\Q consists of four elements €j,j = 0, 1,2,3 for p ^ 2 and of eight elements £j ,jj — 0 , 1 , . . . ,7 for p — 2. 2
2
V
2
p
p
Note that for p = 3 (mod 4) as a number n can be taken —1 because (see [204]) =
( y )
(
1
-
,
" = -
1
;
<
4 J )
for p = 3 (mod 8) or p = 5 (mod 8) as a number n can be taken 2 because (*) = —1 owing to the formula
2
7. Jn order that the necessary and sufficient solutions of this equation 8. Forp= 3 (mod 4)
equation x = —1 has a solution in Q , it is that p = 1 (mod 4); in addition there exist two which we denote by ± r . the equality p
2
2
\x + y \
p
1
2
2
= nu>x(\x\ ,\y\ ) p
(4.8)
p
2
is valid. Jn particular, from x + y = 0, it follows that x = y = 0. • I t is sufficient to verify Eq. (4.8) only for the case \x\ = \y\ . I f it would be | i + j/ |p < \x \ = \y\j, then the congruence x% = —yl (mod p) would be solvable and thus —1 would be quadratic residue modulo p which contradicts to the formula (4.7). • p
2
2
p
2
p
5. Polar
Coordinates
and Circles
in the Field
Q (y/e) p
2
Any element of the field <$p(y/e),e £ Q* , is uniquely represented in the form z = x + -y/iy, x, y 6 (see Sec. 1.4). The numbers (x, y) are called the Cartesian coordinates of the element z\ the element z — x — -Jcy is called conjugate to 2; the set of elements z 6 Qp(i/?)i which satisfy the equation 2
2
zz = x - ey = c, is called "circle" in Q (^/7) P
cf 0
(with the center at 0).
(5.1)
Analysis on the Field of p-Adic Number* 13 I t is clear that the equality 22 = 0 is equivalent to 2 = 0. Denote by Q a subgroup of the group Q* which consists of numbers c of the form (5.1). By the Lemma of Sec. 1.4 the number c can be represented in the form: either c — r or c = nr where r £ Q , and K q\ Q* , hence K — cr~ £ Q i.e. K = VV,V£ Qp(y/e). p £
2
2
2
2
p
P j [
-1
i
1
A pair (p, &) with either (p = r,
is
I t is clear that (—p, —tj) is also polar coordinates of the point z. The unit "circle" in Q (%/e) 22 = 1, which is denned by the equation (5.1) for c = 1 play a special role. Elements of this "circle" form a multiplicative group which we denote by C . p
e
Let us find a parametric representation of the "circle" C , £
^ - r f s l . By introducing the parameter t =
(5.2)
from the equation (5.2) we obtain
2
1 -+- rt
It
• = IT5i-
»= I = 3 *
J
£
( 5
*
'
3 )
C is compact in Q ( \ / £ ) e
p
• C is obviously closed, we shall prove its boundedness (see Sec. 1.3). For | E ( | > 1 we have [1 ± e t | = | e i | and owing to (5.3) | x | - 1; for | e t | < 1 we have | 1 ± E t | = 1 and owing to (5.3) j x | = 1; for | e i | = 1 we shall prove that j l - e f | = 1 and owing to (5.3) \x\ = |1 + e ( | < I . In all three cases | x | < 1, and thus from the equation (5.2) it follows that |y|p < Ie:I~ , and C is bounded. e
2
2
p
2
p
2
p
p
2
2
p
p
p
p
2
2
p
p
p
p
1
c
2
2
I t remains to prove that from |e( | = 1 it follows that [1 — e t | = 1. Let p 2. I n Sec 1.4 it was proved that as e can be taken the numbers n,p and np. But £ = p or E = rjp contradict to the equality |ef | = 1. Therefore e = n. But the equation 1 - n i = 0 (mod p) is not solvable otherwise rjn would be a quadratic residue modulo p what contradict to the Lemma of Sec 1.4.Thus |1 - n t | = 1. For p = 2 a proof is similar, but more complicated. I p
p
2
p
0
o
2
p
6. Q
p
and R
We shall construct a one-to-one continuous mapping
P
14 p-Adic Analysis and Mathematical Physics Let us define the function ip : Q —• K+ by the formula p
£
x~
,
21
kP
(6.1)
0
where numbers x*, 0 < xt 0 that yo = x ,yi = X\,..., j/y_ 1 = X j _ i , Xj
p
p
p
p
u
p
L e m m a . f(x)
> tp(y) if and only if x > y.
• Sufficiency. Let x > y and | x | > \y\ ^ 0, i.e. | x [ > p | y | . We have p
p
W ' " < ( F - I H E P "
Yi
P
p
0<<;
p
M
-
1=0
Therefore
*>(*) -
> 1*1, * p ^ y l f l p > P l f l
Let now \x\ = | y | , x some j > 0. Then p
p
0
= y, x 0
:
P
(1 - ^ 7 )
=
> 0 •
= y,,... ,x _i = y ^ ;
x
u
> kip E **p~ * i 0
P
E
!/*p-
2t
+ l!/l
£ : P T
ip"
2 0
'
+ 1 )
P
0<Jc<j 0
P
+
1
Therefore 2
^ ( x ) - \x\ p~ ' p
(xt
—j-j-j > 0 .
}
> yj for
Analysis on the Field of ji-Adic Numbers 15 Necessity. I f
x,y € Q
p
P
which can be proved in a similar way to the Lemma. Let us note K =
I
The structure of the set K is given by the
T h e o r e m . The set K is a countable union of disjoint perfect nowhere dense sets of the Lebesgue measure zero. • As Q = U p
K =
[J
(see Sec. 1.3). Then
K,
=
y
K
y
n Ky
=4,
*f4 i
,
(6.3)
Let us study a structure of the set Ko (structure of the remaining sets K is similar). As 1 <
F
\0<j
'
0<j
}
r» = 0 , 1 . 2 , . . . , e = (£ ,£i,.
,£„),
0
i = l,2,..,,n-l,
0 < E
£ ^ 0, 0
£ ^Pn
Denote I
n
= {jl
nl
e
J=
[J 0
The Lebesgue measure of the set / is equal to
1,
In-
1 •
16 p-Adic Analytia and Mathematical Phyiiei
€
=
(
p
_
I
)
P ( P - I )
(
_ _ l _ )
1
+
(
_
p
1
)
!
j
/
- v - , (
_ _ l
1
)
T
0-1)
,
P+l
P+l
^
By using the Lemma one can prove that J f l Kn — <j>, i.e. there are no x £ SQ, and n = 0 , 1 , . . . for which the inequalities
E *T* + j £ < <
E
^p-
2 i
+p-
2 n
take place. Besides, using the Lemma on imbedded segments we establish that Ko Ul = [1, p) • From here and from (6.4) i t follows that the Lebesgue measure of the set K is equal to 0. The process of construction of the set Ko = [ l , p ) \ J coincides, within unessential details, with the known process of construction of the Cantor canonical perfect set on the segment [0,1]. Therefore proofs of the remaining properties of the set K are similar to the proofs of corresponding properties of the Cantor set. • 0
a
7.
Space
= Q p X Q p X . . . x Q consists of points x = (x ,x ,... j 1 = 1 , 2 , . . . , n . The norm on is p
1
ky=mf„
,x ),
2
n
Zj G Q , p
x
l jlp.*eQp* .
(7.1)
This norm is a non-Archimedean one as \x + y\ < maxd^lp, \y\ ), p
•
l*j +
p
<
i,jieQ
n
(7.2)
p
m^maxdijlpjj/jlp)
= max(max | ^ | p , max \ \ ) yj p
=
m^x(\x\ \y\ ) pt
p
Analysis
on the Field of p-Adic Numbers
The space QJJ is complete metric locally-compact and totally nected space. We introduce the inner product (x,y)
= i , ! / ! + xiyi + ... + x y ,x,y n
17
discon-
€ Qj; .
n
The following inequality is valid H*,y)\p<\*\p\y\p,*,ye®Z
7
-
( -3)
7
Denote by B ( a ) the ball of radius p with center at the point a G Qp and by S (a) its boundary (sphere); B ( 0 ) = B and S (0) = S , 7 6 Z (cf. 1.3). S ( a ) and S ( a ) are closed-open sets in QJ). 7
y
7
7
7
7
7
I t is easy to verify: if a = (01,02,... , a „ ) then B (a) S (a ) x ... x B (o ) . y
7
2
7
7
= B (ai) x 7
n
I I . A n a l y t i c Functions In this section we consider analytic functions in the field of p-adic numbers. The bases of this theory are presented in books by H.Koblitz [121], J.-P. Serre [190] and W . H . Schikhof [186]. 1 . Power
series
First we consider a numerical series in the field of p-adic numbers
£
ft,
<** G Q
P
.
(1.1)
0
Denote by 5 „ the n-th partial sum of the series (1.1) S
n
=
£ at, 0<*
n = 0,1,....
The convergence of the series (1-1) to a p-adic number S means that \S — Sp| —» 0, n —• 00; we call S its sum and n
S=
£ 0<*<M
a . k
18
p-Adic Analytit
and Mathematical
Physics
Some properties of series in the field Q essentially differ from those in the field of real (or complex) numbers. For example, there is only absolute convergence of series. More precisely the following Lemma is valid. p
L e m m a t. The series (1.1) converges tf and only if lofclj —> 0,
k —> oo.
• Necessity of the condition is obvious: K I p = l-S* - S -i\ k
— ' 0, k - » co.
p
To prove the sufficiency we use the Cauchy criterion. As K | —• 0, k —* oo then for any e > 0 there exists a N = N such that for all it > N the inequality K | < E is fulfilled. Hence for any integers n > N and m >N p
e
p
\Sm
Sri \p
—
E
<
n
Thus the sequence {S ,n series (1.1) converges. n
max b L <
" n<*<m
E .
r
—* oo} of partial sums converges and hence the •
From the Lemma 1 it follows that the sum of series (1.1) does not depend on the order of summation. Now we examine a p-adic power series m =
E M
(1.2)
0<*
which define a p-adic valued function f(x) for those x € Q for which it converges. p
D e f i n i t i o n . A number R = R(f) is called the radius of convergence of the series (1.2) i f it converges for all \x\ < R and diverges for | x | > R. We note that R may take values 0 and p , 7 £ 2. In the last case the series (1.2) converges uniformly in (open-closed) disc B as by the Lemma 1 f
p
1
7
E n
A-*
<
max m
k
\ftR \
0,
m n —* 00, t
Analysis
on the Field of p-Adic Numbers
19
and defines a continuous function f(x) in B . For determination of the radius of convergence of the series (1.2) like in the real case we introduce a number r — r(f) by the formula y
-=
Hm l/ftl*/*
(1.3)
or denoting r = p" t r = 5 = i Km i l n l A I , , . lnp t—oo i
(1.3')
r
L e m m a 2. T/ie series (1.2) converges for all \x\ < r and diverges for \x\ > r. p
p
U Let [ajjp < r. Then = (1 - 2e)r where 0 < e < 1/2. From (1.3) i t follows that there exists N = N such that for all k > N the inequality t
r(l-e) is valid, and hence
= (HiM
/ f c P
)
1
) ' — o,
k
* oo.
By Lemma 1 the series (1.2) converges at the point x. Let now \x\ > r . Then | x | = (1 + 2e)r where e > 0. From (1.3) i t follows that there exists a subsequence {n%, k —* oo) such that p
p
lim \fn \l = rl
k~-oo
Therefore the inequality
is valid for some N = N . e
Hence
kr
/nk
20
p-Adic Analysis
and Mathematical
Physics
and by Lemma 1 the series (1.2) diverges. Note that r is not necessarily an integer power of p, i.e. the number c in (1.3') is not always an integer (see examples in Sec. 2.4). Relations between numbers R(f) and r ( / ) are given by the following lemma which follows from Lemmas 1 and 2. L e m m a 3. R(f) R{f)
< r(f),
y
moreover if p~> < r(f)
T
= P i >f K / ) - P ik™ "ther R(f) = p
7
1
< pf* , 7
y € 1 then 1
or R(f) = p " .
7
Note that in the case r(f) — p , 7 £ 2 convergence of the series (1.2) on circumference \x\ = p requires a special investigation like in the real case. 7
p
2. Analytic
functions
D e f i n i t i o n . A function f{x) is called analytic in a disc B i f it can be represented by a power series convergent in B . (Obviously, one can always assume R(f) = p , see Sec. 2.1.) Analytic functions in a disc possess some usual properties, for instance, they form a ring with respect to ordinary operations of addition and multiplication. However there are some differences, for instance, superposition of analytic functions may turn out to be a non-analytic function (see [186]). We introduce the series for n = 0 , 1 , . . . y
y
7
/(">(*)= E *(*-!)---(*-« + 1 ) / ^ * " " ,
(2.1')
n
o i ^ ^ X * *
U
2
) +
'
(2.1")
which are obtained from the series (2.1) by termwise differentiation and integration respectively; f(x) = f^(x). These functions are called derivative and primitive of order 11 respectively. It ie clear that radii of convergence of the series (2.1) satisfy inequalities ( n)
R(f - )
{n)
< R(f) < R(f ),
n =
1,2,...
(2.2)
Analyiis
on the Field oj p-Adic Nurnbtrt
21
To obtain a more detailed information about radii of convergence of the series (2.1) we shall prove |Jbfp > \ ,
fc€Z+,
lim
it
|fc|i'* = L
k—•oo,ibeZ
m
• In fact, let k G S + , k = p fco, \k\ = p~
m
(2.3)
y +
where m and ka are integers,
p
m > 0, 1 < fc < P - 1- Then 0
m
=
In & — In &o In i > 1— Inp Inp
and thus
=
lim
p
* =
*—co,JteZ+
lim
p
t—eo,*eZ
'!"
= 1.
•
+
By using the formula (1-3) to the series (2.1) and the relation (2.3) we obtain the equalities (
,
(
n ,
' • ( / " ) = >-{/) = r ( / - ) ,
0 = 0,1,... .
(2.4)
From here and from Lemma 3 of Sec. 2.1 it follows that i f p p then R(f) = p and 7 + 1
7
< r(f) <
7
(n)
«(/ )
- RU) =
«=0,1,...;
7
if r(f) = p then two cases are possible: 1) R(f) equalities (2.5) are valid or n
p i r ; ( / f - > ) = R(f) for some «o > 1; 2) R{f)
= P
1 - 1
O I
n > no
7
then either the
(2.5')
then either the equalities (2.5) are valid 1
/£(/<"">) = R(f) for some «o > 1-
= R(fW),
= p
(2.5)
= mjW), P
n > n
0
(2.5")
22
p-Adic Analysis
and Mathematical
Physics
The formulas (2.5) can be interpreted by the following way. I t is possible to differentiate and to integrate an analytic termwise any times; by differentiation a radius of convergence may increase in p times, but by integration this radius may decrease in p times. As we see the situation somewhat differs from the case of real numbers. 3. Algebra
of Analytic
Functions
We denote by A a set of analytic functions in the unit disc B . Such functions are the only ones denned by the series (1.2) for which the condition \fk\ — • 0. k -*• oo is fulfilled (see Sec. 2.1). The set A is linear over the field . On the set A we introduce norm ] | / | | by the formula 0
P
p
11/11 = f « J / * l p .
/ € A
0
(3.1)
The functional (3,1) is in fact a norm, besides the non-Archimedian • Let ll/H = 0, / € A, i.e. max \f \
= 0 then f
t p
one.
