ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 855–863.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
A.A. Boichuk, A.A. Pokutnyi, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 882–890.
PARTIAL DIFFERENTIAL EQUATIONS
Theory of Bifurcations of the Schr¨odinger Equation
A. A. Boichuk
Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 01004 Ukraine
Received June 23, 2016
Abstract—We obtain solvability conditions and a representation of solutions for a boundary
value problem for a linear nonstationary Schr¨odinger equation in a Hilbert space as well as
suﬃcient conditions for the bifurcation of solutions of this equation.
Methods of perturbation theory, whose foundations were laid by Poincar´e and Lyapunov, are
a powerful tool in applied mathematics and permit one to obtain approximate analytical represen-
tations of solutions of rather complicated boundary value problems. Most of these methods arose
when solving speciﬁc problems of mechanics, celestial mechanics, and physics [1–4]. Numerous ex-
amples of various problems that can be studied by operator methods of perturbation theory  can
be found in the monograph . The analysis of these papers and papers concerning boundary value
problems in critical (resonance) cases [7–10] permitted studying perturbation theory of an abstract
Schr¨odinger equation with an unbounded operator on a Hilbert space. This equation is related
to various classes of partial diﬀerential wave equations [12, 13] as well as to quantum functional
analysis, which is actively developing nowadays .
The present paper uses the theory of pseudo-inverse operators [8, 11] to construct perturbation
theory for the Schr¨odinger equation in resonance cases. Note that the spectral perturbation problem
for the Schr¨odinger equation in the nonresonance cases is presented, e.g., in the monograph 
(see also the bibliography therein) and goes back to work by Schr¨odinger himself.
STATEMENT OF THE PROBLEM
Consider the following boundary value problem on a Hilbert space H :
lϕ(·)=α + εl
Here the unbounded operator H(t) has the form H(t)=H
+ V (t)foreacht ∈ J := [a, b] ⊂
is a self-adjoint operator with domain D = D(H
) ⊂H, the operator functions V (t)
(t) are strongly continuous, l and l
are bounded linear operators that specify the boundary
conditions and act from a Hilbert space H to a Hilbert space H
. We seek conditions
for the solvability of the boundary value problem (1), (2) for the case in which the unperturbed
boundary value problem (ε = 0) has no solutions.
To obtain the main assertions, we present a number of auxiliary results about the solvability of
the unperturbed boundary value problem
= −iH(t)ϕ(t)+f(t), (3)