EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1441-1454
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

The Smoothness of Schrödinger Operator With
Electromagnetic Potential

Yahea Hashem Saleem1, Hadeel Ali Shubber1,2,∗

1 Department of Mathematics, College of Education for Pure Sciences, University of Basrah,
Basrah, Iraq
2 Department of Mathematics, College of Education for Pure Sciences, University of
Thi-Qar, Thi-Qar, Iraq

Abstract. In this paper, we prove that the Feynman-Kac Itô formula of the Schrödinger operator
with electromagnetic Ψ(t, x) in equation (1) in [8] which defined as

Ψ(t, x) =

∫
dµt

x(ω)exp

(
−i
∫ t

0

b(ω(s))dω − i

2

∫ t

0

divbω(s)ds−
∫ t

0

V (ω(s)ds

)
ϕ(ω(t))

is differentiable of the variable t, and so establish that the infinitely differentiable in a region,
therefore, investigate smoothness of this function.

2010 Mathematics Subject Classifications: 35J10, 35B65

Key Words and Phrases: Schrödinger operator, electric potential, magnetic potential, smooth-
ness.

1. Introduction

The problem of the self-adjoint operator is central in the quantum machine (the
Diracvon Neumann formulation of quantum mechanics, in which physical observables such
as position, momentum, angular momentum).

Kato [5] who showed on the basis of his elegant inequality that, if V (x) ≥ 0 and
V ∈ L2

loc, then the Schrödinger operator is essentially self-adjoint on the set of infinitely
differentiable finite functions. Nextly, Gaysinsky, Goldstein [4] they proved smoothness
of the Schrödinger operator which is one important step to prove self-adjointness must
be smoothness. After that, Adam Ward [1] investigated the essential self-adjointness of
Schrödinger operator.

Many researchers studied self-adjoint operator were done, for example [2], [6], [7], [9].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3515

Email addresses: yahea−h@mail.ru (Y. H. Saleem), hadeelali2007@yahoo.com (H. A. Shubber)

http://www.ejpam.com 1441 c© 2019 EJPAM All rights reserved.



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1442

We consider the Schrödinger operator with electromagnetic potentials

H =
n∑
j=1

1

2
(i∂j + bj(x))2 + V (x),

in L2(Rn) where, bj(X), j = 1, 2, ..., n and V (x) are real-valued functions on Rn, V ∈
L1
loc(R

n), b ∈ C2(Rn), ∂j = ∂
∂xj

and i =
√
−1.

We proved in [8] the Feynman-Kac Itô formula of the electromagnetic Schrödinger
operator Ψ(t, x) which define as the equation (1) in [8]

Ψ(t, x) =

∫
dµtx(ω)exp

(
−i
∫ t

0
b(ω(s))dω − i

2

∫ t

0
divbω(s)ds−

∫ t

0
V (ω(s)ds

)
ϕ(ω(t))

converges and is an analytic function of the variable t.
In this work, we prove that the Feynman-Kac Itô formula of the Schrödinger operator

with electromagnetic potentials Ψ(t, x) in equation (1) in [8] is differentiable of the variable
t, and we have ∂

∂tΨ(t, x) = − < e−tH , Hh > . Then, we discuss the infinite differentiability
of the function Ψ(t, x) in Rn\A where the potential V = +∞ on a set A. Finally, we
investigate the smoothness of this function Ψ(t, x).

2. Statement of the problem and the main result

In [8] we proved that Ψ(t, x) converges and has an analytic extension for a variable t.
Now, we prove that the smoothness to achieve this goal, we will follow the steps below.

Proposition 2.1. If H =
∑n

j=1
1
2(i∂j +bj(x))2+V (x) is the Schrödinger operator defined

on the interval [α, β]n with zero boundary conditions (V (x) is a continuous function defined
on [α, β]n), φ, h ∈ C∞0 , then < e−tHφ, h > is a differentiable.

∂

∂t
< e−tHϕ, h >= − < e−tHϕ,Hh > . (2.1)

Proof. Let F (t, V ) be the analytic extension defined in [8] as

F (t, V ) =

∫
Rn
Ψ(t, x)h(x)dx, (2.2)

and let

Fα,β(t, V ) =

∫
Rn
Ψα,β(t, x)h(x)dx (2.3)

be the same as in [8] , where

Ψα,β(t, x) =

∫
dy exp(−tHα,β) < x, y >, (2.4)



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1443

we define the operator H on the interval [α, β]n which denoted by Hα,β we have

lim
αn→−∞
βn→+∞

‖Fαn,βn(t, V )− F (t, V )‖ = 0,

uniformly by t ∈ G, where G is compact subdomain of {t = τ + iθ, τ ≥ τ0 > 0} . By the
Weierstrass theorem

lim
αn→−∞
βn→+∞

∥∥∥∥∂Fαn,βn∂t
− ∂F

∂t

∥∥∥∥
L2(Rn,dV )

= 0.