= 0, k = 0 , 1 , . . . and
k
hence / = 0. Let a G Qp, a £ 0. Then M
= max|o:/*| - | a | m a x | / , | p
P
p
= | | ||/||. a
p
Finally, i f f,g 6 A then +
= m a x | / +g \ t
t p
< max max
| , p
|„] < max[||/j|, | | | | ] . ?
•
Theorem. The space A is a Banach algebra. • Prove completeness of A. Let a sequence {/",« —* oo}, / " £ A be fundamental. As fn
" -n\=™jft-fjr\p then the sequences {/£, n —. oo) are fundamental for every fc = 0 , 1 , . . . , thus they converge to some f G Q uniformly with respect to k (see Sec. 1.3) and hence k
lim f -co
k
p
= lim l i m f? = l i m lim f
k
t—oo n--oo
n—-oo I—.oo
= 0.
Analysis on ihe Field of p-Adic jVumiera
23
Therefore the function
/*»= J2 h> 0
belongs to A and lim ||/" - /II = lim n-tca
max
ti-.cc 0
Let f,g £ A and h = fg. Then h £ A (see Sec. 2.2) and
0<j
Hence \\h\\ = max |/ijtL = max 0<j
< f% ^J/jlj>l0*-jl m
m
;
< max | / j | max |
P
p
f f t
| = l l / l l l|ff| P
R e m a r k . The algebra A is called algebra of bounded power series and is a special case of Tate's algebras. x
4. F u n c t i o n s e , l n ( l + x), s i n s , cosa: These functions like in real case are defined by the series
"
(4.1)
jfc' '
0<);
m(l + x ) =
£ l
E
(4.2) (-1)* (2* + l ) f
(4.3)
0<Jt
(4.4)
cos x =
To study convergence of these series we need to estimate | n ! | for any n GZ . p
+
24
p-Adic Analyiii
and Mathematical
Let n S Z+, n < p in the form
) + 1
Phytici
— 1, s 6 Z+. Then n can be uniquely represented
n = n + niP + .-- + n . p ' , a
0 < nj < p - 1,
j = 0 , 1 , . . . , s.
(4.5)
Denote (4.6) 0<;<j
From (4.5) it follows that lim
(4.7)
— = 0.
Now we shall prove the equality (4.8)
A power M ( n ) in which factor p enters in n! is equal to
1J 1
+
n"
P.
n'
+ ...+ -P
=
M(TJ).
7
Here [a\ is the entire part of a number a. By using the representation (4.5) and the notation (4.6) we get i»i \ "-"o M[n) =
, n-no-njp n - n - f » i p - . . . - n,p* ^ +•••+ P 1-p"1-p- — 1-p1 - p -1 = n n ; rt p-1 p-1 p-1 ' p-1 0
r
2
1
1
:
0
l
n - s„ 7i —p-'p - l p - - i1
1
+ (no + n
l P
+ ... + n , p ' ) ^ — = ' p - 1 pP - 1 P—1
from which the equality (4.8) follows. x
1
Function e is denned by the series (4.1) from which i t follows ( e ) ' = ef. Taking into account the equalities (4.7) and (4.8) for the radius r ( e ) (see Sec. 2.2) we get r
Analysis
For p
on the Field of p-Adic Numbers
25
2 we obtain from (4.9) r
- < r ( e ) < 1. P Therefore Rtf?)
x M
= - , n £ l P
x
- 1
= R((e ) )
(4.10)
For p = 2 we get from (4.9) r(e ) = 2 . Let us investigate the convergence of the series (4.1) on the circle \x\ = 2 . Let x = 2 and k = 2". Then «t = 1 and owing to (4.3) - 1
2
1
= 2-*2*- = 2 -
1 7
4 0,
fc-oo.
2 l
Hence on the circle \x\ = 2~ the series (4.1) diverges, and by Lemma 3 of Sec. 2.1 R(e ) = 2 , and then (see Sec. 2.2) 2
x
- 2
x
R(e )
(n)
2
= R((e') )
= 2" ,
n£Z.
(4.11)
x
So for any p the series (4.1) for e can be termwise integrated and differentiated any times in the same radius of convergence. Denote by G the disc of convergence of the series (4.1); G is the additive group \x\ < p for p ^ 2 and \x\ < 2 for p = 2. By using the series (4.1) we verify the relation p
p
_ 1
- 2
p
2
1
e'e^e ^,
x, eG . y
(4.12)
p
Now we prove the equalities x
x
\e - 1\ = \x\ , P
\e \
p
P
= l
xeG .
l
(4.13)
p
• The second one follows from the first one. To prove it we establish beforehand the inequality
(1
c
< \x\ p -V ', P
p
x6G
p i
fce2 , +
(4.14)
26
p-Adic Analysis and Mathematical
Physics
where e = j j ^ f , p / 2 and e = 1- In fact, it follows from the equality (4.8) that p
2
|fc!|.
IF
as owing to (4.6) s > 1; for p = 2 the proof is similar. k
Now by using the inequality (4.14) we obtain (4.13) : „*-i x
\e -K
=
= x „ max
fcj
KKoo
.i-i
fc!
< (i-t)<- < p
l i
fc — 2 . 3 , . . . ,x £ G . p
Function l n ( l + x ) is defined by the series (4.2). Its radius r owing to (2.3) is equal to r ( l n ( l + * ) ) = l i m |fc|>" = 1. I—too
r _ 1
But at the point x — 1 the series (4.2) is divergent as | i | p > 1, k € !•+Therefore R(ln(l+ i » = - .
(4.15)
The function l n ( l + r ) possesses the property l n [ ( l + x ) ( l + y)} = l n ( l + x) + l n ( l + y),
\x\ < i | j , | < i , p
p
(4.16)
which can be verified with the help of the series (4.2). The following equality is valid | i n ( l + x)|p = | x | , p
x Gp.
(4.17)
6
Taking into account the equality (2.3) \k *| < k we have (4.17) p
|ln(l +
1
= |«Lm«|x|*- |«KKoo
1
Analysis on Ike Field 0/p-Adic
1
1
1
-
f P
Numbers
27
P^ 2
l*^ !* V< { ^ - a ^
p l
<
1
2
'
*= 2,3,...,x G,
•
€
Denote by J
P
is a multiplicative group where it follows from the identity l-xy = l + l-y+(l-x)(l-y). x
From the relations (4.12), (4.13), (4.16) and (4.17) i t follows that Ike function e realizes an isomorphism of the additive group G onto the multiplicative group Jp. The inverse isomorphism is realized by the function In x = l n ( l + x — 1) and the equalities are valtd 1
P
\ne* = x,
x£G,
e
i n x
,
x&J ,
(4.18)
p
which can be verified directly with the help of the series (4.1) and (4.2). Functions sin x and cos a; are defined by the series (4.3) and (4 .4). These series like in the case of the function e converge in G , and the following relations are valid 1
p
| s i n x | = \x\ , p
|cosx| = l ,
p
igG .
p
(4-19)
p
The standard trigonometrical formulas are valid, established by the help of the series (4.3) and (4.4), in particular 2
2
cos x + sin i = l ,
TX
e
= cosx + r s i n x ,
i£Q
(4.20)
p
2
where r = - 1 , r € Q , p = 1 (mod 4) (see Sec 1.4). Functions p
x are squares of p-adic functions are squares in Q
in G
p
x (for p = 2 functions e
x
and
ln
+I
(^ )
- 3
p
only for \x\? < 2 ) .
• These assertion follow from the Lemma of 1.4. In fact, the norms of functions (4.21) are equal to 1 owing to the equations (4.13),(4.17) and (4.19). Moreover their canonical forms are 1 + C (x)p+... ,p^2, 1
3
l + C ( ^ ) 2 - + . . . , = 2. 3
P
(4.22)
28
y-Adic Analysis and Mathematical
Physics
1
Let p £ 2. For e the representation (4.22) follows from (4.13) 1
le - 1L = IxL < - . p
x € G . P
For other functions it follows from the similar estimates which can be obtained by using the estimates (4.14) and (2.3). For p = 2 the situation is similar, more complicated. By using the estimate (4.14) we have for | x | < 2~ 2
2
.2k
I cos x — l|a < max
1
„2*
sin x — 1 < max 2
*
e Z
3
< |*|a2 -" < 2" ,
(2A)!
+ ( 2 * + 1)!
< \x\ max 2
- 1 - 2
p
x
x
2 2
l n
The similar estimate for functions e = (e * ) and t for | x | < 2 . Hence the representation (4.22) follows.
1 + I
- 3
2
G
p
* < 2
- 3
.
k€Z+
) are valid only •
Finally we note that the functions sinx and tanx = onto G . The inverse functions are
map one-to-one
p
inx = arcsin
E ^
0< arctanx =
£ o<*<»
« x — , 2 {k\)
XtGp,
2k
,
2A + 1
x £ G, p
(4.23)
(4.24)
besides jarcsinx|p — |arctanx|
5. Theorem
on Inverse
p
= |x| . p
(4.25)
Function
Let / ( x ) be an analytic function in the disc B {a) and f'(a) y± 0, | / ' ( a ) | = p . Then there exists a disc B (a), p < r such that / maps i t on a disc B (b), b = / ( a ) , one-to-one, the inverse function g(y) is analytic in the disc B (b) and the equality r
n
p
p
p+n
p+n
is valid.
Analysis
on the Field of p-Adic Number!
29
I By using the classical Theorem on inverse function we conclude that there exists a disc B (a) of sufficiently small radius p , p < r which the function / maps one-to-one onto some neighborhood U(b) of the point 6 and the inverse function g{y) is analytic in U(b) and the equality (5.1) holds. It remains to prove that for a sufficiently small p" U(b) = B + (b)p
p
p
n
As the function f(x) is analytic in the disc B ( a ) then it can be represented in the form of a series (see Sec. 2.2) r
f(x)
= / ( a ) + / ' ( a ) ( x - a) + (x - a )
2
£
a (x - a ) * " k
2
(5.2)
2<*
besides oo. Denote 2
max M „ p ' < * - > = p*,
SGZ.
t — £,3,...
Then choosing p < n — s (p < r ) from (5.2) we have \f(x)-b\
= \f'(a)\ \x-a\ ,
p
p
x£B (a).
p
(5.3)
p
From (5.3) i t follows that f(B,{a))
= U{b)cB (b)
(5.4)
f+n
and there exists a point x' € S (a) such that p
\f(*')-t>\
,
=
P
n
p
\f'(*)\p\* -a\p=P + -
1
Hence f(x') = y € S + (b) D U{b). Therefore the power series for the inverse function g(y) = g>(b)(y-b) + ... p
n
a
+
1
converges at the point y € S (b) and thus in the disc B +„(b). Reducing i f necessary the radius p we achieve that the function g(y) also will satisfy the equality (5.3) p+n
p
p+n
|s(y)-a|
P
= ^
\y-b\ , p
y€B . p+n
From here it follows that the inclusion 9(B (b)) p+n
C B (a)
= g(U(b))
fi
which together with (5.4) gives U(b) = B + (b)p
n
•
30 p-Adic Analytit and Mathematical Phyiics I I I . A d d i t i v e a n d M u l t i p l i c a t i v e Characters The field Q is an additive group. multiplicative group. P
1 . Additive
Characters
We denote by Q* = Q \ { 0 } its P
of the Field
Q
P
Additive character of the field Q is called a character of additive group Q , i.e. a continuous (complex-valued) function x( ) defined on Q and satisfied the conditions Ixt )! — 1> P
x
P
P
1
x(* + y) = x(*)x(y).
1
*.S>GQ .
1
C - )
P
a
n
a
Analogously one defines (additive) characters of the field Q ( i / £ ) of subgroup By, y G S, of the group Q . It is clear that every additive character of the field Q is a character of any group B . The function vp(^) = exp(2 ri{^} ) (1.2) p
P
P
y
1
p
for every fixed £ G Q is an additive character of the field Q and the group By. I t follows from the relation for fractional parts (see Sec. 1.2) P
P
{x + y}
p
= {x}
p
+ {y} -N.
AT = 0 , 1 .
p
Our goal is to prove that the formula (1.2) gives a general representation of additive characters of the field Q and the group B . Let x( ) arbitrary additive character. From (1.1) we have the relations x
P
x(0) = l ,
x(-x)
= W)
i
= x- (*l
D e
a
n
7
x H - ^ W f ,
n£Z (1.3)
At first we investigate characters of the group B . Let x £ 1 be such a character. Prove that there exists k G 7L such that y
X(a0 = l ,
xGB
t
.
(1,4)
• By virtue of the conditions x(0) = 1, \x(x)\ = 1 and x(i) is a continuous function on B i t is possible to choose such branch of the function y
Analysis on the Field of p-Adic Numbers 31 l n x ( z ) = t a r g x ( i ) that it will be continuous at 0 and argx(O) = 0. In particular there exists k £ S such that | a r g x ( x ) | < 1 for all x 6 B . Taking into account that nx e Bt for all x £ B and n £ S+ we conclude from (1.3) that k
k
|argx(x)| = |-argc(na0| < n n
nEZ ,
;
x G B
+
k
and thus arg\(a:) = 0 and x(x) = l , x € B .
•
k
We assume that the disc B in (1.4) is maximal so that as x(x) ^ 1 in then k < y. Now we prone for any integer r, k < r < 7, the equality k
B
y
p
+i
v ( p - ) = exp(27rimp-' ),
7
3m = 1 , 2
p "* - 1 ,
(1.5)
where m does not depend on r, • For r = y i t follows from (1.4) and (1.3) as 1 - X(P-*)
k
r
= x(p-^- )
p1
= [x(p- )] ~' •
For k < r < y r
X(p" ) = X ( p -
r +
y
y
^ ) = [X(p- )}" "
= [exp(2 rimp-
7+i
7
k
k
+k
T
r
)]" " .
•
7
Denote £ = p m where | £ | = p ~ * | m | > p- p--> = p " and | f | < p~ . Then owing to (1.2) the representation (1.5) takes the form Xp(p~ ) X p ( p 0 > and thus owing to (1.3) we have p
p
p
k
y
=
-1
T
X(P~ )
r
X (p" O,
=
k
P
k
(1.6)
- 7
for some £ £ Q such that p~ > ]£| > p . Let now x 6 By\B . The following representation is valid p
p
k
{x)
X
= x {Sx),
• Let x £ B-y\Btx = x p~ 0
r
+ xip"
P
3£GQ , P
Ki >P" P
7
-
(1.7)
Such x can be represented in the form r+1
+
x - pr k+1
h+1
+ x',
x' G B , k
xo^Q
32 p-Adic Analysis and Mathematical Physics for some r, k < r < 7. By using (1.6) and (1.4) we get the representation (1.7): r
i o
r + 1
xO) = [ x { - ) ] [ x ( p P
r
k+1
)r'...
r+1
= [xp(p- or\xp(p- tw° r
P
+ x - t:
ap
k+l
•-•
T+1
= x {z - t
k+i
[x (p- z)r'- xp(*'o P
+ ... + ^
lP
i
\x(p- )r'-+ x(*')
x
k
+
l
P
-
x
k
+
i + x'i) = x {*0 P
=
The case £ = 0 is impossible otherwise x( ) contradicts to the definition of the number it.
