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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Solutions of the Schrödinger equation in a Hilbert space

Alexander Boichuk* and Oleksander Pokutnyi

Author Affiliations

Laboratory of boundary value problems of the theory of differential equations, Institute of Mathematics of NAS of Ukraine, Tereshenkivska, 3, Kiev, 01601, Ukraine

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Boundary Value Problems 2014, 2014:4  doi:10.1186/1687-2770-2014-4


Dedicated to the academician Kiguradze I.T.


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/4


Received:1 October 2013
Accepted:11 December 2013
Published:6 January 2014

© 2014 Boichuk and Pokutnyi; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
normally resolvable operator; generalized Green operator; Schrödinger equation

Introduction

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 [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 Lyapunov-Schmidt method and efficient methods of perturbation theory, namely the Vishik-Lyusternik 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 boundary-value problem for the Schrödinger equation in a Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M2">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M3">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M4">View MathML</a>, H is Hilbert space and vector-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M5">View MathML</a> is integrable; for simplicity, the unbounded operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M6">View MathML</a> has the following form [12] for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M7">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M8">View MathML</a>

In a more general case, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M6">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M10">View MathML</a>

where T is a strongly positive self-adjoint operator in the Hilbert space H. Since the operator T is closed, the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M11">View MathML</a> of the operator T is a Hilbert space with scalar product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M12">View MathML</a>. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M6">View MathML</a> is self-adjoint in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M14">View MathML</a> with product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M15">View MathML</a>

the infinitesimal generator of a strongly continuous evolution semigroup has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M16">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M18">View MathML</a> (nonexpanding group), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M20">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M21">View MathML</a>. The mild solutions of equation (1) can be represented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M22">View MathML</a>

for any element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M23">View MathML</a>. Substituting this in condition (2), we conclude that the solvability of the boundary-value problem (1), (2) is equivalent to the solvability of the following operator equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M24">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M25">View MathML</a>. Consider the case where the set of values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26">View MathML</a> is closed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M27">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M28">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M18">View MathML</a>, we can conclude [13] that the operator system (3) is solvable if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M30">View MathML</a>

(4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M31">View MathML</a>

is the orthoprojector that projects the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1">View MathML</a> onto the subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M33">View MathML</a>. Under this condition, the solutions of (3) have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M34">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M35">View MathML</a>, and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M36">View MathML</a>. Then we can formulate the first result as a lemma.

Lemma 1Suppose that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26">View MathML</a>has a closed image<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M38">View MathML</a>.

1. Solutions of the boundary-value problem (1), (2) exist if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M39">View MathML</a>

(5)

2. Under condition (5), solutions of (1), (2) have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M40">View MathML</a>

(6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M41">View MathML</a>

is the generalized Green operator of the boundary-value problem (1), (2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M42">View MathML</a>.

We now show that the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M43">View MathML</a> of Lemma 1 can be omitted and, in different senses, the boundary-value problem (1), (2) is always solvable.

(1) Classical generalized solutions.

Consider the case where the set of values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26">View MathML</a> is closed (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M38">View MathML</a>). Then [5]<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M46">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M47">View MathML</a>, and the set of solutions of (3) has the form [5]<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M49">View MathML</a>, where [5,13]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M50">View MathML</a>

is the generalized Green operator (or it has the form of a convergent series).

(2) Strong generalized solutions. Consider the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M52">View MathML</a>. We show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26">View MathML</a> can be extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M54">View MathML</a> in such a way that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M55">View MathML</a> is closed.

Since the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26">View MathML</a> is bounded, the following representation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1">View MathML</a> in the form of a direct sum is true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M60">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M61">View MathML</a> be the quotient space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M64">View MathML</a> be the orthoprojectors onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M66">View MathML</a>, respectively. Then the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M67">View MathML</a>

is linear, continuous, and injective. Here,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M68">View MathML</a>

are a continuous bijection and a projection, respectively. The triple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M69">View MathML</a> is a locally trivial bundle with typical fiber <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M70">View MathML</a>[14]. In this case [[9], p.26,29], we can define a strong generalized solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M71">View MathML</a>

(7)

We complete the space X with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M72">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M73">View MathML</a>[9]. Then the extended operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M74">View MathML</a>

is a homeomorphism of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M65">View MathML</a>. By the construction of a strong generalized solution [9], the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M77">View MathML</a>

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M78">View MathML</a>, which is called the generalized solution of equation (7).

Remark 1 It should be noted that there exist the following extensions of spaces and the corresponding operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M79">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M80">View MathML</a>

Then the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M81">View MathML</a> is an extension of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M26">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M83">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M84">View MathML</a>.

(3) Strong pseudosolutions.

Consider an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M85">View MathML</a>. This condition is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M86">View MathML</a>. In this case, there are elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M87">View MathML</a> that minimize the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M88">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M89">View MathML</a>

These elements are called strong pseudosolutions by analogy with [5].

We now formulate the full theorem on solvability.

Theorem 1The boundary-value problem (1), (2) is always solvable.

(1)

(a) Classical or strong generalized solutions of (1), (2) exist if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M90">View MathML</a>

(8)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M91">View MathML</a>, then solutions of (1), (2) are classical.

(b) Under assumption (8), the solutions of (1), (2) have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M92">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M93">View MathML</a>is an extension of the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M94">View MathML</a>.