Let Hα,β as above then by equations (2.3), (2.4), we have〈
e−tHα,βϕ, h

〉
= Fα,β(t, V ).

Therefore,

∂Fα,β(t, V )

∂t
= −

∫
Ψα,β(t, x)Hα,βh(x)dx, (2.5)

provided suppϕ, supph ⊂ (α, β)n, h(x) ≡ 0 in the neighborhood of the center x = 0. Since
Hα,βh = Hh, then the right side of (2.5) represents a value of form Fα,β(t, V ), but only
for function

Hh =
n∑
j=1

1

2
(i∂j + bj(x))2h+ V (x)h.

According to the estimates for such functions, we may pass to the limit as α→ −∞, β →
+∞.

We observe that we can determine the functions Ψ(t, x) if the potentials V, b are equal
to +∞ on a set A that might have a positive measure

µ {s : V (ω(s)) = +∞, b(ω(s)) = +∞} > 0,

we set

exp

(
−
∫ t

0
V (ω(s))ds

)
= 0, exp

(
−
∫ t

0
−ib(ω(s))ds

)
= 0, exp

(
−
∫ t

0

−i
2
divb(ω(s))ds

)
= 0.

Then the function Ψ(t, x) satisfies the equation

∂

∂t
Ψ(t, x)ϕ =

n∑
j=1

1

2
(i∂j + bj(x))2Ψ(t, x)ϕ+ V (x)Ψ(t, x)ϕ.

Since Ψ(t, x) is analytical with respect to tha variable t, we prove that Ψ(t, x) is a
smooth function for almost every V, b where x ∈ Rn\A.



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1444

Proposition 2.2. Let V ∈ L2(Rn\A), ϕ, h ∈ C∞0 where A is closed set, V (x) = +∞, bj(x) =
+∞, x ∈ A such that suppϕ ∩ A = ∅. Then Ψ(t, x) is an infinitely differentiable function
of the variable x for almost every V, b for Ret ≥ τ0 > 0.

Proof. From equation (2.5) and definition of F (t, V ) in equation (2.1)

∂

∂t

∫
E(Ψ(t, x))h(x) = −

∫
E(Ψ(t, x))

 n∑
j=1

1

2
(i∂j + bj)

2 h(x) + V (x)h(x)

 dx. (2.6)

Let θ(V ) ∈ L2(Rn, dv), we put

f(t) =

∫
Rn
E (Ψ(t, x)θ(V ))h(x)dx = E(F (t, V )θ(V )).

depending on above that f(t) is an analytic function and

∂f

∂t
= E

(
∂F (t, x)

∂t
θ(V )

)

= −
∫
Rn
E

Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj)

2 h(x) + V (x)h(x)

 θ(V )

 dx

= −
∫
Rn
E

Ψ(t, x)θ(V )
n∑
j=1

1

2
(i∂j + bj)

2

h(x)dx−
∫
Rn
E (Ψ(t, x)V (x)θ(V ))h(x)dx.

On the other hand, f(t) =
∫
Rn f(t, x)h(x)dx, where f(t, x) = E(Ψ(t, x)θ(V )).

We have

|f(t, x)|2 ≤ E(Ψ(t, x)2)E(θ(V )2)

= ‖θ‖2L2(Rn,dV ) × E(Ψ(t, x)2).