1
X (0) = 1 in B p
T
• which •
Hence we have just proved that any additive character of the group B has a form (1.2) where either^ = 0 or | £ | > p ~ ' . Now let x(0 ^ 1 be an additive character of the field Q . Then in a disc Bo it is represented in the form y
T + 1
p
p
( 0
x(*) = x « < ° M .
£ 'eQ ,
P
> 1•
P
(1.8)
We shall prove that in the disc Bi i t is represented in the form ll)
x(«) = Xp(Z *),
=t
w
-Kb,
3*0 = 0 , 1 , . . . , p - 1
(1.9)
• As S i = Bo U Si and Bo (~l Si = <j> v/e shall prove at first the representation (1.9) in the circumference S i . Any point a: £ Si is represented in the form 1
x = p- x
+ x',
0
3 x = 1,2,... , p - l ,
x' <= B
0
Q
.
Then using (1.8) we shall have the representation (1.9) in S i : x(x) = [ x f p - ' P x p ^ V ) = [ x ( i ) ] = M
xp{t *'
< 0 ,
r
/ p
x (^V) P
+ €0*') = x
X p
(tf + 6 ) (0)
P
=
Io/p
(0
x (* V) P
^ ) ^ )
p
X
(*f) (1
+ * ' ) } = x «- >*) P
for some £0 = 0 , 1 , . . . ,p — 1. The representation (1.9) is valid also in B owing to (1.8). •
a
Ano/yiij on ike Field of p-Adic Numbers 33 Continuing this process we obtain in the disc B )
x{*)=xpie *),
2
the representation
+&+&i>
for some t]\ = 0 , 1 , . . . ,p — 1, and so on. As a result in Q representation (1-2) X(x) = Y ( £ ; E ) ,
we obtain the
p
£ = ?<°> + & + Sip + - • • € Q .
p
p
Hence, any additive character of the field Q
has a form (1.2) for some
p
x
s
a
In other words, the mapping £ —• Xp(£ ) ' homomorphism of the additive group of the field Q onto the group of additive characters. This mapping is one-to-one (i.e. from the equality Xp{£i ) = Xp(& ) f ° " x e Q it follows that £ = £3). p
x
p
x
r
a
L
Now we have T h e o r e m . The group of additive characters of the field Q is isomorphic to its additive group Q , and the mapping £ —* Xp{£ ) gives this isomorphism. p
x
p
Let us denote Xoo(i) = e x p ( - 2 ^ ) ,
x e R .
(1.10)
Then the following (adelic) formula is valid
II
X P ( « ) = 1.
*eQ.
(l.ii)
2
ai
a
N ^p - -..p - " P
2
n
where EN G , PJ are prime numbers and N is positive integer not divisible by p j , j = 1,2,... , n. From the theory of congruence i t follows (see [204])* that the number x is represented in the form
We use the following result: if natural numbers c and d are relatively prime then the congruence cx + dy = 1 (mod cd) is solvable; indeed, dy = 1 (mod c) i = * ^ (mod d). f
£
34 p-Adic Analysis and Mathematical Physics J
for some N, G 2 + , 1 < Nj < pJ - 1 and M G Z. From (1.12) i t follows that =
J*7,
{*)P
=
0.
P^P;.
> = 1,2
n
and hence the equality (1.11) is valid
n
n
xr(«) =
=n ^nm^
2
!<)<« { j r } ) = exp(27rta:) .
= exp(2iri ^
•
P i
At last we note that any additive character x of the field Q (y/e~) (see Sec. 1.4) has the form p
x{x + V 5 ) - x (£ix)x (£2y), P
2. Multiplicative
£1,6 e QP
P
Characters
of the Field
Q
(113)
p
Multiplicative character of the field Q is called a character of the multiplicative group Q", i.e. a continuous (complex valued) function n(x), defined on Q* and satisfied the condition p
•x(xy) = x(x)x{y),
x,y e Q ; .
Analogously one defines (multiplicative) characters of the field Q ( \ / E ) , of subgroup S of the group Q (^/e) and of subgroup So of the group Q*. Any multiplicative character ir of the field Q has the form P
c
p
p
*(x)
= IX^MI^P'),
M * % = 1-
x- G S , 0
a GC
(2.1)
where JTQ is a character of the group So; conversely, any multiplicative character of the group So is extended up to a multiplicative character of the field Q by the formula (2.1). p
• Let IT be a multiplicative character of the field Q . Any element x G Q is represented in the form (see Sec. 1.2) p
l
x = \x\ x, p
x' = \x\ x G So , p
p
Anatytie on ike Field oj p-Adic Number) 35 and therefore the representation (2.1) is valid
where i t is denoted | x | = p
_ J V
p
, ff(p) = p
1 _ c
*.
•
We notice that the set of multiplicative characters of the compact group So is discrete (see [174], ch.VI). From (2.1) i t follows that i r ( l ) = 1. Let ?ro(a:) £ 1 be a multiplicative character of the group So- Then there exists such k £ that 0
ir <«) s i ,
x € S_fc(l) = [x G Q i j * - 1| < p-"] .
0
P
(2.2)
P
• A proof follows from the conditions 7T (1) = 1, |ir (a:)| = 1 and 7ro(z) is a continuous function on So, and it is similar to the proof of the corresponding statement for an additive character of the group B (see Sec. 4.1). 0
0
y
•
The lowest k eZ+li {0} for which the equality (2.2) is fulfilled is called a rank of the character ITQ(X). There exists only one character of rank 0 : J T O ( Z ) = 1, as So C Let the rank of a character xo(z) be positive. Then *o(x) So
x
= it (xo + iP+ 0
depends only on
TVO(X)
XQ,XI,..,
k
BQ(1).
1
••• + x -iP ~ )
(2.3)
k
,xjb_i.
• In fact a number x € So is represented in the form (see Sec. 1.2) _ 1
k
X = x + xip -f-... = (»o + * & + • • - + £ * - i p * ) ( l + tp ) 0
k
where \t\ < 1 so 1 + tp € fl-t(l). and the equality (2.3) follows. p
1
Then owing to (2.2) m , ( l + i p ) = 1 •
Now we prove the equality: if the rank k > 2 then
E 0<x„-i
*o(*o + xjp + . . . +
1
XM-IP*' )
= 0 .
(2.4)
36
p-Adic
Anaiysit
and Mathematical
Physici
• There exists a number ( = 0, l , . . . , p - l such that 1
p = *o(l+ <j>*- )#l . (Otherwise the rank of the character 7T() would be less then k.) Therefore using (2.4) we get the equalities r
x
53
x
*o( 0 + * i P + •• • +
k
1
t-iP ~ )
0
= P
53
+ tp^Mto
+ tip + • • • + zk-\p
k
l
)
0<**-i
= -
53
^ 0
*„[(*„+
+...+^.xp'-^a+tp*- )] 1
X l P
k
0
=
e
53
, :
1
To(^o + x i p + . . . + i i _ i P " ) t
from which the equality (2.4) follows. Examples of multiplicative characters. 1) w(x)= \x\ -\M*')= _ / 1, 2) r ( x ) = sgn x = -1,
1
p
if z € Q p >
(
if**Q; , i C
P^2,
3
where s £ Q* (see Sec. 1.4) and Q* is defined in Sec. 1.5: £
®;
it
= [*e%:x
0,66 0,,].
2
= a -eb\
(2.5)
To be convinced that the function sgn x defines a multiplicative character of the field Q it is sufficient to prove that £
p
rank (Qp/Qp ) = 2, i£
rank (Q^/Q*,*) = 2 .
(2.6)
• Let p ^ 2. In Sec. 1.4 it was shown that the rank of the group Qp/Qp' is equal to 4. Therefore owing to inclusions Q* C QJ C Qp it remains to prove that t
Of j* Qg* 7* QP
•
(
2
7
>
Analysis
on tke Field
of p-Adic
I t is necessary to consider three cases: z = J),np,p, where n Sec. 1.4). We shall prove mt%,
n t % w
m
vt%,
Numbers
Q*
•
P
37
a
(see
(2.8)
which means that Q* / Q*. it
2
2
Let conversely prj G Q* , i.e. prj = a — nb for some a,6 £ Q , 6 ^ 0 . But the last equality is impossible for none of a and 6 ^ 0 . Indeed rewriting it in the form iP
we see that for 1) \y\
p
p
< 1 the number 1 + j - is a square of a p-adic number
(see Sec. 1.4) and then pn would be a square, 2) | y \ > 1 the number 1 + ^ j p
is a square of a p-adic number and then n would be a square, 3) \y\p = 1 is impossible. Other statements (2.8) are proved similarly. Now we prove that Q " ^ Q"*. Let z = n. Then rj G QJ by the Lemma of Sec. 8.2. Let c = p,pv. Then - e G Q as i t is represented in the form (2.5) with a = 0 and b = 1. • (
>fl
P i £
R e m a r k . For p = 2 (2.6) takes the form rank ( Q J / Q " ^ ) = 2,
rank ( Q ^ / Q ^ ) - 4 ,
see [37] Sec. 6. 3. Multiplicative
Characters
of the Field
Qp(i/£)
According to Sec. 1.5 every element z of the field Q p ( v ^ ) is represented in the form either z ~ r
Let ir{z) be a multiplicative character of the field Q [^/e). Let us denote by and TT2 its restrictions on the field Q and on the "circle" C respectively. Then we shall have p
p
n(rcr) = jri.(r)jr2(
7r(ircr) = •n(v)ir(rtr)
.
(3.1)
From the equality per = ( - p ) ( - f ) it follows the condition Ti(-l)7r (-l) = l . 3
(3.2)
c
38
p-Adic
Antxlyti)
and Mathematical
Phyiici
Further as 2
v =
i>i> e Q* r
v
1/
- £ C
L
we have the equality — Tt\{yv)Tr-i(ylv)
2
Tr (f)
•
(3-3)
Conversely, let Ti and T3 be arbitrary multiplicative characters of the field Qp and the "circle" C respectively satisfied the condition (3.2). Then the function rr{z) defined on the multiplicative group Qp(i/^) by the formulas (3.1) and (3.3) for some v € Q p ( v ^ ) . & £ QJ multiplicative character of the field Qp(yF). e
v
, s
a
R e m a r k . The set of multiplicative characters W2 of the compact group C (see Sec. 1.5) is discrete. c
I V . Integration Theory In this section the theory of integration of complex valued functions of p-adic arguments will be presented and some examples of calculations of specific integrals will be given. 1 . Invariant
Measure
on the Field
Q
p
As the field Q is a locally-compact commutative group with respect to addition then in Q there exists the Haar measure, a positive measure dx which is invariant with respect to shifts, d(x + a) = dx (see [163]). We normalize the measure dx such that p
p
J
dx = 1 .
(1.1)
Under such agreement the measure dx is unique. For any compact K C Q the measure dx defines a positive linear continuous functional on C(K) by the formula J f(x)dx, f e C{K). Here p
K
C(K)
is the space of continuous functions on K with the norm
Analyti) on the Field of p-Adic N-umbert 3 9 A function / € L \ there exists Jim
is called integrable on Q
o c
f m m =
lim
^
p
(improper integral) i f
/ f(*)d*
.
(1.2)
This limit is called an integral (improper) of the function / on Q , and it is denoted by / f(x)dx so that p
ff(x)dx
=
53
/ f(z)dz.
(1.3)
Analogously one defines the improper integral with respect to a point a(=Q : i f / € Ll (q„\{a}) then p
c
/ f(x)dx
2. Change
of Variables
= Jim
in
53
/
f(x)dx.
(1.4)
Integrals
A t first we shall prove the formula d{xa) = \a\ dx, p
aEQt.
(2.1)
• For any a € Q£ the measure d ( i a ) is invariant with respect to shifts, and therefore i t differs from the measure dx by a factor C(a) > 0, d(xa) = C(a)dx. From here it follows that C(a) is a continuous function satisfying the condition C{ab) = C(d)C{b) i.e. C is a multiplicative character of the group Q*, and hence it has a form (2.1) of Sec. (3.2). 1
C(a) = \a\ '- ir {«')> p
|»o(o')| = 1 ,
0
«' € S, 0
«eC.
But C(a) > 0 and therefore w (a') = 1. Hence C(a) = | a | £ ~ \ a e C. We shall find a number a. As 5o is a union of disjoint sets pBo + k = 5_i(fc), k = 0,1,... ,p — 1 whose measures are equal then measure Bn — p measure B-i and hence d(xp) = ^dx i.e. C(p) = * = | p | . Thus a = 2, C(a) — |a| and (2.1) is valid. • 0
p
p
0 p-Adic Analyait and Mathematical Physics By using the formula (2.1) to an integral we get j f(x)dx
= \a\ j
f(ay + b)dy,
p
^ 0 .
•
Let us show how to use this formula to do simple integrals. Example 1.
j dx = p , y
y£Z.
It follows from the formulas (2.1) and (2.2) that j
j
dx=
I«l»
d(p-> ) y
j'dy
= \p-\
W\,
= ^ . P
B„
E x a m p l e 2. fdx
= p-'(l--),
J
yeZ. P
5,
It follows from the formula (2.3)
Jdx
=
J dx- j B-,
S-,
7
dx = p~< - p "
1
B-,-i
Thus the "area" of a circle in Q is positive! p
E x a m p l e 3. From (1.3) and (2.4) it follows that ff(\x\ )dx= p
E x a m p l e 4. In particular,
f i - i ]
Y,
f
W
.
Analysis on Iht Field of p-Adic Numbers 41 E x a m p l e 5.
[\ \x\ dxP= . -1
J Bo j
R
(2.6)
p
ln|*| dx = - ( l - I ) l n p
£
P
TP- =
-
^
-
Here we have used the equality
"(P-1)
0<7
2
R e m a r k . A n invariant measure da;* of the multiplicative group Q* of the field Q has the form p
tTx
=
1
Wr * .
• It follows from (2.1) d>x) = M "
1
^ * ) = I x f M x = d*x,
a 6 Q* .
•
G e n e r a l change o f variables i n integrals. /,<
p
P
• As an integral is a linear continuous functional on C(K) (see Sec. 4.1) then it is sufficient to prove the formula (2.8) for locally-constant functions on K, the set of which is dense in C(K) (see Sec. 6.2). Hence i t is sufficient to prove the equality (2.8) for / ( x ) = 1, x (j K i.e.
Jdx = J \o-'(y)\ dy . p
K
K,
42
p-Adic Analysis and Mathematical Physics
We cover compact K% by a finite number of disjoint discs B (yk) Sec. 1.3) of a sufficiently small radius p" such that
{see
p
\
rk
P
=p,
y€B (y ) p
k
and the disc B (yt) is mapped onto the disc B + {xi:), x = c{y ) according to the Theorem on inverse function of Sec. 2.5. From here using the formula (2.3) we get the equality (2.9) p
p
K(»)I*=E
=E / * *,(*»)
/
ri
K
k
(
»
* B,<su.)
)
k
I
F
*
•
Ki
Examples. I f a function / is integrable on then (see (1.4)) p
(
/ ' « * - / ! * ' ( ; ) * • • •
2
1
0
!
The following formulas are valid (see Sec. 2.4) j f{x)dx
= j
f( my)dy
/ /f>)
f(\ny)dy,
G, 3. Some Examples
J
S
= J f(t aay)dy .
p
of Calculation
j J
f{x)dx
= J f(e»)dy C,
r
of
(2.11)
. 2.12)
Integrals
In addition to Sec. 4.2 we shall list some examples of calculation of integrals in explicit form (see [206]). E x a m p l e 6.
