ISSN 1064-5624, Doklady Mathematics, 2012, Vol. 86, No. 3, pp. 750–755. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © A.A. Zlotnik, I.A. Zlotnik, 2012, published in Doklady Akademii Nauk, 2012, Vol. 447, No. 2, pp. 130–135.
750
The Schrödinger equation plays an important role
in quantum mechanics, nuclear and wave physics,
nanotechnologies, other areas. Frequently, it has to be
solved in unbounded domains. For this purpose,
numerous approaches are available that make use of
approximate transparent boundary conditions (TBCs)
on artificial boundaries (see the review in [1]). Among
the best TBCs are discrete ones that exhibit no reflec-
tion from the artificial boundaries, provide stable
computations, and have a mathematically rigorous
foundation. Such TBCs were developed and analyzed,
in particular, in [2–9]. At the same time, no discrete
TBCs for the finite element method (FEM) have been
developed. The present paper fills this gap.
Consider a generalized formulation of the initial–
boundary value problem for the nonstationary
Schrödinger equation with variable coefficients on the
half-line
+
= (0, ∞) with a zero Dirichlet boundary
condition at x = 0: find a complex-valued function ψ∈
C( ; (
+
)) with D
t
ψ ∈ C( ; L
2
(
+
)) that, for all
t > 0, satisfies the integral identity
(1)
and the initial condition ψ|
t =0
= ψ
0
(x) on
+
. Here,
i is the imaginary unit; = const > 0; and we use the
standard complex Lebesgue and Sobolev spaces, the
subspace (I) := {w ∈ H
1
(I); w(0) = 0}, and the Her-
mitian sesquilinear form
+
H
D
1
+
i ρ D
t
ψ · t , ( )ϕ , ( )
L
2
+
( )
+ ψ · t , ( )ϕ , ( ) =
for all ϕ H
D
1
+
( ) ∈
H
D
1
for the interval I =
+
(and, below, for I = Ω = (0, X)).
The coefficients ρ, B, V ∈ L
∞
(
+
) are real-valued and
satisfy ρ(x) ≥ > 0 and B(x) ≥ > 0, while D
t
=
and D = .
Additionally, assume that, for some X
0
> 0,
(2)
We introduce a grid 0 = x
0
< x
1
< … < x
J
= X < … on
and the elements Δ
j
:= [x
j –1
, x
j
] of length h
j
:= x
j
–
x
j –1
, j ≥ 1. Assume that x
J –1
≥ X
0
and h
j
= h for j ≥ J.
Let
n
( ) be the space of polynomials with complex
(real) coefficients on of degree at most n, where n ≥ 1.
Let (and ) denote the spaces of continuous
piecewise polynomial functions ϕ on (and )
such that ϕ(0) = 0 and ∈ for 1 ≤ j ≤ J (and
j ≥ 1).
We introduce a uniform grid with nodes t
m
=
mτ, where m ≥ 0 and τ > 0. Define := {t
m
. Let
:= Y
m –1
, and Y := , Y := . Let
H( ) denote the space of functions Φ: → ,
Φ
0
=0 with the inner product (Φ, Y :=
(Y
m
)*τ, where z* is the complex conjugate
of z ∈ .
I
w ϕ , ( ) := BDw Dϕ , ( )
L
2
I ()
Vw ϕ , ( )
L
2
I ()
+
ρ B
∂
∂ t
---
∂
∂ x
----
ρ x () ρ
∞
, Bx () B
∞
0 , Vx () > V
∞
, = = =
ψ
0
x () 0 on X
0
∞ , ( ) . =
+
n
H
h
n ()
H
h ∞ ,
n ()
Ω
+
ϕ
Δ
j
n
Δ
j
ω
τ
ωM
τ
}
m 0 =
M
Y
m
›
∂
t
Y Y –
τ
----------
›
s
t
Y Y +
2
----------
›
ωM
τ
ωM
τ
)
ω
M
τ
Φ
m
m 1 =
M
∑
Finite Element Method with Discrete Transparent
Boundary Conditions for the One-Dimensional
Nonstationary Schrödinger Equation
A. A. Zlotnik
a, b
and I. A. Zlotnik
b
Presented by Academician B.N. Chetverushkin April 23, 2012
Received April 25, 2012
DOI: 10.1134/S1064562412060130
a
Department of Higher Mathematics, Faculty of Economics,
National Research University Higher School of Economics,
Myasnitskaya ul. 20, Moscow, 101000 Russia
b
Department of Mathematical Modeling, Moscow Power
Engineering Institute (Technical University),
Krasnokazarmennaya ul. 14, Moscow, 111250 Russia
e-mail: azlotnik2007@mail.ru, ilya.zlotnik@gmail.com
MATHEMATICS