Therefore, ∫
Rn
f(t, x)2dx ≤ const ‖θ‖2L2(dV )

i.e. f(t, x) ∈ L2(Rn, dV ). Further,∣∣∣∣∫
Rn
E(Ψ(t, x)V (x)θ(V ))

∣∣∣∣h(x)dx ≤
(∫

Rn
h(x)2dx

) 1
2
(∫

Rn
E(Ψ(t, x)V (x)θ(V ))2dx

) 1
2

≤ ‖h‖L2(Rn,dx)

(∫
Rn
E(Ψ2(t, x)V 2(x))E(θ2(V (x)))dx

) 1
2

≤ ‖h‖L2(Rn,dx) ‖θ‖L2(Rn,dV )

(∫
Rn
E(Ψ2(t, x)V 2(x))dx

) 1
2



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1445

≤ const ‖h‖L2(Rn,dx) ‖θ‖L2(Rn,dV ) ,

where we have used the estimates∫
Rn
E(Ψ(t, x)2 |V |m)dx ≤ const, (2.7)

where m = 1, 2, ... and the constant depends on m. We estimate the value ∂f
∂twith the

help Cauchy Schwartz inequality for drivatives of analytic function:∣∣∣∣∂f∂t
∣∣∣∣ ≤ const.max

|z−t|
|f(z)| .

Further,

|f(z)| = |E(F (z, V )θ(V ))|

≤ ‖θ‖L2(Rn,dV )E(|F (z, V )|2)
1
2

≤ const ‖θ‖L2(Rn,dV ) ‖h‖L2(Rn,dx) .

Thus,∣∣∣∣∣∣
∫
Rn
f(t, x)

n∑
j=1

1

2
(i∂j + bj(x))2 h(x)dx

∣∣∣∣∣∣ ≤ const ‖θ‖L2(Rn,dV ) ‖h‖L2(Rn,dx) .

One may check in just same way that if h1(x), ..., hp(x), θ1(V ), ..., θp(V ) and constats
Ck,l, k, l = 1, ..., p are given, then∣∣∣∣∣∣E
∫

Rn
Ψ(t, x)

p∑
k,l=1

Ck,l

n∑
j=1

1

2
(i∂j + bj(x))2 hk(x)θl(V )dx

∣∣∣∣∣∣ ≤ const
∥∥∥∥∥∥

p∑
k,l=1

Ck,lhk(x)θl(V )

∥∥∥∥∥∥
L2(Rn,dx,dV )

,

the left side equal to∣∣∣∣∣∣E
∫

Rn
Ψ(t, x)

p∑
k,l=1

Ck,l

n∑
j=1

1

2

(
∂2j + i∂jbj(x) + ibj(x)∂j + b2j (x)

)
hk(x)θl(V )dx

∣∣∣∣∣∣ ,
we pass at the left side to Fourier transformation by the variable x.

We get the following expression:-∣∣∣∣∣∣E
∫

Rn
Ψ̂(t, q)

p∑
k,l=1

Ck,l

n∑
j=1

1

2

(
∂2j + i∂j b̂j(q) + ib̂j(q)∂j + b̂j

2
(q)
)
ĥk(q)θl(V )dq

∣∣∣∣∣∣ ,
which equal to∣∣∣∣∣∣E
∫

Rn
Ψ̂(t, q)

p∑
k,l=1

Ck,l

 n∑
j=1

−1

2
|q|2 +

n∑
j=1

i

2
∂j b̂j(q) +

n∑
j=1

i

2
b̂j(q) |−qi|+

n∑
j=1

1

2
b̂j

2
(q)

 ĥk(q)θl(V )dq

∣∣∣∣∣∣



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1446

=

〈
Ψ̂(t, q),

p∑
k,l=1

Ck,l

 n∑
j=1

−1

2
|q|2 +

n∑
j=1

i

2
∂j b̂j(q) +

n∑
j=1

i

2
b̂j(q) |−qi|+

n∑
j=1

1

2
b̂j

2
(q)

 ĥk(q)θl(V )

〉
L2(Rn,dq,dV )

=

〈
Ψ̂(t, q),

p∑
k,l=1

Ck,l

−n
2
|q|2 +

n∑
j=1

i

2
∂j b̂j(q) +

n∑
j=1

i

2
b̂j(q) |q|+

n∑
j=1

1

2
b̂j

2
(q)

 ĥk(q)θl(V )

〉
L2(Rn,dq,dV )

=

〈
Ψ̂(t, q)

−n
2
|q|2 +

n∑
j=1

i

2
∂j b̂j(q) +

n∑
j=1

i

2
b̂j(q) |q|+

n∑
j=1

1

2
b̂j

2
(q)

 ,

p∑
k,l=1

Ck,lĥk(q)θl(V )

〉
L2(Rn,dq,dV )

.

The right side after the passage to the Fourier transform gains the form∥∥∥∥∥∥
p∑

k,l=1

Ck,lĥk(q)θl(V )

∥∥∥∥∥∥
L2(Rn,dq,dV )

.