Analysis on tlie Field of p-Adic Numbers 43 y
r
• For [£|p < p~ we have \t]x\ < 1 and therefore x p ( £ ) = I - Hence owing to (2.3) p
J X (£x)dx
= j dx=f
P
.
7 + 1
If KIP > P " then for some x' G 5 , we have |c>'| = \t\ \x'\ > p and therefore Xp(i ') ^ 1. Then performing the change of variable x = y — x' we get that the desired integral is equal to zero: p
p
p
x
j
X
p
m d x =
B-,
j
,
Xp{i{y-x ))dy
j {cly)dy.
= (-ix-) Xp
Xp
B,(*<)
B ,
Here we have used the fact that B^(x')
1
= B i f \x'\ < p" (see Sec. 1.3). 7
p
•
E x a m p l e 7.
7
P (l"f),
JXp{£x)dx
=
\t\
1
-f- ,
KIP = P "
0,
1 + 1
3
>
y+
m >p- \
2
( - )
7 CX-
P
lt follows from the formulas (3.1) as = J Xv(i')dx
/ Xp(t*)dx
E x a m p l e 8.* I f
£ 0<7
I / O
-
7
) ! ?
-
7
-
j
X
p{fx)dx
.
< oo then for £ f 0
/ f(\x\ )x (tx)dx P
V
W
P
0<
* About improper integrals see Sec. 4.1
(3.3)
44
p-Adic Analytii
and Mathematical
Pkytici
• I t follows from the formulas (1.3) and(3.2). Denoting | £ | = p have p
Q,
-oo<7«»
P
N
we
I
-oo<7<-iV
P
0<-,
Example 9. x>«»)«ir = 0,
0 .
(3.4)
fli t follows from (3.3) for / = 1. Example 10. „£I-1
1
It follows from (3.3) for / = I s ] " - :
P
Q,
0<7<~
netrV^iet** = (rrp= - p ° ' ) let* • 1
Example 11.
j ln\x\ (Cx)d pXp
x
1
= -J^iet .
f / 0 .
(3.
Analyiie on the Field of p-Adic Numbers 45 I t follows from (3.3) for / = In \x\ : p
J
M
x
\ \pX (£x)dx P
OF 1
= ( -;)ia
_1
P ^ - T ^ P - l n l ^ + l^'Ona-lnp)
£
0<7
0<7
E
\t\
f
.x\j
2
+m
2
2
,-7
P
2
- P '
2
7
7- (3-7)
1
It follows from (3.3) for / = {\x\ + m ) " :
/
XPIZX) dx x\l m
+
2
= a-i)ifiP
=
V ~ P)
e
1
y o
k 0
£
l 0
>
"
2 7
^ l^p
1
2
(p-
+
2 7
m
+
Z
m 2
2
p ia
let _ 2
1
+ ™
2
3
iei? " i ^+ ^ l e l . )
From the formula (3.7) it follows an asymptotics
[xtt + m
2
2
p++ P
lm*m'
HI, - » oo .
(3.8)
46
p-Adic Analysis and Mathematical
lim I, —oo
=
mI r
J
f i _ I ^ i _
v
Physics
y
y
H r n
/
p-*h?-p-**)(i-
,
p
2
" \
l c l 2
0
p / tn \ 1 — p 4
1 — -p3
1
• Note that in the real case the similar integral is equal to OO
/ y
DO
X - « f L T + m
2
I
/ "P(- ^') j x +m
=
2
2
2
— CO
&
=
2
ZL -^Htl m e
(3.9)
—CO
and hence exponentially decreases as |£| —> CO. E x a m p l e 13. jf
y
i a r = p -\
7 e 2,
t = 1,2, -,p~l
(3.10)
• By virtue of invariance of the measure dx with respect to shifts from the formula (2.4) we have
and the formula (3.10) follows. E x a m p l e 14.
7
|
(te=p (i--J,
S,,i ** 0
T
es,
i = i,2,...,p-i.
(3.ii)
Anttfti*
on the Field of p-Adic
Number!
47
It follows from the formulas (2.1) and (3.10):
J
dx = j
5,^0^1
S
dx-
j S-,,x
T
a
dx=p*
|f
f l -
- p
T _ 1
.
=k
E x a m p l e 15. I = 1,2,...
J
dx = p'-
1
,
(l -
7 e S,
k = 0,1,2
(3.12)
p-1,
E x a m p l e 16. i = 0,1,2,...
y
j
dx=p -'-\
yeZ,
0
to?£0.
K
S,i*a **>*i=*i<— >*i = *r
(3.13) E x a m p l e 17. ( = 0 , 1 , 2 , . . .
j S
x
dM^pt-'-ifl-j),
7€Z,0<^
x :4
T' O = *Di l '*l
-.=*|.*I+1**J + 1 io^O.
(3.14)
E x a m p l e 18.
J
l
l
- < ~
l
d
x
=
l(i - ~°v 2
P
P
9
i
Q
>
0
'
(
3
1
5
)
48
p-Adic
Analysis
and Mathematical
• By using the formulas
J\\-z\ - dx 1
+
j
=
p
J
(3.11)
n-*lf**+
1
|i-*r*- %+...
/
S ,* =l,*i^0 e
+
(3.15):
S„,10 = 1,11=0,1:^0
So.co^l
V
and ( 3 - 1 4 ) we get
|1
Se,»g=li'i#0
- l - I
Physics
S , r o = l,*i=D.*3i*0
o
± ( l - i )
^ ( t - i )
+
p / i - p - *
0
...
+
p
E x a m p l e 19.
/
5
ln|l-x| rfx= - - ^
(3.16)
p
P - 1
0
Acting as in Example 1 8 we get j In |1 - x\ dx p
= In 1
y
- In
4 P
y
dfe + h i -
/
(fx + . . .
2
p V
Here we have used the equality
P 7VP
p
(2.7).
2
' 7
P P ( I - P
-
1
) •
4. J n t e g r a i i o n i n Q " The invariant measure dx on Q in a standard way is extended up to an invariant measure dx = dxidx? • • • dx on QJJ (see Sec. 1 . 7 ) , and all which was said in 4 . 1 for Q caries over to (KJ. A reduction of multidimensional integration to a one dimentional one is given by the following theorem (see p
n
p
[163]).
Analysis cti ike Field 0] p-Adic
Numbers
49
Fubini's theorem (on a change of order of integration). Let a function f(x,y}, x G QJJ, y G Q™ be such that an iterated integral
J I
dx
\f(*,v)\dv
exists. Then the function f is integrable on Q p iterated integrals of f, and they are equal
j
J
f(x,y)dy
dx=
j
j
f(x,y)dxdy=
+ m
, and there exist all
J
f(x,y)d
Conversely, if a function f is (absolutely) integrable on
+m
dy.
(4.1)
then all in-
Change of variables in integral. Let x = x(y) (i.e. Xi = Xi(jfi, jftj ..., y ),i = 1,2,... , n) be a komeomorpkic mapping of an open compact Ki C Qp onto (open) compact K C Qp, and also functions Xi(y) are analytic in K\ and n
d e t
$ m = dy
Then for any f G L\K) J f(x)dx K
dxi d
e
* 0,
t
y G K1
•
the equality (4.2) ts valid, =
j ^ t
d
x
(
y
dy
)
f(x(y))dy
.
(4.2)
K,
I A proof of the formula (4.2) carries out by induction on n like in a real case using the one dimensional formula (2.8). I We note also passage under {fk,b —* oo} of Qp (with respect
an analogy of the Lebesgue Theorem on limiting the sign of an integral (see [163]); If a sequence functions ft G L ( Q p ) converges almost everywhere in to the measure dx) to a function f,
h(x)^f(x),
1
fc^co,
zGQ;
a.e. ,
50
j.
Analytis and Mathematical
Phytic! 1
1
and inert exists a function yj £ L ( Q ) suck that p
lM*)|
l
* = 1.2
xeL (Q?)
then the equality is true Km
/ f (x)dx k
= J f(x)dx .
Of
E x a m p l e 20. The function a»i£+i»*£+...+i*ngr' is locally integrable in QJJ i f 2a < n. • By using the formula (2.4) we have 1
= j J-•• J{\xi\l
+
\X2?
P
+...+ K5)""
flg fig £J(J ^
P
'
-™<1i,-,7.<0
- ( ' - ; ) "
E 0
J
E 0
^ 3
(p- " + ... + p - ' - U p -
I W
...+v.
Note the equality min
( <*! + . . . + Or„_i +
O < Q , < I V^ )
=
r
min ( n — l ) a + °<»
a ...a _i 1
B
„- p2N2N
a"
-
- 1
np
Continuing our estimates we get for 0 < a < 2n
Q
£ 0
p - W f l - ^ - l < ^ ,
"
t
" '
+ 2 B
Analyiit on the Field 0/ p-Adic Numbers 51 E x a m p l e 2 1 . p = 3 (mod 4). /
4
/((x.xj^ae.z^rfn^a
-|-*r 2
l(£.£) l (4-4) if £
p"
2 7
i/(p
_ 2 l
)i<^-
0<7
• Let us denote the desired integral by ./(£). Owing to its symmetrey on £ = (£1,^2) is sufficient to prove the formula (4.4) for \£i\ < |£2|pDenote folp = p~ , |£ | = P~ , N < M. Using the equality p
N
M
2 p
\{x,x)\
= \x\ +
p
xl\ = ^W\lMtf ) p
m
p
(see Sec. 1.1) and the formulas {3.1)—(3.4) we get the equalities
J(t) = j
j
f(\xy\l)Xp(S2X2)d*2
/
/(l*j£)Xp(&* )
|islp>lnlp = 1
/(k,| )ki| Xp(£^i)fi(l£2^il )^i 2
P
P
Q +1
/{|x | )Xp(£2X2) • 2
2
p
j
X
p(tixi)dxidxi
\pdxi
- j
l
H l ^ l ^ l p x A ^ ^ ^ i x ^ d x ,
52 p-Adic Analysis and Mathematical Physio =
j
j
l
f(\x \l)\x \ dx + 1
1 P
1
f(\x \l)\x \„dxi 2
2
M>SI&I>'
+
I
s
r
/
+ " P
=
(
f(\x2\l)\^\pXp(Z2X2)dX7
;T
r
y
£
2y
P f(p )
AT+l<7
1
-
^
>
)
[ Xp(t2*2)d*2 §
£
P
- W / (
P
^ ^ ) - V
W
+
1
/ ( P
2
J
V
+
3
)
P
' 0<7
He
2y
£
2y
2
2
p- f(p- \h\; )-f(p*\b\ - ) P
0<7
from where the formula ( 4 . 4 ) follows if we note that fol IK,e)lp when
2
2
2
= |£ + £ ] = p
< Ifsl,,.
•
In particular, for the propagator G = {\(x,x)\
p
2
+ m )-'
(4.5)
(cf. ( 3 . 7 ) ) the formula ( 4 . 4 ) takes the form
/
Xp((*») \(x,x)\ + m* p
2
P / <^„
VP"
0
2 7
2
+
P + m'KCOB ;
( 4
.6)
From here i t follows as in the case of the propagator ( 3 . 7 ) that the righthand part of the equality ( 4 . 6 ) is positive and have an asymptotks
2
P
+
lm*\(U)\j'
l«.OI*
(4.7)
By another method the integral ( 4 . 6 ) has been calculated in Bikulov [ 3 5 ] .
Analyiii on the Field of p-Adic jVumierJ S3 E x a m p l e 22. p = 1 (mod 4).
J f((x,x)) {{f,x))dx Xp
H)
0
(4.8)
if £
TP I / ( P 7
T
) | < oo .
0<-|
• There exists a solution r £ Q of the equation T = — 1 (see Sec. 1.4). Change of variables in the integral p
yi = K * ! - TX ), VI = §(*i + ™2), 2
Xi = J/i + J/2, x = T(yi - y ), 2
2
gives 2
\(x,x)\j, = \xl+x \
2 p
(€.*) =
= |2(l-T)
t f l ! f t
| = knlplsal, ,
+&*a - £ i ( ! / i + j/2)+6''(!(i -y2) = Ciyi + C2y2 - (C,y)
where i t is denoted Ci=Ci + r 6 ,
C2 = 6 - r&
2
(C1C2 =
+ £ - (£.£)) • 2
Therefore the desired integral / ( £ ) takes the form (see (4.2)) J(0
= j
f(\yi\p\y2\p)x {«.y))dy P
X (
(4-9)
54
p-Adic Analysis and Mathematical Physics
By using the formula (3.3) twice to the iterated integral in (4.9) we get the formula (4.8):
^
V
L ^
V = 0
1
/
7=0
1
-/(p-^lCilp- ^!; )^!;
7+1
1
1
1
1
1
2
1
1
1
•/(p- Kil - K2l " )Kilp" IC2| - + /(p lCil - IC2| - )Ki| - lC2lp p
F
p
p
p
1
p
=IC C2| - [ ( i - J ) ' £ ( 7 + i J p - V ^ I C i ^ l p - ) 1
1
-
2
f
=idc ir 3
V.
1
P
1
7
1
T
1
2
1
- - ) £ P " / ( P - K I < 2 | - ) + /(P ICIC2| - ) p
i -
p
^)£(t^)^^"W)
T h e Gaussian Integrals 2
Integrals of the form / x ( a z + bx)dx are called Gaussian ones. Here we shall consider the cases when sets of integration are the circumference , the circle B-, and the whole space Q . p
p
In particular, the important formula will be obtained (see (3.1) of Sec. 5.3)
J (ax
2
Xp
l
+ bx)dx = X (a)\a\; " p
Xp
,
a# 0
Here we used an important number-theoretical function A : Q* —* C by p
Anatysii on the Field of p-Adic Numbers 55 the following way: if 7(a) is even, if 7(0) is odd and p = 1 (mod 4), if 7(a) is odd and p = 3 (mod 4); W
A
1+
f 7f[
1
f l
if7(«)iseven,
1
(- ) ' l
<
1
I ^•' "(- )
if7(i)isodd.
a a
We note the simplest properties of the function A (a): p
| A , ( a ) | , = 1,
A ( ) A ( - n ) = 1, p
a
2
A (ac ) = A ( ) , p
p
a
n^O,
p
a,
C ?
£0.
The furthest properties of the function A (a) will be given in Sec. 5.4. Calculation of the Gaussian integrals for p / 2 is based on the Gauss sums (see [37]) p
0<*
if
p — 1 (mod 4) ,
if
p = 3 (mod 4)
where m is an integer not divisible by p, ( y ) is the Legendre symbol (see Sec. 1.4). In terms of the function A (a) the Gauss sum (*) takes the form p
= A (m),/p, p
;jc< -i 0<Jt
v
y
/
P
1. The Gaussian
Integrals
on the Circles
S
y
E x a m p l e 1 . ptjk 2, |e| = 1, (see [215,218]) p
T
2
/ X {e(?-V) )dy t
=
-p'-'Xric* ), 0, 1, I 0,
7<0,
XL -= p„-y+i 1+2
>P~ : *IP
7
= P>
7 < 0, 7<0,
7>1, (1.1)
56 p-Adic Analysis and Mathematical Physics • For 7 < 0, \x\ < p - \ y e S we have \e(y - 2xy)\ 1xy)\ < max(|j/| , \2xy\ ) < m a x ^ , ? ^ - ) = 1 and hence 2
P
y
p
2
= |e| |(j/
2
-
p
1
p
p
2
(e(x
- y) ) = (e(y
Xp
2
= (£(y
2
2
-
Xp
2xy)+£x )
2
2
- 2xy)) (ex )
Xp
= (£x )
Xp
.
