Cauchy-Neumann Problem for Second-Order General Schrödinger Equations in Cylinders with Nonsmooth Bases

The main goal of this paper is to obtain the regularity of weak solutions of Cauchy-Neumann problems for the second-order general Schrödinger equations in domains with conical points on the boundary of the bases.

Cauchy-Dirichlet problem for general Schrödinger systems in domains containing conical points has been investigated in [1, 2]. Cauchy-Neumann problems have been dealt with for hyperbolic systems in [3] and for parabolic equations in [4–6]. In this paper we consider the Cauchy-Neumann problem for the second-order general Schrödinger equations in infinite cylinders with nonsmooth bases. The solvability of this problem has been considered in [7]. Our main purpose here is to study the regularity of weak solution of the mentioned problem.

The paper consists of six sections. In Section 1, we introduce some notations and functional spaces used throughout the text. A weak solution of the problem is defined in Section 2 together with some results of its unique existence and smoothness with the time variable. Our main result, the regularity with respect to both of time and spatial variables of the weak solution of the problem, is stated in Section 3. The proof of this result is given in Section 4 with some auxiliary lemmas. In Section 5 we specify that result for the classical Schrödinger equations in quantum mechanics. Finally, some conclusions of our results are given in Section 6.

Let be a bounded domain in and denote the closure and the boundary of in . We suppose that is an infinitely differentiable surface everywhere except the coordinate origin and coincides with the cone in a neighborhood of the origin point where is a smooth domain on the unit sphere in We begin by introducing some notations and functional spaces which are used fluently in the rest.

Denote , is the closure of , . For each multi-index , set , .

In this paper we will use usual functional spaces: , where (see [1, 2] for the precise definitions).

Denote is a space of all measurable complex functions that satisfy

—a space of all measurable complex functions that have generalized derivatives up to order with respect to and up to order with respect to with the norm

—a space of all measurable complex functions with the norm

—a weighted space with the norm

Let be a Banach space. Denote by a space of all measurable functions with the norm

In this paper we consider following problem:

where is a formal self-adjoint differential operator of second-order defined in :

and

is the conormal derivative on is the unit exterior normal to is a given function.

Set

Throughout this paper, we assume that the coefficients of are infinitely differentiable and bounded in together with all their derivatives. Moreover, suppose that are continuous in uniformly with respect to for all In addition, assume that is —coercive uniformly with respect to that is,

where is a positive constant independent of and

The function is called a weak solution in the space of the problem (2.1)–(2.3) if , satisfying for each

for all test functions , for all .

Now we derive here some our obvious results of the unique existence and smoothness with respect to time variable of the weak solution of the problem (2.1)–(2.3) as lemmas of main results.

Lemma 2.1.

The solvability of the problem, (see [7, Theorems 3.1, 3.2]). There exists a positive number such that if then for every the Cauchy-Neumann problem (2.1)–(2.3) has exactly one weak solution in , that satisfies

where the constant does not depend on , .

The constant depends only on the operator and the dimension of the space

Lemma 2.2.

The regularity with respect to time variable of the weak solution (see [7, Theorem 4.1]).Let be a nonnegative integer. Suppose that for all and if then for all for all Then for every , the weak solution of the problem (2.1)–(2.3) has generalized derivatives with respect to time variable up to order , which belong to moreover

where is a constant independent of , .

Let be the principal homogenous part of We can write in the form

where is an arbitrary local coordinate system on , is a linear operator with smooth coefficients.

Denote is an eigenvalue of Neumann problem for following equation:

It is well known in [8] that for each the spectrum of this problem is an enumerable set of eigenvalues.

Recall that is the positive real number in Lemma 2.1. Now, let us give the main result of the present paper.

Theorem 3.1.

Let be a nonnegative integer. Assume that is a weak solution in the space with of the problem (2.1)–(2.3) and if , if . In addition, suppose that in the strip

where or according to or there is no point from the spectrum of the Neumann problem for the equation (3.2) for all . Then we have and the following estimate holds

where is a constant independent of .

By using the same arguments as in [1, 2] and Lemmas 2.1, 2.2, we can prove following lemma.

Lemma 4.1.

Let arbitrary. Assume that is a weak solution of the problem (2.1)–(2.3) in the space and . Then for almost all the equation

holds for all functions .

Now we surround the origin by a neighborhood with a sufficiently small diameter such that the intersection of and coincides with the cone We begin by proving some auxiliary lemmas.

Lemma 4.2.

Let be a weak solution in of the problem (2.1)–(2.3) such that outside . Moreover, we assume that . Then for almost all one has

(i)if then ,

(ii)if then where arbitrary.

Proof.

Because from Lemma 2.2 we have or for almost all . Following Lemma 4.1, is a solution of the Neumann problem for elliptic equation

where for almost all .Denote , Let be large enough such that . By choosing a smooth domain such that , from the theory of the regular of solutions of the boundary value problem for elliptic systems in smooth domains and near the piece smooth boundary of domain (see [9] for reference), we have for almost all and the following inequality holds

where is a positive constant independent of . It follows

By choosing and setting , one has

Return to the variable , we get

where the positive constant is independent of

Case 1 ().

Then

It follows from (4.6) that

where does not depend on Taking sum with respect to , one has

This implies

Because in out of a neighborhood of conical point is a smooth domain, so we have

for all almost all From (4.7), (4.11) and we receive for almost all

Case 2 ().

Since so for almost all one has This implies where arbitrary, is a positive constant. Because outside , so we have

For all we have , so it follows from [8, Lemma  7.1.1, page 268] that

From the inequality (4.6), for all one gets

where does not depend on By using analogous arguments used in Case 1, from (4.13), (4.14) we have

for all almost all That is . The lemma is proved.