From this we get∥∥∥∥∥∥
−n

2
|q|2 +

n∑
j=1

i

2
∂j b̂j(q) +

n∑
j=1

i

2
b̂j(q) |q|+

n∑
j=1

1

2
b̂j

2
(q)

 Ψ̂(t, q)

∥∥∥∥∥∥
L2(Rn,dq,dV )

≤ const.

(2.8)

In particular

∫
Rn

−n
2
|q|2 +

n∑
j=1

i

2
∂j b̂j(q) +

n∑
j=1

i

2
b̂j(q) |q|+

n∑
j=1

1

2
b̂j

2
(q)

2 ∣∣∣Ψ̂(t, q)
∣∣∣2 dq < +∞, (2.9)

for almost every V and b, i.e. Ψ(t, x) ∈ W1 for almost every V and b. Besides, ‖Ψ‖2W1
is

an integrable function of V.
Further, in just the same way as it was as done while deducing (2.6) one can show that

∂2

∂t2

∫
E(Ψ(t, x))h(x)dx =

∫
E(Ψ(t, x))

 n∑
j=1

1

2
(i∂j + bj(x))2 + V (x)

2

h(x)

 dx.

∂2

∂t2

∫
E(Ψ(t, x))h(x)dx =

∫
E(Ψ(t, x))[

 n∑
j=1

1

2
(i∂j + bj(x))2

2

h(x) +
n∑
j=1

1

2
(i∂j + bj(x))2 V (x)h(x)

+ V (x)
n∑
j=1

1

2
(i∂j + bj(x))2 h(x) + V 2(x)h(x)]dx.



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1447

Let us multiply the above equation by θ(V ) and integrate over dV, extract the ex-

pression containing
(∑n

j=1
1
2 (i∂j + bj(x))2

)2
h(x) and estimate the other terms in this

equation.
The term

∂2

∂t2

∫
E(Ψ(t, x))h(x)θ(V )dx

is estimated in just the same way as in the case of the first derivative, i.e. with the help
of the Cauchy integral formula.

The term ∫
E(Ψ(t, x)h(x)V 2(x))dx

admits application of the estimates see equation (2.7). Let us write

n∑
j=1

1

2
(i∂j + bj(x))2 V (x)h(x) + V (x)

n∑
j=1

1

2
(i∂j + bj(x))2 h(x)

=

n∑
j=1

−1

2
∂2j (V (x)h(x))+

n∑
j=1

i

2
∂jbj(x)(V (x)h(x))+

n∑
j=1

i

2
bj(x)∂j(V (x)h(x))+

n∑
j=1

1

2
b2j (x)(V (x)h(x))

+V (x)

n∑
j=1

−1

2
∂2j (h(x))+V (x)

n∑
j=1

i

2
∂jbj(x)(h(x))+V (x)

n∑
j=1

i

2
bj(x)∂j(h(x))+V (x)

n∑
j=1

1

2
b2j (x)(h(x))

=
n∑
j=1

−1

2
∂2j (V (x))h(x) +

n∑
j=1

−1

2
V (x)∂2j (h(x)) + 2

n∑
j=1

(
−1

2
∂(j)(V (x))

)(
−1

2
∂(j)(h(x))

)

+
n∑
j=1

i

2
∂jbj(x)(V (x)h(x))+

n∑
j=1

i

2
bj(x)∂j(V (x))h(x)+

n∑
j=1

i

2
bj(x)V (x)∂j(h(x))+

n∑
j=1

1

2
b2j (x)(V (x)h(x))

+V (x)

n∑
j=1

−1

2
∂2j (h(x))+V (x)

n∑
j=1

i

2
∂jbj(x)(h(x))+V (x)

n∑
j=1

i

2
bj(x)∂j(h(x))+V (x)

n∑
j=1

1

2
b2j (x)(h(x))

=
n∑
j=1

−1

2
∂2j (V (x))h(x) +

n∑
j=1

−V (x)∂2j (h(x)) + 2
n∑
j=1

(
−1

2
∂(j)(V (x))

)(
−1

2
∂(j)(h(x))

)

+
n∑
j=1

i∂jbj(x)(V (x)h(x))+
n∑
j=1

i

2
bj(x)∂j(V (x))h(x)+

n∑
j=1

ibj(x)V (x)∂j(h(x))+
n∑
j=1

b2j (x)(V (x)h(x)).

