Factorization of the Non-Stationary Schrodinger Operator
aa r X i v : . [ m a t h - ph ] J un Factorization of the Non-Stationary Schr¨odingerOperator
Paula Cerejeiras Nelson Vieira
Department of Mathematics,University of Aveiro,3810-193 Aveiro, Portugal.E-mails: [email protected], [email protected]
November 21, 2018
Abstract
We consider a factorization of the non-stationary Schr¨odinger ope-rator based on the parabolic Dirac operator introduced by Cerejeiras/K¨ahler/ Sommen. Based on the fundamental solution for the parabolicDirac operators, we shall construct appropriated Teodorescu and Cauchy-Bitsadze operators. Afterwards we will describe how to solve the nonlinearSchr¨odinger equation using Banach fixed point theorem.
Keywords:
Nonlinear PDE’s, Parabolic Dirac operators, Iterative Meth-ods
MSC 2000:
Primary: 30G35; Secundary: 35A08, 15A66.
Time evolution problems are of extreme importance in mathematical physics.However, there is still a need for special techniques to deal with these problems,specially when non-linearities are involved.For stationary problems, the theory developed by K. G¨urlebeck and W.Spr¨oßig [10], based on an orthogonal decomposition of the underlying functionspace in terms of the subspace of null-solutions of the corresponding Dirac op-erator, has been successfully applied to a wide range of equations, for instanceLam´e, Navier-Stokes, Maxwell or Schr¨odinger equations [6], [10], [11], [2] or[13]. Unfortunately, there is no easy way to extend this theory directly to non-stationary problems.In [7] the authors proposed an alternative approach in terms of a Witt basis.This approach allowed a successful application of the already existent tech-niques of elliptic function theory (see [10], [6]) to non-stationary problems intime-varying domains. Namely, a suitable orthogonal decomposition for the1nderlying function space was obtained in terms of the kernel of the parabolicDirac operator and its range after application to a Sobolev space with zeroboundary-values.In this paper we wish to apply this approach to study the existence anduniqueness of solutions of the non-stationary nonlinear Schr¨odinger equation.Initially, in section two, we will present some basic notions about complexi-fied Clifford algebras and Witt basis. In section three we will present a factoriza-tion for the operators ( ± i∂ t − ∆) using an extension of the parabolic Dirac oper-ator introduced in [7]. For the particular case of the non-stationary Schr¨odingeroperator we will present the corresponding Teodorescu and Cauchy-Bitsadzeoperators in analogy to [10]. Moreover, we will obtain some direct results aboutthe decomposition of L p -spaces and the resolution of the linear Schr¨odingerproblem.In the last section we will present an algorithm to solve numerically thenon-linear Schr¨odinger and we prove its convergence in L -sense using Banach’sfixed point theorem. We consider the m -dimensional vector space R m endowed with an orthonormalbasis { e , · · · , e m } . We define the universal Clifford algebra Cℓ ,m as the 2 m -dimensional associa-tive algebra which preserves the multiplication rules e i e j + e j e i = − δ i,j . A basisfor Cℓ ,m is given by e = 1 and e A = e h · · · e h k , where A = { h , . . . , h k } ⊂ M = { , . . . , m } , for 1 ≤ h < · · · < h k ≤ m . Each element x ∈ Cℓ ,m will be represented by x = P A x A e A , x A ∈ R , and each non-zero