Abstract
Necessary and sufficient conditions for the existence of a solution of a boundaryvalue problem for the Schrödinger equation are obtained in the linear and nonlinear cases. Analytic solutions are represented using the generalized Green operator.
Keywords:
normally resolvable operator; generalized Green operator; Schrödinger equationIntroduction
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 boundaryvalue 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 [14]. Therefore, for the investigation of periodic problems for the Schrödinger equation, we develop the technique of generalized inverse operators [58] 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 [9]. 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 [10].
For the analysis of a weakly nonlinear Schrödinger equation, we develop the ideas of the LyapunovSchmidt method and efficient methods of perturbation theory, namely the VishikLyusternik method [11]. 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
Consider the following boundaryvalue problem for the Schrödinger equation in a Hilbert
space
where
In a more general case, the operator
where T is a strongly positive selfadjoint operator in the Hilbert space H. Since the operator T is closed, the domain
the infinitesimal generator of a strongly continuous evolution semigroup has the form
for any element
where
where
is the orthoprojector that projects the space
for
Lemma 1Suppose that the operator
1. Solutions of the boundaryvalue problem (1), (2) exist if and only if
2. Under condition (5), solutions of (1), (2) have the form
where
is the generalized Green operator of the boundaryvalue problem (1), (2) for
We now show that the condition
(1) Classical generalized solutions.
Consider the case where the set of values of
is the generalized Green operator (or it has the form of a convergent series).
(2) Strong generalized solutions. Consider the case where
Since the operator
where
is linear, continuous, and injective. Here,
are a continuous bijection and a projection, respectively. The triple
We complete the space X with the norm
is a homeomorphism of
has a unique solution
Remark 1 It should be noted that there exist the following extensions of spaces and the corresponding operators:
where
Then the operator
(3) Strong pseudosolutions.
Consider an element
These elements are called strong pseudosolutions by analogy with [5].
We now formulate the full theorem on solvability.
Theorem 1The boundaryvalue problem (1), (2) is always solvable.
(1)
(a) Classical or strong generalized solutions of (1), (2) exist if and only if
If
(b) Under assumption (8), the solutions of (1), (2) have the form
where
(2)
(a) Strong pseudosolutions exist if and only if
(b) Under assumption (9), the strong pseudosolutions of (1), (2) have the form
where
1 Main result (nonlinear case)
1.1 Modification of the LyapunovSchmidt method
In the Hilbert space
We seek a generalized solution
To find a necessary condition for the operator function
where q is a positive constant.
The main idea of the next results was used in [15] for the investigation of bounded solutions.
Let us show that this problem can be solved with the use of the following operator equation for generating amplitudes:
Theorem 2 (Necessary condition)
Suppose that the nonlinear boundaryvalue problem (10), (11) has a generalized solution
Proof If the boundaryvalue problem (10), (11) has classical generalized solutions, then, by Lemma 1, the following solvability condition must be satisfied:
By using condition (5), we establish that condition (13) is equivalent to the following:
Since
To find a sufficient condition for the existence of solutions of the boundaryvalue
problem (10), (11), we additionally assume that the operator function
This problem can be solved with the use of the operator
where
Theorem 3 (Sufficient condition)
Suppose that the operator
(1) The operator
(2)
Then, for an arbitrary element
This solution can be found by using the following iterative process:
1.2 Relationship between necessary and sufficient conditions
First, we formulate the following assertion:
CorollarySuppose that a functional
Remark 2 If the assumptions of the corollary are satisfied, then it follows from its proof
that the operators
In this way, we modify the wellknown LyapunovSchmidt method. It should be emphasized that Theorems 2 and 3 give us a condition for the chaotic behavior of (10) and (11) [16].
1.3 Example
We now illustrate the obtained assertion. Consider the following differential equation in a separable Hilbert space H:
where T is an unbounded operator with compact
We find the solutions of these equations in the space
for all pairs of constants
Then we can obtain the next result.
Theorem 4 (Necessary condition for the van der Pol equation)
Suppose that the boundaryvalue problem (16), (17) has a bounded solution
Remark Similarly, we can study the Schrödinger equation with a variable operator and more general boundary conditions (as noted in the introduction).
Consider the differential Schrödinger equation
in a Hilbert space H with the boundary condition
where, for each
In this case,
A detailed study of the boundaryvalue problem (18), (19) will be given in a separate paper.
Competing interests
The authors did not provide this information.
Authors’ contributions
The authors did not provide this information.
References

Vainberg, MM, Trenogin, SF: The Branching Theory of Solutions of Nonlinear Equations, Nauka, Moscow (1969)

Malkin, IG: Some Problems of the Theory of Nonlinear Oscillations, Nauka, Moscow (1956) (in Russian)

Grebenikov, EA, Ryabov, YA: Constructive Methods of Analysis of Nonlinear Systems, Nauka, Moscow (1979) (in Russian)

Boichuk, AA, Korostil, IA, Fec̆kan, M: Bifurcation conditions for a solution of an abstract wave equation. Differ. Equ.. 43, 481–487 (2007)

Boichuk, AA, Samoilenko, AM: Generalized Inverse Operators and Fredholm BoundaryValue Problems, VSP, Utrecht (2004)

Moore, EH: On the reciprocal of the general algebraic matrix (abstract). Bull. Am. Math. Soc.. 26, 394–395 (1920)

Penrose, RA: A generalized inverse for matrices. Proc. Camb. Philos. Soc.. 51, 406–413 (1955). Publisher Full Text

Pokutnyi, AA: Linear normallyresolvable equations in Banach spaces. J. Comput. Appl. Math.. 107, 146–153 (in Russian) (2012)

Klyushin, DA, Lyashko, SI, Nomirovskii, DA, Semenov, VV: Generalized Solutions of Operator Equations and Extreme Elements, Springer, Berlin (2012)

Tikhonov, AN, Arsenin, VY: Methods for Solving IllPosed Problems, Nauka, Moscow (1979) (in Russian)

Vishik, MI, Lyusternik, LA: The solution of some perturbation problems for matrices and selfadjoint or nonselfadjoint differential equations. Russ. Math. Surv.. 15, 3–80 (1960)

Reed, M, Simon, B: Methods of Modern Mathematical Physics, Mir, Moscow (1979)

Biletskyi, BA, Boichuk, AA, Pokutnyi, AA: Periodic problems of difference equations and ergodic theory. Abstr. Appl. Anal.. 12, Article ID 928587 (2011)

Boichuk, AA, Pokutnyi, AA: Bounded solutions of weakly nonlinear differential equations in a Banach space. Nonlinear Oscil.. 11, 158–167 (2008). Publisher Full Text

Chueshov, ID: Introduction to the Theory of InfiniteDimensional Dissipative Systems, Acta, Kharkiv (2002)