According to the definition of the potential V (x),

n∑
j=1

1

2
(i∂j + bj(x))2 V (x) = ξj−1,m

n∑
j=1

1

2
(i∂j + bj(x))2 Vj−1,m (x− (aj−1,m1, aj−1,m2, ..., aj−1,mn))



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1448

+ ξj,m

n∑
j=1

1

2
(i∂j + bj(x))2 Vj,m (x− (aj,m1, aj,m2, ..., aj,mn))

for x ∈ Πn
d=1[aj,md, aj−1,md) hence the term∫

E(Ψ(t, x)θ(V )h(x)
n∑
j=1

1

2
(i∂j + bj(x))2 V (x))dx

also admits application of the estimates see equation (2.7). Let us pass in the expression∫
E(Ψ(t, x)V (x)θ(V )

n∑
j=1

1

2
(i∂j + bj(x))2 h(x))dx

E(Ψ(t, x)

n∑
j=1

−1

2
∂j(V (x))θ(V )

n∑
j=1

−1

2
∂j(h(x)))dx

to the Fourier transform over the variable x. Consider, for example, the second one. It
will have the following form:-

E

∫
Rn
Ψ̂(t, q)ĥ(q)V̂ (q)

 n∑
j=1

−1

2
|q|2 +

1

2

n∑
j=1

i∂j b̂j(q) +
1

2

n∑
j=1

ib̂j(q)| − iq|+
1

2

n∑
j=1

b̂2j (q)

 dqθ(V )


and its absolute value is less or equal to the expression

const

E (∫
Rn
|Ψ̂(t, q)|2

)
×

 n∑
j=1

−1

2
|q|2 +

1

2

n∑
j=1

i∂j b̂j(q) +
1

2

n∑
j=1

ib̂j(q)| − iq|+
1

2

n∑
j=1

b̂2j (q)

2

dq


∥∥∥ĥ(q)

∥∥∥
L2(dx)

‖θ(V )‖L2(dV ) .

Now we get from the above estimate,

∫
Rn
E

Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj(x))2

2

h(x)θ(V )

 dx ≤ const ‖h‖L2(Rn,dx) ‖θ‖L2(Rn,dV ) .

We apply also calculations to a random variable of the form
∑p

k,l=1Ck,lhk(x)θl(V ).

We get the estimate of the form (2.7) where
∑n

j=1
1
2 (i∂j + bj(x))2 hk(x) is replaced by(∑n

j=1
1
2 (i∂j + bj(x))2

)2
hk(x). From this, we get the estimates (2.8) and (2.9) where q is

replaced by q2. Thus Ψ(t, x) ∈ W2 for almost every V, b. Besides, ‖Ψ‖2W2
is an integrable

function of V.



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1449

We can continue this arguments. As a result, we get that Ψ(t, x) ∈ Wm for all m =
1, 2, ..., and for almost every V and ‖Ψ‖2Wm

is an integrable function of V. Therefore, Ψ(t, x)
is an infinitely differentiable function of the variable x for almost every V. In addition, the
function Ψ satisfies, in the classical sense, the following differential equation

∂Ψ

∂t
=

n∑
j=1

1

2
(i∂j + bj(x))2 Ψ(t, x)− V (x)Ψ(t, x)

for almost every V.
Let us consider an initial condition which is satisfied by the function Ψ. Since Ψ(t, x)

is defined for t > 0, we have to find limt→0Ψ(t, x). We record

Ψ(t, x) =

∫
Rn
dy

(∫
dµtx(ω) exp

(
−i
∫ t

0
b(ω(s))dω − i

2

∫ t

0
divb(ω(s))ds−

∫ t

0
V (ω(s))ds

))
ϕ(y),

where the integral converges for almost every V, b, x.∫
Rn
E (Ψ(t, x)− ϕ(x))2 dx =

∫
Rn
dxE(

∫
Rn
dy(

∫
dµtx(ω) exp(−i

∫ t

0
b(ω(s))dω− i

2

∫ t

0
divb(ω(s))ds−

∫ t

0
V (ω(s))ds))φ(ω(t))−

∫
p(x, y, t)ϕ(x)dy)2

=

∫
Rn
dx

∫
Rn

∫
Rn
dydz

∫ ∫
dµtx,y(ω)dµtx,z(η))E(exp(−i

∫ t

0
b(ω(s))dω− i

2

∫ t

0
divb(ω(s))ds−

∫ t

0
V (ω(s))ds))ϕ(y)−ϕ(x)(exp(−i

∫ t

0
b(η(s))dη− i

2

∫ t

0
divb(η(s))ds−

∫ t

0
V (η(s))ds)ϕ(z)−ϕ(x)),

Now

exp

(
−i
∫ t

0
b(γ(s))dγ − i

2

∫ t

0
divb(γ(s))ds−

∫ t

0
V (γ(s))ds

)
= 1−

∫ 1

0

(
−i
∫ t

0
b(γ(s))dγ − i

2

∫ t

0
divb(γ(s))ds−

∫ t

0
V (γ(s))ds

)