Lemma 4.3.

Let , and for . Assume that is a weak solution in of the problem (2.1)–(2.3) such that outside . In addition, suppose that the strip

where or according to or , does not contain any point of the spectrum of the Neumann problem for the equation (3.2) for all . Then .

Proof.

We can rewrite (2.1) in the form

If then by applying Lemma 4.2 we have . In another way, because are continuous in uniformly with respect to for all then , for all and is a constant independent of . Therefore, from the hypotheses of this lemma one gets for almost all . Since in the strip there is no spectral point of the Neumann problem for the equation (3.2) for all , then following results of the work [9], one gets and satisfies

for almost all , where is a positive constant. Using the same arguments in the proof of Lemma 4.2, we have

for almost all . Multiplying this inequality with , then integrating with respect to from 0 to , from Lemma 2.2 one gets

Then is a function in the space

If then following Lemma 4.2 we have for almost all . This and the property of the functions continuous in uniformly with respect to follows . Because the strip does not contain any spectral point of the Neumann problem for (3.2), so from results of the work [9] we have satisfying

Repeating the proof in the case we achieve , too.

Now differentiating (2.1) with respect to , we have

where . From the hypotheses of the operator and Lemma 2.2 we have for almost all . Repeating arguments used for function we receive or .

In another way, it follows from Lemma 2.2 that

From (4.23) and the assertion that both and are in the space we have . This lemma is proved.

Lemma 4.4.

Let be a nonnegative integer number, be a real number satisfying , be a weak solution in of the problem (2.1)–(2.3) such that outside . Assume that , and for . Moreover, suppose that the strip

does not contain any point of the spectrum of the Neumann problem for the equation (3.2) for all , where or according to or . Then , satisfying

where the constant is independent of

Proof.

We use the induction by . For then we had Lemma 4.3 with noting that . Assume that lemma's assertion holds up to , we need to prove this holds up to . It means that we have to prove following inequality:

for , where is a positive constant.

Since for , so for . In another way, for . Then from Lemma 2.2 we have for all Hence, by using similar arguments in the proof of Lemma 4.3 we get . This means that (4.26) holds for .

Assume that (4.26) holds for . By putting (by inductive hypothesis) and differentiating (2.1) -times with respect to , we have

where Following the assumptions of the induction of and the hypotheses of the function one has . It follows . In another way since so we have for almost all . Because the strip does not contain any point of the spectrum of the Neumann problem for (3.2) for all , then following results of the work [9], one gets . This implies . Note that then by applying [8, Theorem 7.3.2] one gets satisfying

where is a positive constant. In another way, it is easy to see that

Hence from the inductive assumptions we receive

where is a constant independent of . It means that (4.26) is proved. Finally we only need to fix in (4.26) to complete the proof of this lemma.

Now let us prove Theorem 3.1.

Proof.

Denote , where and in a neighborhood of coordinate origin. The function satisfies

where is a linear differential operator order 1. Coefficients of this operator depend on the choice of the function and equal to outside Denote . It is easy to see that is equal to in a neighborhood of conical point. Therefore we can apply the theorem on the smoothness of a solution of elliptic problem in a smooth domain to this function to conclude that for almost all By applying Lemma 2.2 we receive and

Now, let us prove Theorem 3.1 by induction by When then functions satisfy the hypotheses of Lemma 4.3. So It follows that is in Assume that the theorem holds up to then we have By using analogous arguments in the proof of Lemma 4.4, with note that (from the hypothesis of induction), we can prove that . So The inequality in Theorem 3.1 can derive from inequality (4.25) (for ) and inequality (4.32). The theorem is proved completely.

In this section we apply the previous result to the Cauchy-Neumann problem for classical Schrödinger equations in quantum mechanics. It is shown that the smoothness of the weak solution of this problem depends on the structure of the boundary of the domain, the right hand side and the dimension of the space

The classical Schrödinger equation in quantum mechanics has the form

where is the Laplace operator. Now we consider the Cauchy-Neumann problem for (5.1) in infinite cylinder with the initial condition

and the boundary condition

where is the unit exterior normal to

The Laplace operator in polar coordinate in can be written in the form

where is the Laplace-Beltrami operator on the unit sphere Therefore, the corresponding spectral problem for (3.2) is the Neumann problem for following equation:

The regularity of the weak solution of the problem (5.1)–(5.3) can be stayed as follows.

Theorem 5.1.

Let be a weak solution in the space of the Cauchy-Neumann problem (5.1)–(5.3) and if , . Then .

Proof.

Note be nonnegative eigenvalues of the Neumann problem for equation

Then are eigenvalues of the Neumann problem for (5.5). It is easy to see that when the strip

does not contain any eigenvalue of the Neumann problem for (5.5). By applying Theorem  3.1 we have . The theorem is proved.

The Schrödinger equation has received a great deal of attention from mathematicians, in particular because of its application to quantum mechanics and optics. It is therefore important to research boundary value problems for it. Such problems have been previously proposed and analyzed for Schrödinger equations whose coefficients are independent of the time variable and in finite cylinders (see, e.g., [10]). In infinite cylinder , the first initial boundary value problem for this kind of equation with coefficients depend on both of time and spatial variables has been considered (see [1, 2]). In this paper, for a general Schrödinger equation in infinite cylinder with conical points in the boundary of base, we proved regularity property of solution of second initial boundary value problem. As a special application of these new results, we received the regularity of solution of a classical Schrödinger equation in quantum mechanics when the dimension of space . The similar questions for the case can be answered after researching the asymptotic of solution in the case the strip contains eigenvalues of the associated spectral problem. This is also the aim of our future research.

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