vector x = P mj =1 x j e j ∈ R m has a multiplicative inverse given by − x | x | . We denoteby x Cℓ ,m the (Clifford) conjugate of the element x ∈ Cℓ ,m , where1 Cℓ ,m = 1 , e jCℓ ,m = − e j , ab Cℓ ,m = b Cℓ ,m a Cℓ ,m . We introduce the complexified Clifford algebra Cℓ m as the tensorial product C ⊗ Cℓ ,m = ( w = X A z A e A , z A ∈ C , A ⊂ M ) where the imaginary unit interacts with the basis elements via ie j = e j i, j =1 , . . . , m. The conjugation in Cℓ m = C ⊗ Cℓ ,m will be defined as w = P A z A C e ACℓ ,m . Let us remark that for a, b ∈ Cℓ m we have | ab | ≤ m | a || b | .We introduce the Dirac operator D = P mj =1 e j ∂ x i . It factorizes the m -di-mensional Laplacian, that is, D = − ∆. A Cℓ m -valued function defined on anopen domain Ω , u : Ω ⊂ R m Cℓ m , is said to be left-monogenic if it satisfies Du = 0 on Ω (resp. right-monogenic if it satisfies uD = 0 on Ω).A function u : Ω Cℓ m has a representation u = P A u A e A with C -valuedcomponents u A . Properties such as continuity will be understood component-wisely. In the following we will use the short notation L p (Ω), C k (Ω), etc.,2nstead of L p (Ω , Cℓ m ), C k (Ω , Cℓ m ). For more details on Clifford analysis, see[5], [12], [4] or [9].Taking into account [7] we will imbed R m into R m +2 . For that purpose weadd two new basis elements f and f † satisfying f = f † = 0 , ff † + f † f = 1 , f e j + e j f = f † e j + e j f † = 0 , j = 1 , · · · , m. This construction will allows us to use a suitable factorization of the time evo-lution operators where only partial derivatives are used.
In this section we will study the forward/backward Schr¨odinger equations,( ± i∂ t − ∆) u ( x, t ) = 0 , ( x, t ) ∈ Ω , (1)where Ω ⊂ R m × R + , m ≥ , stands for an open domain in R m × R + . We remarkat this point that Ω is a time-variating domain and, therefore, not necessarily acylindric domain.Taking account the ideas presented in [1] and [7] we introduce the followingdefinition Definition 3.1.
For a function u ∈ W p (Ω) , < p < + ∞ , we define the forward(resp. backward) parabolic Dirac operator D x, ± it u = ( D + f ∂ t ± i f † ) u, (2) where D stands for the (spatial) Dirac operator. It is obvious that D x, ± it : W p (Ω) → L p (Ω).These operators factorize the correspondent time-evolution operator (1),that is ( D x, ± it ) u = ( ± i∂ t − ∆) u. (3)Moreover, we consider the generic Stokes’ Theorem Theorem 3.2.
For each u, v ∈ W p (Ω) , < p < ∞ , it holds Z Ω vdσ x,t u = Z ∂ Ω [( vD x, − it ) u + v ( D x, + it u )] dxdt where the surface element is dσ x,t = ( D x + f ∂ t ) ⌋ dxdt, the contraction of thehomogeneous operator associated to D x, − it with the volume element. We now construct the fundamental solution for the time-evolution operator − ∆ − i∂ t . For that purpose, we consider the fundamental solution of the heatoperator e ( x, t ) = H ( t )(4 πt ) m exp (cid:18) − | x | t (cid:19) , (4)3here H ( t ) denotes the Heaviside-function. Let us remark that the previousfundamental solution verifies( − ∆ + ∂ t ) e ( x, t ) = δ ( x ) δ ( t ) . We apply to (4) the rotation t → it . There we obtain( − ∆ − i∂ t ) e ( x, it ) = − ∆ e ( x, it ) + ∂ it e ( x, it ) = δ ( x ) δ ( it ) = − iδ ( x ) δ ( t ) , i.e., the fundamental solution for the Schr¨odinger operator − ∆ − i∂ t is e − ( x, t ) = ie ( x, it )= i H ( t )(4 πit ) m exp (cid:18) i | x | t (cid:19) . (5)Then we have Definition 3.3.