. expα

(
−i
∫ t

0
b(γ(s))dγ − i

2

∫ t

0
divb(γ(s))ds−

∫ t

0
V (γ(s))ds

)
dα (2.10)

We get from (2.10) and from the estimates in (proposition 2.1 in [8]).

|E[

∫
dy

∫
dµtx,y(ω)

[
exp

(
−i
∫ t

0
b(ω(s))dω − i

2

∫ t

0
divb(ω(s))ds−

∫ t

0
V (ω(s))ds

)
(ϕ(y)− ϕ(x))

]



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1450

.[

∫
dz

∫
dµtx,z(ω)

[
exp

(
−i
∫ t

0
b(ω(s))dω − i

2

∫ t

0
divb(ω(s))ds−

∫ t

0
V (ω(s))ds

)
(ϕ(z)− ϕ(x))

]
|

≤ (

∫
p(x, y, t)ϕ(y)dy−ϕ(x))(

∫
p(x, z, t)ϕ(z)dz−ϕ(x))+const|

∫
p(x, y, t)ϕ(y)dy−ϕ(x)|t

+const|(
∫
p(x, z, t)ϕ(z)dz − ϕ(x))|t+ const.t2

since ϕ(x) ∈ C∞0 , we have ∫
Rn
E(Ψ(t, x)− ϕ(x))2dx→ 0.

Thus Ψ(t, x) satisfies the equation ∂Ψ
∂t =

∑n
j=1

1
2 (i∂j + bj(x))2 Ψ(t, x) − V (x)Ψ(t, x)

with the initial condition Ψ(t, x) → ϕ(x) in L2(Rn, dx, dV ) as t → 0. Repeating the
same estimates, we can show that Ψ̃1(t, x) → Hϕ(x) in L2(Rn, dx, dV ) as t → 0 where
Ψ̃ corresponds Hϕ(x), and also Ψ̃m(t, x) → Hmϕ(x) in L2(Rn, dx, dV ) as t → 0 where Ψ̃
corresponds Hmϕ(x).

First, we note that since the function Ψ(t, x) satisfies the estimates equation (2.7) and,
by lemma (3.1) in [8], may be analytically extended into the mentioned band, then we can
repeat literally all the arguments of this section for x ∈ Rn\A and show that the function
Ψ(t, x) is infinitely differentiable if x ∈ Rn\A.

We now consider the case x ∈ A. We assume, that the function V (x) in a neighborhood
of x ∈ A satisfies the following requirements considered in the work of M.D. Gaysinsky
(see[3],p.23):

(I) There exists ε > 0, δ > 0, k,N are some constant such that 0 < V (x)− d(x,A)−2−ε <
kd(x,A)−N , if 0 < d(x,A) < δ; where x ∈ Rn\A, d(x,A) is the distance between x
and closed set A.

(II) For each α = (α1, α2, , αn) there exists δα > 0 such that

| ∂α∂xαV (x)| = O(d(x,A)−kα , if 0 < d(x,A) < δα where kα are some constants.

We will show now that, in this case, Ψ(t, x) is infinitely differentiable at zero for almost
every V, b, and also estimate the derivations of Ψ(t, x) in a neighborhood of x ∈ A if the
support of the function ϕ(x) is disjoint with the closed set A. First, we show that Ψ(t, x)
fast decreases as x approach to the set A.

Proposition 2.3. Let V ∈ L2(Rn\A), ϕ, h ∈ C∞0 where A is closed set, V (x) = +∞, bj(x) =
+∞, if x ∈ A, 0 < V (x)−d(x,A)−2−ε < Kd(x,A)−N , if 0 < d(x,A) < δ; where x ∈ Rn\A,
d(x,A) is the distance between x and closed set A, ε > 0, δ > 0, k,N are some constant,and
let for each α = (α1, α2, , αn) there exists δα > 0 such that
| ∂α∂xαV (x)| = O(d(x,A)−kα , if 0 < d(x,A) < δα where kα are some constants. Let

ϕ ∈ C∞0 such that suppϕ ∩A = ∅. Then Ψ(t, x) is an infinitely differentable function.