Xp
The desired integral owing to (2.4) of Sec. 4.2 is equal to Jdx
2
{cx )
Xp
For 7 < 0, jxjp = p ~ (e(x
-
Xp
2
y) )
7 + 1
we have \ey \
y
X
p(ex
2
+ £y
2
-
-
P
2
, y 6 S 2
=
± ) x (ex )
= p>(l-
lexy)
=
P
= p
2 7
< 1 and hence
(£x )x (-2£xy) 2
Xp
P
.
The integral owing to (3.2) of Sec. 4.3 is equal to j
2
(cx )
Xp
For 7 < 0, \x\ = p ,
2N
0
+ e,p + . . . )(x
+ • • - + (*w-i
-
we have
y
{e(x - y) } = | p " (e p
= -p^'xp^x )
N > - 7 + 2, y € S
N
p
2
2
(-2exy)dy
Xp
+ p+...+
0
(**_, -
Xl
1
~ y ^ - i j p ^ "
yelp*
- 7
+ •• - ? \ Jp
2 - L(y ,y\,... 0
-
, yN+1-2)
-e xlyx -,-i 0
+
where L does not depend on yN+-,-i- Then taking into account the formula (3.13) of Sec. 4.3 for I = JV + 7 - 1 we have for the integral
p-
N
£ E
E
-
E 2
exp f - — £ i y w 0
+ 7
-p(2-i) -i J= 0 .
Analysis on tke Field of p-Adic Numbers 57 Acting similarly we are convinced that in cases y > 1, | x j ft p* the desired integral equal to 0, and in the case 7 > 1, | x | = p i t is equal to p
1
p
j
2
fcOO
- y) )dy
S ,y =*o T
Xp
S ,yo=x ,... p
X
M*
- y?)dy
= •••
•S,,yo=io,yi=ii
0
j =
J
=
J
2
(e(x
~ y) )dy
dy
,y-,-i=x -,
y
0
7
-( -l)-l
y
7
=
l
i
owing to the formula (3.13) of Sec. 4,3 for I = 7 — 1. E x a m p l e 2 . p # 2, |e| = 1, 7 € Z (see [215,218]) p
2
P^l-^Xpiep* ),
*lp < P "
7 + 1
, 7<0,
z| = p -
7 + Z
. 7<0,
*IP>P~
7
.
P
/
2
x (ep(x-y) )dy
0,
=
P
*\P
0,
*\p >P >
Ap(erp)Vp\
4>=P >
2
T
0,
3
7 = 1. 7>2,
7
x|p^P , 7 > 2 . (1.2)
• Similar to the Example 1. Some peculiar are: For 7 = 1, | x | < 1, y € Si we have p
2
2
|ep(x - 2 x y ) | = | e | | p | | i - 2 x y | p
P
P
p
2
< ^ max(|*| , |2xj/| ) < i max( 1,p) = 1 . p
Therefore 2
2
2
X (cp(3! - y) ) = Xp(epy )x [£p{z p
P
- 2xy)} = x (epy) . p
But {spy }
= | i ( e + £ip + .. .)(yo + yip + • • • ) | a
p
0
7<0,
K{£p)y/P-Xp(£px ),
2
+
=
58 p-Adic Analysis and Mathematical Physics and owing to (3.10) of Sec. 4.3 and (*) the integral is equal to
M«f)*-jc 5
r
(^f)
l
i
0
=
E
e x p ( 2 ^ ) - l
0
P
^
o
= A ( p)Vp-l p
£
'
For 7 = 1, k l = P. y € 5 i we have P
2
{ e p ( * - y ) } p = {^(eo + £iP + •••)[*
-yi)p+ ...]
:
and the integral is equal to
e x
/ e x p ( — £ o ( x - y o ) M dy = 0
i
£
P (~ V
i
P
0
=
Y\
exp ( — e ( ^ o - yo) ) - exp ( — £ x\ 2
£ 0<*
e
x
p
( \
P
2
,
' r
i
V P 2
— ) -ex 27ri{epz } P / P
1
For 7 > 2, j « | = p , y £ S p
7
p
2
j
a
0
V
O
=
x
°( ° ~ f°) ) '
/
= A (£p)^p p
x (epx ) P
2
we have
2
{£p(z-y) } 2
= {p* "
r + 1
p
} ( £ + £lP + . . . ) [ x - J f e + fci - * i ) p + • • 0
0
2 7
Z
2
+ ( * 2 - 2 - y 2 - z ) p " + . . . ] }}p, 7
7
As in Example 1 the integral over that part of S where Xo ^ yo is equal to 0. Therefore the integral is equal over that part of S where xn = j/o. y
7
Analysis on the Field of p-Adic Numlert 59 and so on. So we have J
2
X > M x - y) )dy
j
=
2
x {ep(x
- y) )dy
P
=„ .
•ST.SD=IO
j
Xp(ep(x -
L
0<S-,-i
2
y) )dy
P
Yl *p [ " - * _ = ^p(ep)VP • 0<Jt<po<*<»-i Here we have used the formulas (3.13) of Sec. 4.3 for / = y — 1 and (*) e
=
2 5 r i
a
p
E x a m p l e 3. p = 2, |e| = l,y€Z
(see[239,218])
2
ss|a
2
V-^iex ), 2
-2^ {ex ) X2
/
2
- y) )dy
1
T<0,
i|2 = 2 - ^ ,
T<0.
+ !
3
0, X2(e(z
+1
I
| >2-^+ , 3
7
< 0 ,
z|2
7=1,
i,
z|a = 2,
T=1,
o,
a| >4,
7=1,
=
2
V2X (e),
*|2=27,
7>2,
0,
*b*2\
7>2.
3
(1.3) • Similar to Examples 1 and 2. Special cases are the following. For 7 = 1, | x | < 1, y € Si we have 2
\e{x
2
- 2xy)\
2
2
< m a x ( | x | , \2xy\ ) < m a x ( l , - | y | j ) = 1 , 2
2
{ey }2-{\(l
+ ei2 + --.)(l
+
y i
2+...f^
and the integral is equal to j X2^y )dy 2
Si
= j exp [ 7ri Q + | 2
dy
5 ,
—KM
= <(-!)'
= \ +^
,
60 p-Adic Analysis and Mathematical Physics For 7 — 1) \z\v = 2, y E Si we have 2
\z(x~y) \
= ^[(*i-!/i)2 + (* -i/ )4 + ...]
2
= \(x-y) \
2
2
< 1
:
2
2
and the integral is equal to
Si
For 7 > 2, | i | H*
= 2 \ y e S we have
2
7
= {2
- yfh
_ 2 T
( 1 + M + . . . ) [ ( * , - !fi)2 + £ i - y )4 + .. . ] } 2
2
a
.^37-4) - ^ ( ^ i - ai)»3r-8 1
= L(y ,yi,... 0
and the desired integral is equal to J
xM*
f
- y?)dy =
xM*
2
- y) )dy
= •••
y i=»i, sg =13
•S„yi=ii
/
v-.-J = ' i - a
For the last integral we have a
{ e ( * - y) >2 =
+ Ei2 + . • • H K - j - y - i ) +
- *,)2 + . . . ] '
7
and hence the desired integral is equal to y
=
w»
2 , r i
x
(^+Y)( 7-i-!/T-i)
e x p ^ i Q + ^ ^ T - i - ^ - i ) 2«t k=0,l
a
;
dy
1
+
(M)]—h(l t)]
= l + i ( - i ) * * = v^A (e) . 2
Analysis on ike Field of p-Adic Numbers 61 Here we have used the formular (3.13) of Sec. 4.3 for I = y — 2.
•
E x a m p l e 4. p = 2, |e|a = *> 7 G Z (see [239,218]) M»<2~ 1
2
J X (2e(x - y) )dy 2
5
1
|xb > 2 - » + ,
-1,
Ma < I .
7=1,
1,
Ma = 2,
7=1.
<0,
7
<0,
7<0,
A (2e),
Ma = .
7=1,
0,
Ma>8,
7 = 1,
2A (2e),
Ma<2.
7 = 2,
0,
Ma > *.
7 = 2,
4
2
T
7
4
o,
=
2
.
k b = 2-r+»,
3
-2^- v (2« )
7 + 3
2
2A (2£),
Ma =
7 > 3,
0,
Ma#2\
7>3.
2
(1.4) • Similar to Examples 1-3. Special cases are the following. For 7 = 1, |z|2 = 4, y G Si we have
2
{2 (x - y) ) £
2
= j ^(1 + 2 + £ 4 . . . )[1 + (xi - 1)2 + (x - )4 £ l
3
a
(«, " l )
1
+
4
+
2
Vi
+
, *1 -
2 '
and the integral is equal to
= exp Si
1 + i. .•"(-ir
a
= A (2£) . 2
•4
62
v-Aiic
Antlytii
and Mathematical
For 7 = 2,
N
= 1 ,N <\,yeSi
= | i ( l +£ 2 + 4...)[-(l1
=
/ I
8
+
£i
+
4
Phytict
e 2
£
+
+
2
2 +
^ + ^
we have
) 2 + ... + ( l - y _ ) 2 2
I / l
2
N
N
2
+ ...] J^
l
1
and the integral is equal to / e x p [ 2 « ( 1 + ^ + f ) ] rf = 2exp [ft* g + § + f ) ] = 2A (2 ) V
2
£
For 7 > 3, |ir| = 2"", 7 € S we have 2
2-
Z l + 3
7
( l + E I 2 + - . - ) [ ( * ! " » ) + (as - » ) 2 + . . .
+ («7-a + !/7-3)2 " + • - • + (Z2-,-4 - y 2 7 - 4 ) 2 " + .. / 7
4
21
5
and the integral is equal to
*T*1"1
r
!.-l- T-l
In the last integral we have {2e(*-y)% = | j ( l + e 2 + c 4 4 " ) [ « T - a - 1/7-2 + 1
2
= £ [('7-3-Vr-a)*+4(«T-i - * r - i ) ' +
- ^-1)2+.. . ] J 2
- * r - i ) ( « r - a - 1*7-2)]
Analyst! on the Field of p-Adic Numbers 63 and the integral is equal to
S ,yi — c,,...
/
JTI, e x p
- irT-3) +
y
E
y
P< j[(z7-2-!/7-2) + («7-l-S/7-l!
- s f y _ i ) ( % _ - y - 2 ) + (2ci + e ) ( : r , _ - y - 2 ) ] !
3
e
!
= 1 + e" + 2 e x
7
A
k
i +^
2
+
e
7
2
2
*)* ]} '
[ — ( 1 + 2ci + e ) = 2 ^ < - l ) ' ' i ' > = 2A (2£)
P
a
a
E x a m p l e 5. p # 2, | o | > p
2 - 2 7
p
, 7 € Z
l/
I
4
a
E E *p { T l * + ^ + *=0,1; = 0,1
_ j X (a)\a\; \ bx)dx = p
+
y-i-i)
3
e X
E
+ 4(%_i
Xp(ax
2
-
,y _)=T -
y
=
3
\
(-£),
p
(1.5)
0,
s. 2N
• Let a = o~p~ where either a = e or a = ep, |e| = 1 (see Sec. 1.4). Under the condition |aj > p ' ' , either N > 1—7 (for c — t) or N > 2—j (for IT = ep). Performing in the integral change of variables of integration p~ x = y, dx = p~ dy we get p
2 2 1
p
N
N
I
2
Xp
(p- "cTX
7
+ bx)dx
S, J
N
= p~
2
b p aw 2
=
N
P~ Xp
N
Xp(vy -rl>P y)dy
4(T
)/
6p
64
p-Adic Analysis
and Mathematical
Physics
Using the formulas (1.1) for a = e, N + 7 > 1 and (1.2) for IT = ep, N + 7 > we get for the desired integral the expressions 2
P
~
N x
> ("in")
I
=
M*P)VP
I 0
^
I S
= P
W
+
7
. « =
*
i f otherwise.
By combining these cases and taking into account that 1 = A (0,
P
P
= K\
N
A (n) = A M , p
N
P \^\;
!
p
p
N
- ^ = \a\ >\ p
,
= \a\ H
N
p f~
= |a| i f
we get the formulas (1-5).
I
E x a m p l e 6. p = 2, | « | , > 2 " ' ', 7 e 2. 4
2
1
• Similar to Example 5 by using the formulas (1.3) and (1.4). The formulas (1.5) and (1.6) admit unification. E x a m p l e 7. |4a| > p " ' , 7 € Z. 2
2 7
p
/ v
p
a i '
T
t e ^ = <
7
.. .
E x a m p l e 8. p / 2, \a\ = p p
1 - 2 7
(I- )
, 7 S S. H P < P -
7
+
1
,
H
7
+
2
.
P
> P "
(1.8)
Analysis on ike Field 0/ p-Adic Numbers 2y
1
• As a = p takes the form
e, |e| = 1 then after substitution p
7
l
p
7
65
x = y the integral
2
6p- +i\ l 7
*p
2ep
5, fc
7 + 1
If l l < p "
7 + 2
\p 2
(
I f |6L > p "
< p, and we use the formula (1.2) for 7 = 1,
then ^ ~ I r
P
b \ \
3
{
I
6 p~ '
, + s
Y
> p , and we use the formula (1.2) for 7 = 1, 2
then
2<
P
2
kip 5 P -
s 0
t
n
e
integral is equal to 0.
2. The Gaussian
•
integrals on the discs
B
y
E x a m p l e 9. p ft 2, 7 € 2 ,
f P ^ J W I P ) ,
y * < «
= j
X
p
i
m
;
v
K P
(_£)
%
Q
( p
-.
,
K
p
2
*
t
< i,
>
L
(2.1)
27
2
2
• For |a|pP < 1, y € S we have | a x | = | a | k | < 1 hence X ( a x ) = 1, and the formula (2.1) follows from the formula (3.1) of Sec. 4.3 7
P
P
p
2
P
j x (bx)dx By
7
2 7
m
7
= p fi(p |6|p) .