Given the fundamental solution e − = e − ( x, t ) we have asfundamental solution E − = E − ( x, t ) for the parabolic Dirac operator D x, − it the function E − ( x, t ) = e − ( x, t ) D x, − it = H ( t )(4 πit ) m exp (cid:18) i | x | t (cid:19) (cid:18) − x t + f (cid:18) | x | t − im t (cid:19) + f † (cid:19) (6)If we replace the function v by the fundamental solution E − in the genericStoke’s formula presented before, we have, for a function u ∈ W p (Ω) and a point( x , t ) / ∈ ∂ Ω, the Borel-Pompeiu formula, Z ∂ Ω E − ( x − x , t − t ) dσ x,t u ( x, t )= u ( x , t ) + Z Ω E − ( x − x , t − t )( D x, + it u ) dxdt. (7)Moreover, if u ∈ ker ( D x, + it ) we obtain the Cauchy’s integral formula Z ∂ Ω E − ( x − x , t − t ) dσ x,t u ( x, t ) = u ( x , t ) . Based on expression (7) we define the Teodorescu and Cauchy-Bitsadze op-erators.
Definition 3.4.
For a function u ∈ L p (Ω) we have(a) the Teodorescu operator T − u ( x , t ) = Z Ω E − ( x − x , t − t ) u ( x, t ) dxdt (8) (b) the Cauchy-Bitsadze operator F − u ( x , t ) = Z ∂ Ω E − ( x − x , t − t ) dσ x,t u ( x, t ) , (9) for ( x , t ) / ∈ ∂ Ω . F − u = u + T − D x, + it u, whenever v ∈ W p (Ω) , < p < ∞ . Moreover, the Teodurescu operator is the right inverse of the parabolic Diracoperator D x, − it ) , that is, D x, − it T u = Z Ω D x, − it E − ( x − x , t − t ) u ( x, t ) dxdt = Z Ω δ ( x − x , t − t ) u ( x, t ) dxdt = u ( x , t ) , for all ( x , t ) ∈ Ω . In view of the previous definitions and relations, we obtain the followingresults, in an analogous way as in [7].
Theorem 3.5. If v ∈ W p ( ∂ Ω) then the trace of the operator F − is tr( F − v ) = 12 v − S − v, (10) where S − v ( x , t ) = Z ∂ Ω E − ( x − x , t − t ) dσ x,t v ( x, t ) is a generalization of the Hilbert transform. Also, the operator S − satisfies S − = I and, therefore, the operators P = 12 I + 12 S − , Q = 12 I − S − are projections into the Hardy spaces.Taking account the ideas presented in [7] an immediate application is givenby the decomposition of the L p − space. Theorem 3.6.
The space L p (Ω) , for < p ≤ , allows the following decompo-sition L p (Ω) = L p (Ω) ∩ ker ( D x, − it ) ⊕ D x,it (cid:18) ◦ W p (Ω) (cid:19) , and we can define the following projectors P − : L p (Ω) → L p (Ω) ∩ ker ( D x, − it ) Q − : L p (Ω) → D x, − it (cid:18) ◦ W p (Ω) (cid:19) . roof. Let us denote by ( − ∆ − i∂ t ) − the solution operator of the problem ( − ∆ − i∂ t ) u = f in Ω u = 0 on ∂ ΩAs a first step we take a look at the intersection of the two subspaces D x, − it (cid:18) ◦ W p (Ω) (cid:19) and L p (Ω) ∩ ker ( D x, − it ).Consider u ∈ L p (Ω) ∩ ker ( D x, − it ) ∩ D x, − it (cid:18) ◦ W p (Ω) (cid:19) . It is immediate that D x, − it u = 0 and also, because u ∈ D x, − it (cid:18) ◦ W p (Ω) (cid:19) , there exist a function v ∈ ◦ W p (Ω) with D x, − it v = u and ( − ∆ − i∂ t ) v = 0.Since ( − ∆ − i∂ t ) − f is unique (see [14]) we get v = 0 and, consequently, u = 0, i. e., the intersection of this subspaces contains only the zero function.Therefore, our sum is a direct sum.Now let us u ∈ L p (Ω). Then we have u = D x, − it ( − ∆ − i∂ t ) − D x, − it u ∈ D x, − it (cid:18) ◦ W p (Ω) (cid:19) . Let us now apply D x, − it to the function u = u − u . This results in D x, − it u = D x, − it u − D x, − it u = D x, − it u − D x, − it D x, − it ( − ∆ − i∂ t ) − D x, − it u = D x, − it u − ( − ∆ − i∂ t )( − ∆ − i∂ t ) − D x, − it u = D x, − it u − D x, − it u = 0 , i.e., D x, − it u ∈ ker ( D x, − it ). Because u ∈ L p (Ω) was arbitrary chosen ourdecomposition is a decomposition of the space L p (Ω).In a similar way we can obtain a decomposition of the L p (Ω) space in termsof the parabolic Dirac operator D x, + it . Moreover, let us remark that the abovedecompositions are orthogonal in the case of p = 2 . Using the previous definitions we can also present an immediate applicationin the resolution of the linear Schr¨odinger problem with homogeneous boundarydata.
