Solutions of the Schrödinger equation in a Hilbert space
© Boichuk and Pokutnyi; licensee Springer. 2014
Received: 1 October 2013
Accepted: 11 December 2013
Published: 6 January 2014
Necessary and sufficient conditions for the existence of a solution of a boundary-value problem for the Schrödinger equation are obtained in the linear and nonlinear cases. Analytic solutions are represented using the generalized Green operator.
Keywordsnormally resolvable operator generalized Green operator Schrödinger equation
The Schrödinger equation is the subject of numerous publications, and it is impossible to analyze all of them in detail. For this reason, we only briefly describe the methods and ideas that underlie the approach proposed in this paper for the investigation of the linear and a weakly nonlinear Schrödinger equation with different boundary conditions.
In this work, we develop constructive methods of analysis of linear and weakly nonlinear boundary-value problems, which occupy a central place in the qualitative theory of differential equations. The specific feature of these problems is that the operator of the linear part of the equation does not have an inverse. This does not allow one to use the traditional methods based on the principles of contracting mappings and a fixed point. These problems include the most complicated and inadequately studied problems known as critical (or resonance) problems [1–4]. Therefore, for the investigation of periodic problems for the Schrödinger equation, we develop the technique of generalized inverse operators [5–8] for the original linear operator in Banach and Hilbert spaces.
On the other hand, we use the notion of a strong generalized solution of an operator equation developed in . The origins of this approach go back to the works of Weil and Sobolev. Using the process of completion, one can introduce the concept of a strong pseudoinverse operator for an arbitrary linear bounded operator and thus relax the requirement that the range of its values be closed. In this way, one can prove the existence of solutions of different types for the linear Schrödinger equation with arbitrary inhomogeneities. Thus, one may say that, in a certain sense, the Schrödinger equation is always solvable. There are three possible types of solutions: classical generalized solutions, strong generalized solution, and strong pseudosolutions .
For the analysis of a weakly nonlinear Schrödinger equation, we develop the ideas of the Lyapunov-Schmidt method and efficient methods of perturbation theory, namely the Vishik-Lyusternik method . The combination of different approaches allows us to take a different look at the Schrödinger equation with a constant unbounded operator in the linear part and obtain all its solutions by using the generalized Green operator of this problem constructed in this work. Possible generalizations are discussed in the final part of the paper. By an example of the abstract van der Pol equation, we illustrate the results that can be obtained by using the proposed method.
Auxiliary result (linear case)
Statement of the problem
for , and any . Then we can formulate the first result as a lemma.
- 1.Solutions of the boundary-value problem (1), (2) exist if and only if(5)
- 2.Under condition (5), solutions of (1), (2) have the form(6)where
is the generalized Green operator of the boundary-value problem (1), (2) for .
Classical generalized solutions.
Strong generalized solutions. Consider the case where and . We show that the operator can be extended to in such a way that is closed.
has a unique solution , which is called the generalized solution of equation (7).
These elements are called strong pseudosolutions by analogy with .
We now formulate the full theorem on solvability.
- (1)(a) Classical or strong generalized solutions of (1), (2) exist if and only if(8)
If , then solutions of (1), (2) are classical.
- (b)Under assumption (8), the solutions of (1), (2) have the form
where is an extension of the operator .
- (2)(a) Strong pseudosolutions exist if and only if(9)
- (b)Under assumption (9), the strong pseudosolutions of (1), (2) have the formwhere
1 Main result (nonlinear case)
1.1 Modification of the Lyapunov-Schmidt method
We seek a generalized solution of the boundary-value problem (10), (11) that becomes one of the solutions of the generating equation (1), (2) in the form (6) for .
where q is a positive constant.
The main idea of the next results was used in  for the investigation of bounded solutions.
Theorem 2 (Necessary condition)
Suppose that the nonlinear boundary-value problem (10), (11) has a generalized solution that becomes one of the solutions of the generating equation (1), (2) with constant and for . Then this constant must satisfy the equation for generating amplitudes (12).
Since as , we finally obtain [by using the continuity of the operator function ] the required assertion.
To find a sufficient condition for the existence of solutions of the boundary-value problem (10), (11), we additionally assume that the operator function is strongly differentiable in a neighborhood of the generating solution ().
where (Fréchet derivative). □
Theorem 3 (Sufficient condition)
The operator is Moore-Penrose pseudoinvertible;
Then, for an arbitrary element satisfying the equation for generating amplitudes (12), there exists at least one solution of (10), (11).
1.2 Relationship between necessary and sufficient conditions
First, we formulate the following assertion:
Corollary Suppose that a functional has the Fréchet derivative for each element of the Hilbert space H satisfying the equation for generating constants (12). If has a bounded inverse, then the boundary-value problem (10), (11) has a unique solution for each .
Remark 2 If the assumptions of the corollary are satisfied, then it follows from its proof that the operators and are equal. Since the operator is invertible, it follows that assumptions 1 and 2 of Theorem 3 are necessarily satisfied for the operator . In this case, the boundary-value problem (10), (11) has a unique bounded solution for each satisfying (12). Therefore, the invertibility condition for the operator expresses the relationship between the necessary and sufficient conditions. In the finite-dimensional case, the condition of invertibility of the operator is equivalent to the condition of simplicity of the root of the equation for generating amplitudes .
In this way, we modify the well-known Lyapunov-Schmidt method. It should be emphasized that Theorems 2 and 3 give us a condition for the chaotic behavior of (10) and (11) .
Then we can obtain the next result.
Theorem 4 (Necessary condition for the van der Pol equation)
Remark Similarly, we can study the Schrödinger equation with a variable operator and more general boundary conditions (as noted in the introduction).
A detailed study of the boundary-value problem (18), (19) will be given in a separate paper.
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