P
2
2 7
2
Let now H p ? > 1, \a\ = p " \ JV = 1, 2 , . . . , a = s p " " , |c|p = 1, Performing the change of variable of integration x = p ~ y, dx = p->~ dy, \y\ = p ~ \x\ < p we get p
N
y
N
N
p
j
N
P<-
=
P
7
""
2
y
N
p
N
Xp{zy +p -*by)dy
N
fxp(p -''by)dy+
£
[
2
N y
Xp(ey +p - by)dy (2.2)
66
p-Adic
Analysis and Mathematical
Physics N
For | | | p > pT, i.e. \h\p > \a\ pi = p™~\ we have \p - p. Taking in account that in (2.2) \e\ p ' ' > p , i = 1,2,... ,N we conclude owing to the formulas (3.1) of Sec. 4.3 and (1.7) that all integrals i n (2.2) are equal to 0, and the formula (2.1) is proved in this case. p
p
y
N
N
2 1
p
2
P
For | £ I P < P , i-e- I P * (2.2) takes the form (2.1): 7
7
"
^
<
N
P • the integral in the left-hand side of T6
y
dy
N
P~ X
P
1/2
=
K Xr
because the integrals under sign of sum are equal to 0. The case | | p > 1, \a\ = p ~ ~ \ N = 1,2,..., a = e p ^ " ^ , = 1 is considered analogously, owing to the formulas (3.1) of Sec. 4.3 and (1.8). • 2 7
a
2N
p
l
2
2
1
p
E x a m p l e 10. p = 2, y £ %, JX2(ax
2
+ bx)dx
7
7
2 fl(2 |6| ), 2
A (a)|2«j 2
A ( )| 2
a
2 a
|-
2
1
2
X 2
1 / 2
7
( - £ ) 6(\b\ - 2 " ) , 2
7
X 2
1 / 2 X 2
(-£)fl(2 |6| ), 2
< 1,
|al 2
27
= 2,
( - £ ) 0 (2-7 T»
/
l
i
f
2
|a| 2
27
|«| 2
2 7
2
,
where tfU
27
2
1 / 2
A («)|2a|-
|n| 2
T
H>=P >
3
= 4, > 8,
( 2
3
)
Analysis on the Field oj p-Adic
Numbers
67
• Similar to the Example 9. The cases \a\ 2 "> = 2 and |n| 2 "' = 4 are considered specifically. • 2
2
2
2
The formulas (2.1) and (2.3) admit the inification. Example 11. 7 € 2 2
/
Xp
{ax +bx)dx
f P'toWK), -
\a\ p
{ \ W)l2aS*'% p
3. The Gaussian
2y
p
integrals
m *>
( - £ ) n ( -<\l\ ), P
on Q
< I,
pP
p
> L
{
2
A
)
p
E x a m p l e 12. a / 0
j
XP(™
U«)Mp X* ( m
+ bx)dx =
2
~ S •
(3-1)
I t follows from the formula (2.3) by 7 —* 00. Note that a formula similar to (3.1) is valid in the real case Q
J X o(ax
2
0
+ bx)dx =
l/2
X ( )\2a\Z Xoo^-^y 00 a
m
= IE : (3.2)
:
trial™ = M , X » ( * ) = e x p ( - 2 i r i x ) and Aoo(a) = exp I - ' 7 signal = < V 4 / { Ifi , f
a<
0.
(3.3)
• The formula (3,2) follows from the classical formula
y e x p ( - t a a - ibx)dx = A ( a ) | 2 a | " ' e x p J^i—J , 2
l
DO
2
a/ 0.
—oo • (3.4)
68
p-Adic Analysis
4. Further
and Mathematical
Properties
Physio
of the Function
X (a) p
We prove the equality X (a)X (b) p
+ ^j ,
= X (a + b)X (^
p
p
p
a,b,a + b€%.
(4.1)
It follows from the formula (3.1)
,-1/2
J Xp{h )dy
2
2
/x (°-x )dx P
= J x (*(x
2
= JX {*z )dx
j
2
P
2
a
= X( p
a
p
=J j
[(a + b)y
2
x
( - ~ )
Xp
2
h ]dxdy
dx
j (^^) ^ Xp
1/2
fc)|2 -r2i.| - A a
M * - y ? +
-2axy]dydx
+ b)\2a + 2b\;^
p
= X„(a + b)\2a + Zbf
p
P
2
Xp
= j X (ax )X (a
= A ( +
X (a)X (b)\2a\-^\2b\-
- y) )xp(h )d*dy
P
P
=
p
+ b)X ( ^ ± * ) P
p
ab ( j 1
a+b 1
|2a|- /'|26r /
2
Integrals entering in this chain of equalities extend in fact over discs of finite radii (depending on a and 6) (see Sec. 4.1). Hence use of the Fubini Theorem is justified here. •
The following adelic formula is valid Ma) = l , 2
(4.2)
Analysis on tke Field of p-Adic
Numbers
69
• The product (4.2) converges for all rational numbers a -ft 0, as only a finite number of factors in it differs from the unity. Let a rational number a ft 0 have a form ,
a = ±2°"p? pj' » • « * .
0 = 0 , 1 , . . . , n)
where 2 , p i , p , . . . , p „ are relatively prime numbers. 2
Owing to the relations X (a)X (-a) p
2
= 1,
p
A ( n c ) = \{a),
2 < p < oo
p
i t is sufficient to prove the formula (4.2) for numbers a of the form a = 2
0 , 0
pip ...p , 2
n
0*0 = 0 , 1 ,
so that Aoo(a) = exp(—i^). Let aa = 0. Denote by i , 0 < / < n a number of prime numbers of the form AN -f- 3 in the set (pi,P2, - - • ,Po) so n — / is a number of prime numbers of the form AN + 1 in this set. Note that the product of numbers of the form 4JV + 1 is again a number of the same form, but the product of numbers of the form AN + 3 ic of the form 4JV + 1 i f a number of factors are even and it is of the form AN + 3 i f a number of factors is odd. Then we have X (a) = i [ l
+ (-l)'i],
2
(
X (a) = 1 i f p p
?
j
j = 1,2,...,«
i Pi
-
1
)
i f Pj — 1 (mod 4),
if Pj = 3 (mod 4).
Therefore f
£ M«)=«p(-*f)^ti+*(-i)V 2
(4.3)
n l<j
r i p *
V
70
p-Adic Analysis
and Mathematical
Pkysxcs
Taking into account properties of the Legendre symbol i n particular the reciprocity law (see [204])
where p and q are prime odd numbers we get
• n fj=n = n h>"
(4.4)
= (-1)
l<;
From here and (4.3) the formula (4.2) follows A (a) = e x p ( - ^ ) - ^ [ l p
i(-l)']i'(-l)
+
l i l
^
2
= l
(= 0,1,....
1
Let now a = 1. Denotefeyh ^ Js, 0 < l j +fa+ fa < n a number of prime numbers of the form 8/^+3, 8JV+5, &N + 7 in the set (pi,p2. • • • iPn) respectively. Note the product of numbers of the form + 1 is a number of the same form, but the product of numbers of the form 87V -+ j(j = 3, 5,7) is of the form 8JV + 1 i f a number of factors are even and is of the form SN + j(j ~ 3, 5,7) if a number of factors are odd. Therefore 0
l+i
^ ^
Ha) =
^
if
h,b,h
e
v
e
n
OT
i f i i odd J2J3
h,h,h
odd,
even or /1 even, / , / 2
p
2
if / odd, i i , / even or f even, h,l odd; a
2
3
= if Pj = 3 (mod 4),
A (a)=l, p
odd,
if h odd, (1,(3 even or l even, 11,(3 odd,
if P> = 1 (mod 4), A »
3
i£p?%
PyiPi,
j = l,2,...,«.
2
(4.5)
Analyiii
on ike Field of p-Adic
jVumSerj
71
Thus acting like in (4.4) and using the formula (4.7) of Sec. 1.4 we get
H
n M*)=> '>n +
fejte)
-^fe-u* n f
= t''+^(-i)'-+''(-i)'''" '"
1+fc
) (
n(£] n (
+
a
1
' ''''' = (_i)'i+'=+id+' ) _ 3
From this formula using (4.5) we obtain the formula (4.2) A,(a) = exp ( - 4 )
A (a)(-1)'^H« 2
1 +
M'= 1 .
•
2
R e m a r k . A function similar to A (a) has been considered by A. Weil [225] for locally compact fields. Its particular expressions for the fields Qp are contained in papers [215,239,3,182,151]. The function A (a) is connected with the Hilbert symbol (see [37]}. p
p
By definition the Hilbert symbol (a, b), a,b € Q* is equal to + 1 or - 1 subject to i f the form o r + by — z represents 0 in the field Q or not, The Hilbert symbol obeys the properties (see [37]): 5
2
2
p
2
2
( a « , & / , ) = (a,&), (aja2.ii) = ( a i , 6 ) ( « 2 , t )
and for p ^ 2 (e.d) (a,-S)
(p,e) = f ^ j ,
= 1, = sgn a, f
|e| = [Slip = 1 , p
s
£ £ Q ; see (2.4) of Sec. 3.2).
The following formula is valid: A (a)A„(&) = (a,&)A (a&), p
p
a,6e<£ .
(4.6)
• Let p ft 2. Owing to the properties of the Hilbert symbol and the function A ( a ) , i t is sufficient to prove the equality (4.6) for the following cases (see Sec. 1.2): p
72
p-Adic
Analysis and Mathematical
1) a = b = £, 2) a = 6 = p, Z)a = b = ep,
4) a = e, 5) a = £, 6) a = p,
Physics
b = p, 6 = ep, b = ep,
\e\ = 1, p
that can be done by means of elementary calculations. From the formula (4.6) for b = — ea it follows the formula \{a)\ {-ea)
= sgn a\ (-e),
p
t
(4.7)
p
obtained in paper [182]. 5.
Example
= ( -j) 1
(see [206])
13. p±2
N;
1 / a
1
5 (ifll; ,
(5.h)
,7(«) even, -1/2
P 7(a) odd,
(5-l ) 2
1
3
if | . | < l a j " ' ; p
1
3
if N|p > l < ' .
(5.1 ) 3
Here the function S"(a,g) is defined by the formula
t!fl_o»+i) v
0<*
'
*
k|
(5.2)
oo .
(5.3)
0
From the equality (5.1a) i t follows the asymptotes
j e-K ( ( -y) )dy 2
Xp a x
4
3
P +P p + 1l * l p * > g ' ) + 0(|x|- ), a
3
s
-
Analysit
• Let 7(a) be even, |a| = p , The integral is equal to 2N
N € 2 . Then |a| > p
p
E
/ Js
-oo<7
= X (ax )
73
if 7 > -7V+1.
X (a(x-yf)dy P
y
e - ' " / »<-2a«i,)<.y
1-AT<7<«
1 / 2
2 - 2 7
p
£
P
on the Field of p-Adie Numbers
y
N
2
For | J : | < |a| = p~ we have x ( a * ) = 1, x ( - 2 a z y ) = 1 , » C 7 < -TV, j ^ l p = | z | p , 7 > —JV + 1, and by virtue of the formulas (5.4) and (1.7) the integral is expressed by the formula (5.1i): p
p
P
p
7
p
-co<7<-JV
-H) v
v
/
r
'
e-'V
-eo<7<-«
E -'-V^=(i-i),-" E 7
^
v
N
= ( I - ; ) W P -
1
/
2
E
r
^"V
/
0<7<™ 7
0<7
1 / 2
N
M
N+l
For | x | > \a\ = p~ we have |2ax| = p > jr , and by virtue of the formulas (5.4), (1.7) and (3.2) of Sec. 4.3 the integral is equal to (5.1 ); p
p
p
3
74
p-Adic Analysis and Maihtmaiical
E
" jdy
Physics
J
+ e " ""
M )
|
x f-2Mj*)rf» p
< 7 < - M
. — oo
+2-Af E<7<-W ,-p + E
3i
y X (ay -
,-p
2
2axy)dy
P
l-N<7
= (I-
X p
(az )
£ ^^P -co<-,<-M
2
M
- Xp(^ )p- e
7
^
2
+ |ax| V p f a ^ )
7
5
p- e-IHr-^_ -P l-l,-
:
e
0<7 p " and |a| = p ' ' " if 7 = —A . The integral is equal to 2 N + l
2
p
1-
2 7
p
1
if 7 >
+ 1
r
p
2
X (a* ) P
y
-p
2
Xp{ay -
2axy)dy
(5.5) 1-JV<7
For | i [ < |a(p p
XP(-2aiy)
1 / 2
i.e. | i |
= 1, y e S , T
p
< p"""
5^
1
7<-JV-l,
2
we have ^ ( a x ) = 1, 2ai
By virtue of the formulas ( 1 . 7 ) and ( 1 . 8 ) the integral ( 5 . 5 ) is expressed
Analysis on the Field of p-Adic
jVumterj
75
by the formula (5.1a): 1 a
2
M )x (-o* ) - ^= P
-oo< <-W-l 7
1 M«) w
M
w
—
2
Tor | i | > p - we have |2az| = p > p + . By virtue of the formulas (1.7), (1.8) and (3.2) of Sec. 4.3 the desired integral ( 5 . 5 ) is expressed by the formula (5.1a): p
p
£ e"* L-co<7<-JVf
5
1
fdy ^
+ e-'* -"
^ S!_
x (-2axy)dy P
M
2 - M <7
2
7
*>
L v +
e
M
Mi
1-i) E^""V -p- e-^-
)
M
T=M 3
-K^(^| |-^ 0
2
l
X
(
,(-oa
r
-xp(™ )\™\; e-^™ ''
•
N
N+l
For \x\ = p~ we have | 2 a i | = p , and owing to the formulas (1.7) and (1.8) the integral ( 5 . 5 ) is expressed again by the formula (5.I3): P
p
1
E
2
X (az ) P
eWl-lV+e-'- " /
. -oo<7< -><-"-
1
+
e
"
P
K
J X (iy P
2
V
-
CM
J
2axy)dy
-W+l<7
2
+
e
-Kv ( * )|a|rV3 p
a
M°)Xp{-ax ) 2
-
— VP]
{ay*
Xv
-
2axy)dy
76
v-Adic Analysis and Mathematical
Phyiici
as
E x a m p l e 14. p = 2 (see [2391) 2
/
e-M'> 2(a{z-y) )dy X
to
= l-iri\a\?>\-W
( K
+ i\ \- ^S a
= \a\?'%~W
2
2
if 7(a) is even;
1/2
a
a
l
«
l
= 5 $ f M « W * * ( * F . j ) 7(0)
a
|*U -
2
(
,
1 / a
I) , |x| =
( . b i ^ M i "
a
W - (5.60
, |.|, <
+ N/2A ( )|a| - e- l"l>', !
if
^ )
(|«| -S
J(-l)*|-e«»5
+
1
5
1
-1/2
2|a
(5.6 ) 2
" .
.
(5.6 ) 3
6
)
4
l » b - V W ,
is odd;
1/a
1
1 6
a
= v^A (a)|a|2 « - > - ' 3 + M ^ l t " ' ) [ s f ^ M j , ± ) - 2 e - l « l > ] , 2
l*b > 2 | a | 3
1/2
.
(5.6 ) 6
The integrals (5.6) are calculated in a similar way to integrals (5 1) for
Analysis on ike Field oj p-Adic
Numbers
77
From the formula (5.6e) i t follows the asymptotics
K|a-*.oo .
6. Analysis
of the Function
(5.7)
S(a,q)
The function S(a, q) is denned by the formula (5.2). The function $(a,q) is entire on a, real for real a (and positive q > 0) and satisfies the functional equation e-° = S(a,q)-qS(c
.
for
(6.1)
asymptotics S(a,q)-.e-° S(a,q)
o
,
+ 0(e- '> l
~ -^C(ln«
a -
-oo, 5
l 9
) + 0(e-"/' ),
( | j < 1), S
a -
(6.2)
oo(0 < « < 1) , (6.3)
ifAere zAe function C(x,q)
1 2
= e'
l
£
S
e
e- *«",
- o o < z < oo
(6.4)
— 0O<jfc<00
is rea/ analytic positive 2| In g|-penWic. I The listed properties of the function 5 ( a , g ) , except the asymptotics (6.3), follow directly from the representation (5.2). We shall prove the asymptotics (6.3). Let us represent the function 5 in the form S( ,q)= a
£ - DO
**«-
a , W
-
E l
r*«-*"" •
<">
78
p-Adic Analysis and Mathematical
Physio
If we set a = e* and use the function C ( r , g ) (see (6.4)) then we get from (6.5) S{a,q) = -Lc{\na,q)-1>( )
(6.6)
a
where the remainder term
KKoo
satisfies the estimates -e-
a / j 3
< 0(a) < C , e -
o / j 5
,
a >2
and the function C(x, q) is T — 2|ln g|-periodic, C(* + T , ) = e ^ 9
+
£
g
* -*- V* e
— oo
= -/ e
a
£
,'-v«
a
,
"-"
!