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1451

Proof. Following the works of M.D. Gaysinsky [3], we will say that a random variable
Ṽ (x) has A−property if

E| ∂
m

∂xm
Ṽ (x)|r ≤ O(d(x,A)−km,r ,

for d(x,A) < δm,r where δm,r, km,r > 0 are constant, m = 1, 2, ...

E| ∂
m

∂xm
Ṽ (x)|r ≤ exp(Cm,rx

2),

where Cm,r > 0 is some constant, m = 0, 1, 2, ... .
It is evident that the derivations of a function (Ṽ (x)), having the A−property, also

have the A−property; the product of functions, having the A−property, also has the
A−property. We prove, by induction, that for any random function Ṽ with A−property
of the function Ṽ Ψ(t, x) belongs to the Sobolev space Wm,m = 0, 1, 2, ..., for almost every
V, b. Consider a sequence of smooth function λν(x) such that

(a) λν(x) = 0 if d(x,A) < 1
ν or d(x,A) > ν

(b) λν(x) = 1 if 2
ν < d(x,A) < ν − 1

(c) max|α|≤2
∣∣ ∂α
∂xαλν(x)

∣∣ ≤Mνs, where M, s are constant.

Let M = 0. We record

E|Ψ(t, x)Ṽ (x)|2 ≤ (E(Ψ(t, x))4)
1
2 (E(Ṽ (x))4)

1
2 ≤ β̃0, (2.11)

where β̃0 is some constant.
Now, it follows from (2.11) and corollary(2.2)in [8] that

E

(∫
Rn
|Ψ(t, x)Ṽ (x)|2

)
dx ≤ β0,

where β0 is a constant. We can write for any h(x), θ(V )

E

∫ Ṽ (x)Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj)

2

h(x)θ(V )

 dx

= lim
ν→∞

∫ Ṽ (x)Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj)

2

λνh(x)θ(V )

 dx

= lim
ν→∞

(

∫
Ṽ (x)Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj)

2

 (Ṽ λνh(x)) + V Ṽ λνh(x)

− V Ṽ Ψ(t, x)λνh(x)



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1452

−Ψ(t, x)λνh(x)

 n∑
j=1

1

2
(i∂j + bj)

2

 Ṽ−2Ψ(t, x)

 n∑
j=1

−1

2
∂j(Ṽ )

 n∑
j=1

−1

2
∂j(λνh)

)θ(V )dx.

Since the support of the functions h̃(x) = Ṽ (x)λνh(x) is disjoint with the neighborhood
of the point x ∈ A, we can repeat literally for h̃(x)θ(V ) all the arguments which we have
stated in the case when V (x) has no singular points. We have then

E

(∫
Ψ(t, x)Hh̃(x)θ(V )

)
dx = E

(∫
Ψ (1)(t, x)h̃(x)θ(V )

)
dx,

where Ψ1(t, x) corresponds to the function ϕ(1)(x) = Hϕ(x). Thus, we have

lim
ν→∞

E

∫ Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj)

2

 (Ṽ λνh(x)) + V Ṽ λνh(x)

 θ(V )

 dx

= lim
ν→∞

E

(∫
Ψ (1)(t, x)(Ṽ λνh(x))θ(V )

)
dx.

In addition,

|E(

∫
V Ṽ Ψ(t, x)h(x)θ(V ))dx| ≤ const

∫
E(V (x)2|Ψ(t, x)|)

1
2E(Ṽ (x)2|Ψ(t, x)|)

1
2dx;

E(Ṽ (x)2|Ψ(t, x)|) ≤ E(Ṽ (x)4)
1
2E(Ψ(t, x))2)

1
2 ≤ const exp

(
−

l−1∑
k=0

(tk+1 − tk)(bk)2
)

exp

(
−(x− α)2

ct

)
,

where c > 0 is constants. The similar estimate is true if we replace Ṽ by
∑n

j=1
1
2∂j Ṽ or(∑n

j=1
1
2(i∂j + bj)

2
)

(Ṽ ) (we use the A−property).