Theorem 3.7.
Let f ∈ L p (Ω) , < p ≤ . The solution of the problem ( − ∆ − i∂ t ) u = f in Ω u = 0 on ∂ Ω is given by u = T − Q − T − f. roof. The proof of this theorem is based on the properties of the operator T − and of the projector Q − . Because T − is the right inverse of D x, − it , we get D x, − it u = D x, − it ( Q − T − f ) = D x, − it ( T − f ) = f. In this section we will construct an iterative method for the non-linear Schr¨odingerequation and we study is convergence. As usual, we consider the L − norm || f || = Z Ω [ f f ] dxdt, where [ · ] denotes the scalar part.Moreover, we also need the mixed Sobolev spaces W α,βp (Ω). For this weintroduce the conventionΩ t = { x : ( x, t ) ∈ Ω } ⊂ R m Ω x = { t : ( x, t ) ∈ Ω } ⊂ R + . Then, we say that u ∈ W α,βp (Ω) iff u ( · , t ) ∈ W αp (Ω t ) , ∀ tu ( x, · ) ∈ W βp (Ω x ) , ∀ x Under this conditions we will study the (generalized) non-linear Schr¨odingerproblem: − ∆ x u − i∂ t u + | u | u = f in Ω (11) u = 0 on ∂ Ω , where | u | = P A | u A | . We can rewrite (11) as D x, − it u + M ( u ) = 0 , (12)where M ( u ) = | u | u − f . It is easy to see that u = − T − Q − T − ( M ( u )) (13)is a solution of (12) by means of direct application of D x, − it to both sides ofthe equation.We remark that for u ∈ W , (Ω), we get || D x, − it u || = || Q − T − M ( u ) || = || T − M ( u ) || .
7e now prove that (13) can be solved by the convergent iterative method u n = − T − Q − T − ( M ( u n − )) . (14)For that purpose we need to establish some norm estimations. Initially, wehave that || u n − u n − || = || T − Q − T − [ M ( u n − ) − M ( u n − )] ||≤ C || M ( u n − ) − M ( u n − ) || , (15)where C = || T − Q − T − || = || T − || .We now estimate the factor || M ( u n − ) − M ( u n − ) || . We get || M ( u n − ) − M ( u n − ) || = ||| u n − | u n − − | u n − | u n − ||≤ ||| u n − | ( u n − − u n − ) || + ||| u n − − u n − | u n − ||≤ m +1 || u n − − u n − || (cid:0) || u n − || + || u n − |||| u n − − u n − || (cid:1) , We assume K n := 2 m +1 (cid:0) || u n − || + || u n − |||| u n − − u n − || (cid:1) so that || u n − u n − || ≤ C K n || u n − − u n − || . Moreover, we have additionally that || u n || = || T − Q − T − M ( u n − ) ||≤ m +1 C || u n − || + C || f || (16)holds.In order to prove that indeed we have a contraction we need to study theauxiliary inequality 2 m +1 C || u n − || + C || f || ≤ || u n − || , that is, || u n − || − || u n − || m +1 C + || f || m +1 ≤ . (17)The analysis of (17) will be made considering two cases Case I:
When || u n − || ≥