S
c(«, ). 5
— oo< t
From the formulas (5.11), ( 5 . 1 ) , as \a\ - * 00 2
(5,6i), (5.63)
i t follows the asymptotics
1
p
1 f ( l - j ) C (in |a|-», i ) + O f e - ^ H r ' ) , \
( l - i ) C (in K
1
7(a) even,
1,2
- Inp, I ) + 0(\a\; e-^-,'),
7
( ) odd. a
(6.7)
V I . Generalized Functions ( D i s t r i b u t i o n s ) The theory of generalized functions on any locally compact group was presented by F. Bruhat, [43] and on a locally compact disconnected field by I.M.Gelfand, M.I.Graev and I.I.Pjatetskii-Shapiro [82]. Here we expose the following [206] bases of this theory adapted to the field Q (and to the p
Anatyiis
on the Field of p-Adic Number)
79
space Qp). This theory in many aspects is similar to the corresponding theory on the space E™ but there are some essential distinctions. 1. locally
Constant
Functions
A complex-valued function f(x) denned on Q is called locally-constant if for any point r 6 Q there exists an integer l(x) £ 2 such that p
p
f(x + z')=f(z),
k'|p.
The set of locally-constant functions on Q we denote by £ = €(Q ). It is clear that any function from £ is continuous on Qp. The set £ is linear over the field C. p
P
L e m m a 1. Let f € £ and K be a compact in Q . I € 2 such that
Then there exists
p
f(z + z') = f(z),
W\ , P
z€K
.
• Let K is contained in a disc Bfj. I t is sufficient to prove the Lemma 1 for the compact B?j. By the Heine-Bore! Lemma (see Corollary 5 from the Lemma 3 of Sec. 1.3) from the covering { S | ( ) ( z ) , x € BN} of the compact BN it is possible to choose a finite disjont subcovering r
{S
1 ( t t )
k
Let us denote I = min l(x ).
( x * ) , fc = l , 2 , . . . , M } . k
Then for any point x € Bn t^(x ) x
and for all
x' € Bi we have f(x + x') = f(x
k
+x-x
k
+ x') = f(x)
as I * - * * + x'\
p
< ffiaxfj* - z%
\x'\ ) p
r
< m a x ^ ' V ) = p'< "> .
•
Examples. 1. | x | € £ . 2. |a:|ptt(p |;r|p) £ £ , 7 ^ 2 , where f!(() = 1, 0 < i < 1, 12(0 = 0 , 0 1. p
7
80
-p-Adic Analytit
and Mathematical
Phyeict
2
3. xM> x (* ) e £ 4. 6(xa - k) £ £, k = 1,2,... ,p - 1, where S(x - k) — I , x = k, 6(x -k) = 0, xayik. 5. Let K be a clopen set in Q (see Sec. 1.3) and 9 j r ( ) be its characteristic function: 0jr(x) = ± 1 , * € iff, 0 (:c) = 0, x £ i f . Then P
0
0
0
z
p
K
1
Denote by At(x) Sl(p- \x\ ),k€Z.
the characteristic function of the disc Bt • A t f ) =
k
p
L e m m a 2. Every function f from £ in every disc fljv is represented in the form f(x)=
v
£ f(«")Mz-* ), i
x€B
(1.2)
N
p
v
where I G Z and a" € 5jv sucA that the discs Bi(a ), form the canonical covering of the disc Bp/-
v = 1,2
N
p ~'
• By the Lemma 1 there exists a number / € Z, such that f(x + x') = f(x),
x&B ,
x'EB,.
N
1
By / = N the statement of the Lemma is obvions (by a = 0). Let / < N. Owing to Sec. 1.3 (the example 2) the disc Bjy can be covered by disjont discs Bi(a"), v = 1,2,... ,p ~' with centers a" of the form N
1
a = 0,
v
r
a = p- (a
0
+a + lP
r
1
.. . + a _,_ p -^ ) r
l
r = N, N - 1 , . . . , / + 1
l
(the canonical covering of the disc Bfj). Therefore l, l
U
^ >
xeB x$
N
B. N
Hence the equality (1.2) is valid: f(x)
=
E /(«)*<« -•**) = E
I
A
/ K ) < ( * - a"), w
i
>e 5
W
.
Analysis
on the Field of p-Adic Numbers
81
Convergence in S we define by the following way: ft —> 0 , k —• oo in S if for any compact K C Q P
f (x)
'Mo,
h
2. Test
k^oo.
Functions
n = 1. We call a (est function every function from £ with compact support. The set of test functions is linear, we denote it by U = X>(Q ). Let ip £ T>. Then by the Lemma 1 of Sec. 6.1 there exists / € 2 , such that P
tp(x + x') =
x' e Bf,
x € Q • P
Such largest number I we call the parameter of constancy of a function lM=i(y>), Let us denote by V' = VjvfQp) the set of test functions with support in the circle Bfi and with parameter of constancy > \. The following imbedding is true: N
l
V CV%„
N < N',l < I' .
N
E x a m p l e s . 1. A (x) € V , 7 € 2 . 2. 6(\x\ - p>) e VI- , 7 e Z , where y
y
1
p
-
M
ail
i
3. i ( i - k)6(\x\ - p ) £ l ? , )t = 1,2,... , p - 1, 7 € 2 , where the function 6(x(, — it) is defined by the formula (1.1). 4. I f i f is a clopen set then fljr 6 V. 5 . £ {x) p{x) G P , / = min( ,0), 7 ^ 2 . By the Lemma 2 of Sec. 6.1 every function ip from V' is represented in the form 1
0
h
7 - 1
p
X
7
7
N
*K*)=
E i
-<•").
*€Q
P
(2.2)
p
for some a" (E SJV, which do not depend on
N
82
p-Adic Analysis
and Mathematical
Physics
From here it follows that the space T>' is finite-dimensional, its dimension is p ; the functions N
N-1
N
A,( -a»),
v=l,2,...,p ->
X
form an orthogonal basis in "Dy. Convegence in V we define by the following way: y>k —* 0, k —> oo in V iff (i) 'Pk £ f V ° e r e N and / do not depend on fc, w
(ii) m — ' 0, k - oo. This convergence assigns the Schwartz topology in D. The space T> is complete i.e. for every convergent in itself sequence {ipk, fc —* oo), ipi, £V, , — f \ —• 0, k,l —* oo in V, there exists a function ip G V such that ipi- —* it —» oo in T>. From the definitions it follows directly V = l i m indU^r,
V
N
=
lim mcYD'
(2.3)
N
Now we shall prove: 2?(Q ) is dense in C(K)* p
• Let / (E C(JC) and e be an arbitrary positive number. There exists a number 7 G 2 such that \f(x)-f(a)\ < e if a: G B ( a ) f l K , a G X . As the compact 7f can be covered by a finite number of disjoint discs B (a ) (see Corollary 3 from the Lemma 3 of Sec. 1.3) the characteristic functions A (x — a ) of these discs obey the property 7
v
y
v
y
£ A , ( i - 0 = l,
(2.4)
zGif,
besides A ( x — a") G V (see Example 1). Therefore 7
A(*) = £ / K ) A ( z 7
l a
')ei'
and owing to (2.4) l l / - A I | c ( K ) < max reft
" The definition of the space C ( / f ) see in Sec. 4.1
< e $ > ( * - a " ) = e. 7
Analysis on the Field of p-Adic Numbers
83
Let O be an open set in Q . The space of test functions V{0) is defined as a set of test functions from £>(Q ) — V which supports are contained in O. The space V(O) is the subspace of the space V\ its properties are similar to the properties of V as in the case of the field H (see [205]). p
P
V(0)
p
is dense in L (0),
1 < p < oo.
• I t follows from the facts that V(0) is dense in C(K) and C(K) is dence in L {0) where K is an arbitrary compact contained in O (see Sec. 4.1). p
•
In the space T>(0) the T h e o r e m o n " d e c o m p o s i t i o n o f u n i t y " is valid: let an open set 0 be a union of no more than a countable set of disjoint circles,
Then their characteristic functions composition of unity in O,
A (x yk
k
— a ),
E A ^ ( z - * ) = l, t a
xeO.
k = 1,2,... form a de-
(2.5)
In conclusion we shall prove the following L e m m a . In order thai (fy —# Q jfe —* 00 in T>, it is necessary and sufficient that the condition (i) and one of the conditions ( h i )
0, * - » oo,
k
2
p
k
are satisfied. • The conditions ( h i ) and ( i i ) are necessary. Let us prove their sufficiency. From the condition (i) and ( i i i ) it follows the condition (ii) by virtue of representation (2.2). I t remains to prove that from the conditions (i) and ( i i ) it follows condition ( i i i ) Let it be not the case. By the condition (i)
2
84
p-Adic Analysis and Mathematical
Physics
infinite). Then for the test function ip = Ai(x — a) we have a contradictory chain of equalities: lim
/ tp (x)&i(x
- a)dx = 0 = l i m / tp (x)Sl(p~'\x
k
ki
*-.oo J
= lim 1-.CO
-
a\ )dx p
'-•aoj
I
/
kt
J Btti)
dx = Cp' / 0 .
•
J B|(«)
woo
3, Genera/iied f u n c t i o n s ("Distributions^, n = 1 v4 generalized function (distribution) f : we call every linear functional f onV, f :), p € 2 > . TAis set me denote by P ' = P ' ( Q ) . 2>' is a Jinear set: linear combination A / + / 1 J of generalized functions / and j from I>' (A and u are any complex numbers) is defined by the equality P
(\f
+ ttg,
Convergence in V : we define as the weak convergence of functional: f —* k
0, * — oo in V iff (f ,
- O . i - o o ^ e H .
k
L e m m a . If A is a linear operator from V into a linear topological space M then operator A is continuous from D into M. • Let ip —* 0, fc —* oo in V. Then -p € V to the representation ( 2 . 2 ) of Sec. 6,2 k
k
N
for some JV and / and owing
M * ) = £ M«")M* - O .
-* M
.
B
l
From here by virtue of linearity of the operator A it follows that A is continuous A
£
v
? t ( a ) i 4 A 0 - a") — 0, I
it -» oo in M .
•
From the Lemma it follows that 7?' is the set of linear continuous funct i o n a l on V, i.e. the space V is a strongly conjugate space to the space Z>.
Anatyiit
on the Field of p-Adic
Numbcri
85
Therefore by the study of the space 2?' it is possible to use general theorems of the functional analysis. I n addition the theory is essentially simplified in comparison with the corresponding theory over the field K {cf. [205]): i t is sufficient to verify the linearity of functionals, their continuity follows automatically. The space V
is complete.
• Let a sequence {ft,k —* 00} of functional ft G V converge in itself, / k - / j — 0, t , / — c o i n D ' , i.e.
Hence there exists a number C( ) • k — co
(3.1)
It is clear that the functional C(p) = (f,p), f £ V, tp € T>. The equality (3.1) shows that f -> f, fc 00 in Z>'. • k
Every function / G formula (/,?)
defines a generalized function / G V = j f{x)
tpeV.
The correspondence (3.2) between functions f £ tions f G V is one-to-one.
by the (3.2)
and generalized func-
A degeneralized function / vanishes on an open set O, f(x) = 0, x G O, if ( / , v ) = 0 f ° V £ P ( O ) . Analogously the equality of generalized functions / and g i n 0, f(x) — g{x) = 0, x G O, is defined. Since in 27 a "decomposition of unity" is valid (see Sec. 6.2) then the notion of support of a generalized function / is introduced by the standard way, like to the case of the field ffi. We denote support / by supp / ; x € supp / means that / does not vanish in any neighborhood of the point x. r
a
u
E x a m p l e 1 . The Dirac o-function {6,
1
* The definition of Z,.
see in Sec. 4.1.
>fi<=V .
(3.3)
86
p-Adic Analysis and Mathematical
Physics
It is clear that supp 6 = { 0 } . Conversely, every f £ V for which supp / = {0} is represented in the form f = C6 where C is a constant. • Let tp € V and I be its parameter of constancy (see Sec. 6.2), so that
p
P
(/, v) = if, w) = <M(f, i ) = c(6, v), a=(/,»>).
•
E x a m p l e 2. k
k
M * ) = P n(p \x\ )
-
p
S(x),
k —* co in D'
(3.4)
• Let tp £ V and ( be its parameter of constancy. Then for all k > —I we shall have
(«*,*>) = JP n( \x\ )
k
P
= V(0) J P
p
k
=P
J
dx = (0) . - (6,
R e m a r k . The limiting relation (3.4) is equivalent to the following one j b {x)-p{x)dx l
—
i-^co^eP.
(3.4')
Q, The limiting relation (3.4') is valid also for all functions
Operators
in Z>'
Linear operators in V are defined as conjugate ones to corresponding linear operators in V: Let A be a linear operator from V into V then its conjugate operator A' : V —• V is defined by the formula [A'f,
peP,/eD',
(4.1)
Analysis
It is clear that A'f £ P ' and ,-1* i t is a continuous operator from A specific expression for the (4.1) to functions / £ L]^ after if,A<$
on the Field of p-Adic Numbers
87
is linear and thus by the Lemma of Sec. 6.3 V into V. operator A* we get by using the formula performing the integral (see (3.2))
= j
f(x){Ay>)(x)dx
to the form (A' f,
E x a m p l e 1. a(x)6(x)
=
= (/,«¥>),
V€V.
(4.2)
a{0)6(x).
E x a m p l e 2. I f / £ V and supp f C B
N
then
/ - M * ) / .
(4.3)
E x a m p l e 3. I f / £ V and supp / is a clopen set then / = <W(z)/
(4.4)
where f upp/ is the characteristic function of the set supp / (see Example 5 of Sec. 6.1). Linear change of variable y = ax + 6, a ^ 0 in a generalized function f(y) in accordance with the formula (4.1) is defined by the equality a
(/(«+
=(/(»).
V€V.
E x a m p l e 4. (6(x - x°),tp) =
(4.5)
—• f(—x) in
88
p-Adic Analysis
and Mathematical
Physics
E x a m p l e 5. 6(x) = 6(x) is even. Denote by £' = £'(Q ) the strongly conjugate space to the space £, i.e. the space of linear continuous functionals on £. I t clear that £ ' C W, P
T h e o r e m . In order that a generalized function f from V it is necessary and sufficient that supp / is compact.
belongs to £'
• Sufficiency immediatly follows from the formula (4.3). Necessity, Let / G £' and supp / be an unbounded set in Q . Then there exists a sequence of points {x , k = 1 , 2 , . . . } , x £ supp / , such that —» oo, k —*. oo. I t means that there exist neighborhoods By(x ), fk < 7 and functions {B (x )) such that {f, oo. The contradiction shows that the supp / is bounded. • p
k
k
>l
k
y
Let O be an open set in Q . The space T>'(0) is the set of linear (and hence continuous) functionals on V{0). If 0 is a clopen set then every f G D'(0) admits an extention F G p
P'(QP).