Since λν(x) is bounded and λν(x) 6= 1 whenever d(x,A) < 2
ν or d(x,A) > ν − 1, we

have

E(

∫
−V Ṽ Ψ(t, x)λνh(x)−Ψ(t, x)λνh(x)

 n∑
j=1

1

2
(i∂j + bj)

2

 Ṽ −2Ψ(t, x)

 n∑
j=1

−1

2
∂j(Ṽ )


 n∑
j=1

−1

2
∂j(λνh)

)θ(V )dx− E(

∫
V Ṽ Ψ(t, x)h(x)− Ψ(t, x)h(x)

 n∑
j=1

1

2
(i∂j + bj)

2

 Ṽ

−2Ψ(t, x)

 n∑
j=1

−1

2
∂j(Ṽ )

 n∑
j=1

−1

2
∂jh

)θ(V )dx

≤ const
∫
{d(x,A)< 2

ν
}∪{d(x,A)>ν−1}

exp

(
−

l−1∑
k=0

(tk+1 − tk)(bk)2
)

exp

(
−(x− α)2

ct

)
dx,



Y. H. Saleem, H. A. Shubber / Eur. J. Pure Appl. Math, 12 (4) (2019), 1441-1454 1453

where c > 0 is constants.
Therefore,

E

∫ Ṽ (x)Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj(x))2

h(x)θ(V )

 dx = E

(∫
Ψ (1)(t, x)Ṽ (x)h(x)θ(V )

)
dx

+E(

∫
[V (x)Ṽ (x)Ψ(t, x)h(x)−Ψ(t, x)h(x)

 n∑
j=1

1

2
(i∂j + bj(x))2

 Ṽ (x)−2Ψ(t, x)

 n∑
j=1

−1

2
∂j(Ṽ )


 n∑
j=1

−1

2
∂jh

 θ(V )])dx.

Now, we act in just the same way as in the case of the potential without singularities.
Namely, we make the fourier transform by the variable x at the left side. We get the
following expression:

E(

∫
̂(Ṽ (x)Ψ(t, x))(k)

 n∑
j=1

−1

2
|k|2 +

1

2

n∑
j=1

i∂j b̂j(k) +
1

2

n∑
j=1

ib̂j(k)| − ki|+ 1

2

n∑
j=1

ib̂2j (k)

 h̃(k)θ(V )dk.

(2.12)

Since V (x)Ṽ (x) and
(∑n

j=1
1
2(i∂j + bj(x))2

)
have the A−property, we have∣∣∣E (V (x)Ṽ (x)Ψ(t, x)h(x)θ(V )

)
dx
∣∣∣ ≤ const ‖h(x)θ(V )‖L2((Rn,,dx,dV ) ; (2.13)

∣∣∣∣∣∣E
∫ Ψ(t, x)

 n∑
j=1

1

2
(i∂j + bj(x))2

 Ṽ (x)h(x)θ(V )

 dx

∣∣∣∣∣∣ ≤ const ‖h(x)θ(V )‖L2((Rn,,dx,dV ) .

(2.14)

In the last term we also make the Fourier transform by x. We get the following
expression:

2iE

∫ Ψ(t, x)

 ̂n∑
j=1

−1

2
∂j( ˜(V ))

 (k)h̃(k)(
−nk

2
)θ(V )

 dk. (2.15)

Now, it follows from (2.12)-(2.15) that

|E(

∫
̂(Ṽ (x)Ψ(t, x))(k)

∑n
j=1

−|k|2
2 + i

2

∑n
j=1 ∂j b̂j(k) + i

2

∑n
j=1 b̂j(k)| − ki|+ i

2

∑n
j=1 b̂

2
j (k)

1 + |−nk2 |
h̃(k)θ(V )dk|



REFERENCES 1454

≤ const‖h(x)θ(V )‖L2(Rn,dx,dV ). (2.16)

Further the estimate (2.16) is literally transferred onto functions of the form
∑
Ck,lhk(x)θl(V ).

Therefore, Ṽ (x)Ψ(t, x) ∈W1 for almost every V, b

‖Ṽ (x)Ψ(t, x)‖2W1
≤ β1(V ),

where E(β1(V )) < +∞. Continuting these arguments of induction, we get Ṽ (x)Ψ(t, x) ∈
Wm for almost every V, b

‖Ṽ (x)Ψ(t, x)‖2Wm
≤ βm(V ), (2.17)

where E(βm(V )) < +∞.
Thus, we have proved that the function Ψ(t, x) for almost every V, b is infinitely dif-

ferentiable for all x, in particular, at the point x ∈ A. Besides, for any function with the
A-property, the estimates of the form (2.17) take place.

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