1, we can establish the following inequality inrelation to (17) || u n − || − || u n − || · m +1 + || f || m +1 ≤ || u n − || − || u n − || m +1 C + || f || m +1 . || u n − || − || u n − || · m +1 + || f || m +1 ≤ || u n − || − || u n − || · m +1 + 136 · m +2 + || f || m +1 − · m +2 ≤ (cid:18) || u n − || − · m +1 (cid:19) ≤ · m +1) − || f || m +1 = 12 m +1 (cid:18) · m +1 − || f || (cid:19) . (18)If || f || ≤ · m +1 then (cid:12)(cid:12)(cid:12)(cid:12) || u n − || − · m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ W, where W = q · m +1) − || f || m +1 .In consequence, if16 · m +1 − W ≤ || u n − || ≤ · m +1 + W then we have from (16) the desired inequality || u n || ≤ || u n − || . Furthermore, we have now to study the remaining case. Assuming now that || u n − || ≤ · m +1 − W , we have || u n || ≤ m +1 C (cid:18) · m +1 − W (cid:19) + C || f || ≤ · m +1 − W and || u n − || ≤ · m +1 − W, || u n − || ≤ · m +1 − W so that it holds || u n − − u n − || ≤ (cid:18) · m +1 − W (cid:19) . With the previous relations we can estimate the value of K n K n = 2 m +1 (cid:0) || u n − || + || u n − |||| u n − − u n − || (cid:1) ≤ m +1 "(cid:18) · m +1 − W (cid:19) + 2 (cid:18) · m +1 − W (cid:19) ≤ · m +1 (cid:18) · m +1 − W (cid:19) = 12 − · m +1 W < , (19)9hich implies that || u n − || ≤ R := 13 · m +1 . Finally, we have that || u n − u n − || ≤ K n || u n − − u n − || , with K n < . Case II:
When || u n − || <
1, we can establish the following inequality || u n − || − || u n − || · m +1 + || f || m +1 ≤ || u n − || − || u n − || m +1 C + || f || m +1 . Then, from (17), we have || u n − || − || u n − || · m +1 + || f || m +1 ≤ ⇔ (cid:0) || u n − || − · m +1 (cid:1) ≤ · m +2 − || f || m +1 . (20)Again, if || f || ≤ · m +1 then (cid:12)(cid:12)(cid:12)(cid:12) || u n − || − · m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ W, where W = q · m +2 − || f || m +1 .As a consequence, · m +1 − W ≤ || u n − || ≤ · m +1 + W ⇔ q · m +1 − W ≤ || u n − || ≤ q · m +1 + W leads to || u n || ≤ || u n − || . Again, considering now the case of || u n − || ≤ q · m +1 − W , we obtain || u n || ≤ m +1 C (cid:16)q · m +1 − W (cid:17) + C || f || ≤ r · m +1 − W and || u n − || ≤ q · m +1 − W || u n − || ≤ q · m +1 − W − W || u n − − u n − || ≤ q · m +1 − W . K n K n = 2 m +1 (cid:0) || u n − || + || u n − |||| u n − − u n − || (cid:1) ≤ m +1 (cid:20)(cid:18) · m +1 − W (cid:19) + 2 (cid:18) · m +1 − W (cid:19)(cid:21) = 3 · m +1 (cid:18) · m +1 − W (cid:19) = 12 − · m +1 W < , (21)which implies that || u n − || ≤ R := 13 · m +1 . Finally, we have that || u n − u n − || ≤ K n || u n − − u n − || , with K n < . The application of Banach’s fixed point, to the previous conclusions, resultsin the following theorem
Theorem 4.1.
The problem (11) has a unique solution u ∈ W , (Ω) if f ∈ L (Ω) satisfies the condition || f || ≤ · m +1 . Moreover, our iteration method (14) converges for each starting point u ∈ ◦ W , (Ω) such that || u || ≤ · m +1 + W, with W = q · m +1) − || f || m +1 . Acknowledgement
The second author wishes to express his gratitude toFunda¸c˜ao para a Ciˆencia e a Tecnologia for the support of his work via thegrant
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