψ (y ′ ) . ∂n ∂nX ′
(3.5)
T(X ′ , y ′ ) = 0
and
Thus (1.2) follows from (3.1)–(3.5). 3.2. Proof of Theorem 1.2 We apply (1.2) with R > r = 1 to h = h+ − h− on Tn (Ω ; (−R, −1)), and then obtain W (R) ∂ϕ(X ′ ) dσQ + d1 (−1) + m+ (−R) + h+ (Q)Ψ (−y ′ ) d2 (−1) ′ ∂n V (R) X Sn (Ω,(−R,−1)) ∂ϕ(X ′ ) = m− (−R) + h− (Q)Ψ (−y ′ ) dσQ , ∂nX ′ Sn (Ω,(−R,−1))
(3.6)
′
(R) where Q = (X ′ , y ′ ) and m± (−R) = χV (R) N (h± )(−R). Without loss of generality we can assume R > 2, we have from (3.6) −1 ∂ϕ(X ′ ) W (R) V R2 ′ − m− (−R) + 1 − V (y ) h (Q) dσX ′ dy ′ ′ V (R) W R2 ∂n X −R ∂Ω 2 ∂ϕ(X ′ ) dσQ . m− (−R) + h− (Q)Ψ (−y ′ ) ∂nX ′ Sn (Ω,(−R,−1)) −1 ∂ϕ(X ′ ) ′ + dσX ′ dy ′ + |d1 (−1)| + |d2 (−1)|. . m+ (−R) + (3.7) V (y ) h (Q) ∂nX −R ∂Ω −1 −1 +∞ m+ (−R) Since h ∈ EΩ , we obtain from (1.4) 1 (−y)N (h+ )(y)dy < +∞, which gives that χ′ (R) dR = −∞ V
lim m+ (−R) < +∞.
R→+∞
(3.8)
Combining (3.7) and (3.8), we conclude that h− (Q) satisfies (1.5). If we apply (1.3) in place of (1.2) with R > r = 1 to h = h+ − h− on Tn (Ω ; (1, R)) h− (Q) satisfies (1.6). Hence Theorem 1.2 is proved from |h| = h+ + h− . 3.3. Proof of Theorem 1.3 For any P ∈ Tn (Ω ), define h′ (P ) = h(P ) − PI aΩ [h](P ). Then from Theorem 1.2 and Lemma 2.2 we know that UV (h′ ) = UV (h) and VW (h′ ) = VW (h). Hence, by applying Lemma 2.3 to h′ (P ), we obtain the conclusion of Theorem 1.3. Acknowledgments I would like to thank the anonymous referees for valuable comments and remarks. This paper was written while I was at the Laboratoire de Math´ematiques de Bretagne Atlantique of the Universit´e de Bretagne-Sud, Vannes, France, as a visiting scholar. I am grateful for the support and hospitality of this university, and also would like to express gratitude to Professor Quansheng Liu for his kind and cordial help. In particular, I am most grateful to the China Scholarship Council (Grant No. 201308410183) for giving me the financial support that allowed me to work in UBS for one year.
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