(2)

(a) Strong pseudosolutions exist if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M95">View MathML</a>

(9)

(b) Under assumption (9), the strong pseudosolutions of (1), (2) have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M96">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M97">View MathML</a>

1 Main result (nonlinear case)

1.1 Modification of the Lyapunov-Schmidt method

In the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M1">View MathML</a> defined above, we consider the boundary-value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M99">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M100">View MathML</a>

(11)

We seek a generalized solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M101">View MathML</a> of the boundary-value problem (10), (11) that becomes one of the solutions of the generating equation (1), (2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M102">View MathML</a> in the form (6) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M103">View MathML</a>.

To find a necessary condition for the operator function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M104">View MathML</a>, we impose the joint constraints

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M105">View MathML</a>

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M106">View MathML</a>

(12)

Theorem 2 (Necessary condition)

Suppose that the nonlinear boundary-value problem (10), (11) has a generalized solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M107">View MathML</a>that becomes one of the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M102">View MathML</a>of the generating equation (1), (2) with constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M109">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M110">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M103">View MathML</a>. Then this constant must satisfy the equation for generating amplitudes (12).

Proof If the boundary-value problem (10), (11) has classical generalized solutions, then, by Lemma 1, the following solvability condition must be satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M112">View MathML</a>

(13)

By using condition (5), we establish that condition (13) is equivalent to the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M113">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M114">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M115">View MathML</a>, we finally obtain [by using the continuity of the operator function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M104">View MathML</a>] 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M104">View MathML</a> is strongly differentiable in a neighborhood of the generating solution (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M118">View MathML</a>).

This problem can be solved with the use of the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M119">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M120">View MathML</a> (Fréchet derivative). □

Theorem 3 (Sufficient condition)

Suppose that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121">View MathML</a>satisfies the following conditions:

(1) The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121">View MathML</a>is Moore-Penrose pseudoinvertible;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M123">View MathML</a>.

Then, for an arbitrary element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M124">View MathML</a>satisfying the equation for generating amplitudes (12), there exists at least one solution of (10), (11).

This solution can be found by using the following iterative process:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M125">View MathML</a>

1.2 Relationship between necessary and sufficient conditions

First, we formulate the following assertion:

CorollarySuppose that a functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M126">View MathML</a>has the Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M127">View MathML</a>for each element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M128">View MathML</a>of the Hilbert spaceHsatisfying the equation for generating constants (12). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M129">View MathML</a>has a bounded inverse, then the boundary-value problem (10), (11) has a unique solution for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M128">View MathML</a>.

Remark 2 If the assumptions of the corollary are satisfied, then it follows from its proof that the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M132">View MathML</a> are equal. Since the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M127">View MathML</a> is invertible, it follows that assumptions 1 and 2 of Theorem 3 are necessarily satisfied for the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M121">View MathML</a>. In this case, the boundary-value problem (10), (11) has a unique bounded solution for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M135">View MathML</a> satisfying (12). Therefore, the invertibility condition for the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M129">View MathML</a> expresses the relationship between the necessary and sufficient conditions. In the finite-dimensional case, the condition of invertibility of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M127">View MathML</a> is equivalent to the condition of simplicity of the root <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M128">View MathML</a> of the equation for generating amplitudes [5].

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) [16].

1.3 Example

We now illustrate the obtained assertion. Consider the following differential equation in a separable Hilbert space H:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M139">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M140">View MathML</a>

(15)

where T is an unbounded operator with compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M141">View MathML</a>. Then there exists an orthonormal basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M142">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M145">View MathML</a>. In this case, the operator system (10), (11) for the boundary-value problem (14), (15) is equivalent to the following countable system of ordinary differential equations (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M146">View MathML</a>):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M147">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M148">View MathML</a>

(17)

We find the solutions of these equations in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M149">View MathML</a> that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M103">View MathML</a>, turn into one of the solutions of the generating equation. Consider the critical case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M152">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M153">View MathML</a>. In this case, the set of all periodic solutions of (16), (17) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M154">View MathML</a>

for all pairs of constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M156">View MathML</a>. The equation for generating amplitudes (12) is equivalent in this case to the following countable systems of algebraic nonlinear equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M157">View MathML</a>

Then we can obtain the next result.

Theorem 4 (Necessary condition for the van der Pol equation)

Suppose that the boundary-value problem (16), (17) has a bounded solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M107">View MathML</a>that becomes one of the solutions of the generating equations with pairs of constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M156">View MathML</a>. Then only a finite number of these pairs are not equal to zero. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M162">View MathML</a>, then these constants lie on anN-dimensional torus in the infinite-dimensional space of constants:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M163">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M164">View MathML</a>

(18)

in a Hilbert space H with the boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M165">View MathML</a>

(19)

where, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M166">View MathML</a>, the unbounded operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M167">View MathML</a> has the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M169">View MathML</a> is an unbounded self-adjoint operator with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M170">View MathML</a>, and the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M171">View MathML</a> is strongly continuous. The operator Q is linear and bounded and acts from the Hilbert space H to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M172">View MathML</a>. As in [12], we define the operator-valued function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M173">View MathML</a>

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M174">View MathML</a> admits the Dyson representation [[12], p.311]; denote its propagator by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M175">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M176">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M177">View MathML</a> is a weak solution of (14) with the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M178">View MathML</a> in the sense that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M179">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M180">View MathML</a> is differentiable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/4/mathml/M181">View MathML</a>

A detailed study of the boundary-value 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.

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