• I n this case 8o(x) G £ (see Example 5 of Sec. 6.1), and a required extention F is given by the formula (F,
m
E x a m p l e 6. The function H
e- ''eiUQp\{0}}, (
,
and therefore it belongs to f ' ( Q p \ { 0 } ) . I t admits an extention (regularisation) from V by the formula
(/o,V>)=
f
e-^[p(x)->p(Q))dx-r
j
l
e
-W
This fact does not occur in the case of the field ffi (cf. [205])!
Analyait on the Field of p-Adic
Numbtrt
89
Every generalized function / £ T>'{Q) admits the restriction fe> £ V'(0') on any open set O' C G by the rule
(fo',v) = (M,
Conversely, in the space V'(0) t h e T h e o r e m o n "picewise s e w i n g " is valid: let an open set O he a union of no more than a contable set of disjoint discs B k(a ), k = 1,2,..., and let f £ ^ ( ^ ( a * ) ) . Then there exists a unique f 6 V'{0) such that / B ( O * ) = f , k = 1,2, k
y
k
t
T J J
• I t follows from the Theorem on "decomposition of unity" of Sec. 6.2. The desired function / is denned by the formula f(x)
= ^A *(z-a*)M4, k
x £ O .
y
5. Test and Generalized
Functions
M
(Distributions),
n > 1.
The theory of test and generalized functions in Qjj, n > 1 (see Sec. 1.7) is constructed similar to one in QP, all results in a suitable reformulation remain valid. The ball (see Sec. 1.7) B (a) y
= [x £ Q ; : \x - a\ < p*],
a = (a a ...
p
is the product of discs
u
,a )
7l
n
B-,(a{),
B (a) = B( ) 7
x B , ( a ) x . . . x B (a )
y ai
2
y
n
.
(5.1)
The parameter of constancy I (see Sec. 6.2) is a vector, I = ( l i . / j , . . . ,l ), where fj is a parameter of constancy with respect to the variable Xj, j = 1,2,... , n; / = min lj . The formula (2.2) of Sec. 6.2 takes the form: every / £ T>' , I = (li,h, - •• , I ) is represented in the form n
N
n
v
E
^ )A {xi-a\)..-A (x -a" ), h
K
n
n
x£®"
p
£ '* (5.2) for some a" = (a\ , a , . . . oJJ) € Bjv which does not depend on ip. 2
90
p-Adic
Analyeii
and Mathematical
Phyaici
For l i = / = . . . = /„ = / the formula (5.2) owing to (5.1) takes the form 2
V(*) =
£
^"JAit*-"")
(5.2')
N
i
u
where A | ( z — o ) is the characteristic function of the ball fl|(a"). In the Fourier-transform theory the form x£ must be replaced on the scalar product (x,£) = xrfi + .x„t;„. 6. The Direct
Product
of Generalized
Functions
Let generalized functions / g and g £ 2>'(Qp") be given. Their direct product is defined by the formula < / 0 ) x ?(!,), v») = ( / f r ) , 0 r M . ^ . a J ) ) >
v e ftO"*" ). 1
(6.1)
As by virtue of the representation (5.2) of Sec. 6.5 a test function ip from 2 ( Q p " ' ) is represented in a finite sum of the form 1
+
rt«.¥)
= X>*(*)ifc(v).
P*€D(Q;),
(6.2)
v*€P(Qp"), l+m
then the operator p — (a(y), '(Qp " )- Further owing to (6.1) and (6.2) we have p
1+m
p
l+
1
s
( / ( * ) x g(y% ) 9
= £ ( / , p ) ( , ^ ) = (9(y) x /(»),*»), t
S
p £ P(Q;
+ m
)
(6.3) so the direct product is commutative: f(x)x (y)
(6.4)
= g{y)xf(x)
3
Firom (6.3) i t follows also that the direct product f(x)xg(y) is continuous with respect to the joint factors / and g; i f fk —• 0, k —• oo in Z>'(QJJ) and — 0, k -» oo in f ' ( Q p " ) then f x g -* 0, k — oo in T > ' ( Q £ ) . Note that for g = 1 the equality (6.4) is equivalent to the equality + M
3fc
k
f(x),
J p(x,y)dy l
k
= j(f(x) p(x,y))dy, }t
.
(6.5)
Analysis on the Field of p-Adic Numbers 91 E x a m p l e . 6(x) = 6(xi) x ^ ( x i ) x . . . x S(a;„). The direct product f(x) x g(y) of generalized functions f(x) from V'(0), O C Qp and g(y) from T>'(0'), 0 ' C Q " is denned by the similar way. 7. The "Kernel"
Theorem
Remind that every linear operators and functionals defined on V is continuous {see Sec. 6.3). Every generalized function F from P ' ( Q p ) defines the linear operator : £ D ( Q ^ ) D'fQp") by the formula + m
(A
^€P(Q "), p
VGD(Q
m F
).
(7.1)
The inverse statement is also true. T h e o r e m . Let B((%), V> e P(Qp") be a bilinear functional. Then there exists an unique generalized function F G P ' ( Q p " ) such that +
(F,
= BM),
p € P(Q "), p
^ g 7J(Qp") .
1
(7.2)
+m
• As every * ( x , y) from Z > ( Q p ) is represented in a finite sum of the form (6.2) then the bilineaT functional B(*) k on T > ( Q
1+m p
+m
) , and thus F G V'(Q^ ).
The generalized function F satisfies
the equality (7.2). I t is clear that F is unique.
•
C o r o l l a r y 1. ( S c h w a r t z ' T h e o r e m o n " K e r n e l " ) Let f —- A((Qp ) into T>'(Qp"). Then there exists a unique generalized function F £ 7J>'(Qj| ) such that the equality (7.1) is valid. 1
+m
The generalized function F(_x, y) is called the kernel of the operator A. 1
C o r o l l a r y 2. Every bilinear form B{), >p G PfQp ), i> G PfQp") is continuous.
92
p-Adic Analysis
8.
Adeles
and Mathematical
Physics
As i t is known, the Riemann zeta-function admits the representation in the form of the Euler product over all prime numbers:
C(*)= E V = n - ^ ' 1
N
*.>1.
7
l
H|a|p = l ,
a€®.
p In this section certain facts from analysis on adele group are given. Adeles group. A n adele is a sequence of the form x — (z X2,. • •, x ,...), where is a real number, x is a p-adic number, and also beginning from some p all x satisfy the inequality \x\ < I , The set of all adeles forms the adeles ring A, if addition and multiplication are defined component wisely. The additive group of this ring is called the adeles group. Elements of the adeles ring A which have an inverse element are called ideles, that is a sequence A = (A«i , A j , . . . , X ,...) is an idele i f A / 0 and \X \ = 1 for all p with the exception of a finite quantity. The set of all ideles forms a group with respect to multiplication. On the adeles group A one introduces a natural topology, which respect to which A becomes a locally-compact group. The Haar measure on A is denoted by da. I t can be expressed in terms of measures da on lQ> by the following way: BOI
p
p
p
p
p
p
P P
p
dx = dx dx ca
p
• • • dx
2
....
p
The Haar measure on the ideles group A ' we denote by d*A, it can be expressed in terms of measures d*X on multiplicative group Q* by the way: d*A = d*Aood*A2... d'X . . . . p
p
There exists a parameterization of characters of the additive group of rational numbers Q by means of adeles. For an adele a = (000,02, • • •) we put X (x) a
= exp2fl-i(-aooi: + a x + . . . a x + . . . ) . 2
p
(8.1)
Analysis on the Field of p-Adic Numbers
93
Here the sum is considered modulo integer numbers, in this case only a finite number of summands nonvanishes. The map a —* a(x) is a homomorphism of the adeles group on the group of characters of Q. The kernel of this homomorphism consists of adeles of the form a = (a,a,... , a,...), where a is a rational number, such adeles are called principal adeles. X
Let us consider the function on A: X ) ( a ) = exp 2 ) r i ( - a (0
0O
+ a + ...) 2
.
(8.2)
I t has the properties: 1) X(o)(o) is a character on A; 2) X(o)(°) = 1 i f a is a principle adele, that is
[]
xM=l,
aeQ,
(8.3)
2
where \p is character on Q , see Sec. 3 (Xt»(<*) = exp2iri( —a)). The property 1) follows directly from the definition (8.2). The property 2) is proved in Sec. 3. p
ax
Any additive character on the adeles group A has the form X{o)( )> where a £ A. The Tate formula. The Bruhat-Schwartz space T ( A ) consists of functions
P
where ipp{xp) satisfy the following conditions: 1) poo( oo) is a infinitely differentiable function on Bt, decreasing i f l^oo I ""* oo more rapidly than any inverse degree of |zoo|2) f (x ) £ V(Q ) where the space V(Q ) is defined in Sec. 6.1. z
p
p
P
P
x
3) For almost all p one has V^t^p) = 1 i f \ p\p < 1 and v>p(x ) = 0 i f \x \„ > I. The Fourier transform maps the space T ( A ) into itself. Let JT(A) be a character on the ideles group A * of the form TT(A) = | A | 0 ( A ) , where s is an arbitrary complex number, | A | = I A ^ k , | A | - . . | A | , and 0 ( A ) = ^ 2 ( A ) • - - ^p(-^p) • • • is an arbitrary character on the subgroup of ideles of the form (1, A . . . ) , where | A | = 1 for all p. Let us consider the Mellin transform of a function
p
a
2
2
P
p
2
2 J
P
P
j
A-
94
p-Adic Analyiii
and Mathematical
Physict
The function (8.4) can be analytically continued to the whole plane of the complex variables except the points s = 0 and s = 1. The following relation (Tate formula) is valid * ( M =
(8-5)
where i(0, ) = J ^(A)|A|'f>(A)d'A .
(8.6)
S
4-
Let us show that from (8.5) i t follows the functional relation for zetafunction. We shall choose in (8,4) a function y>(A) of the following form; V(A) = P „ ( A « ) f i ( | A | ) S 2 ( | A | ) . . . , 2
2
3
3
(8.7)
where ^
(
*
)
=
e
7 b
x
p
( - y ) •
and the function f i ( | r | ) is defined in Sec. 6.1. Note, that (8.7) is the product of vacuum vectors in real and p-adic quantum mechanics (see Sec. 11). p
V I I . C o n v o l u t i o n a n d the Fourier T r a n s f o r m a t i o n In this section we study most important linear operations over generalized functions, namely the convolution and the Fourier transform operations and connected with them the multiplication operation. 1. Convolution
of Generalized
Functions
A sequence {tjt, k —> oo} of functions n £ V we call 1-sequence, i f there exists N £ Z such that k
fj ( ) t X
t
= Afcfz) = f 2 ( p - | x | ) ,
k >N .
p
It is clear that "t —* l,fc —' oo in £ . The sequence { A i , f c —» Co) we call the canonical 1-sequence. Let / and g be generalized functions from V. Their convolution we call the functional defined by the equality (f*g, v) = £ t t £ / C * 0
x
s(y), A {x)ip{x k
+ )) y
f*g
(i.i)
Analysis on the Field of p-Adie
Numbers
9S
if the limit exists for all ip G V. The right-hand side of the equality (1.1) defines an linear functional on T>, and thus f+g G V (see Sec. 6.3). Note that the equality (1.1) is equivalent to the following one (/*?,¥>) = J i m (f(x) X g(y),r, (x)
+ y)),
for any 1-sequence {7j*,ifc —» oo}. The convolution g*f is defined by the similar way: (S*/,9) = J i m (g{y) x f(x),A (y)
+ y))
}
(1.1')
T h e o r e m . If the convolution f*g, f,g G V exists then there exists the convolution g*f, and they are equal f*g = g*f .
• Let G V so that
*>(*)= £
v
c„Mx-« ) w
i
for some a" G Bjv and C G C. therefore it is sufficient to prove the theorem for the test functions of the form v
v
.
By the condition of the theorem the convolution f*g exists. I f we apply the formula (1.2) to the 1-sequence
we get (f*9<¥>v) = J i m (f(x) x g(y) A (~x i
k
= l i m ( / ( * ) x g(y),A {-x k
v
+ a )
+ a?)A,(x + y - a")) .
96
p-Adic
Analysts and Mathematical
Physics
Taking into account the easily verifiable identity n ( p - * | - z + a"\ )Q(p-'\z p
+ 1 , - 0 % ) = U(p- \y\ Mp-'\x k
+y -
P
u
a | ) P
for it > I, we proceed with our equalities (/*?,
+
k
=
v
y-a ))
l i m ( / ( z ) x g ( y ) , A ( j / ) ^ ( i : + j/)) = ( s * / . ^ ) • t
Here we have used commutativity of the direct product (see Sec. 6.6) and the formula (1.2). • The following formula is valid: i f f*g exists then f*g = f*g where f(x) = /(—x) is the reflection operator (see Sec. 6.4). If / , g € V and supp g C Sjv then the convolution f*g exists and the representation (f*9,
* g(y), A (y)
+ y)),
-p 6 V
(1.4)
is valid. • We use the definition (1.1) and the formula (4.3) of Sec. 6.4. Then for all ) = Jim ( / ( x ) x g(y), A (y)p(x k
+ y))
= l i m ( / ( x ) x A (y)g{y),A (y)
+ y))
= Jim ( / ( x ) x g{y),A (y)A (y)
+ y))
N
k
N
= ( / ( * ) x g(y), A (y)y>(x N
k
+ y)) ,
as A (y)A (y)
N
+ y) -> A ( j / ) p ( x + y), N
* — oo in X»(Q^) .
•
Analysis on the Field of p-Adic
Numbers
97
The convolution f*g is continuous with respect to the joint factors f and g: if ft —* / , it —• oo in 2?', g —* g, k —• oo in T>', supp g C B J V then ft*9t — /*'. k
1
• I t follows from the representation (1.4) and from the continuity of the direct product, see Sec. 6.6. •
E x a m p l e 1. f*6 = f = 6*f,
f<=V.
(1.5)
• I t follows from the representation (1.4) and (6.1) of Sec. 6.6
(/*«, v) = (fix) x 6(y), &H(yMx + y)) = ( / ( i ) , ( i ( j ) , D ( » M « + »))) = ( / » . f f
•
E x a m p l e 2. I f / € I>' then f*h where the sequence
-*/,fc^ooin25'
(1.6)
—> oo} is defined in (3.4) of Sec. 6.3.
• I t follows from (1.5) and (3.4) of Sec. 6.3: f*6k -* f*S = /,
k — oo in V
owing to the continuity of convolution, see Sec. 6.6.
•
1
If 9 = V € T> then f * yj £ S and the formula (1.4) fates the form -Hi),
ieQp ;
(1.7)
in addition the parameter of constancy of the function f*g does not exceed the parameter of constancy of the function ip.
98
p-Adic Analytit
and Mathematical
Phyitci
• Let 0 e V' . Then by using the formulas (6.5) of Sec. 6.6 and (1.4) we obtain the representation (1.7): N
x
( W , v ) M / W . W ! / ) i ^ ( i / M i + y)))=
= [fi'hj
3 3
—
Ht-*MO
x
)f(£) 6 P(Qp)representation (1.7) E x a m p l e 3. I f / € V
+
d
v) v
j{f{z)Mi-z))
The remaining affirmations follow from the • then
A (f*6 )^f, k
j =
*(v)v(
fc^coinD'
k
,
and also A (f*6 ) £ V. The Example 3 shows that every generalized function is a weak limit of test functions i.e. T> is dense in 77'. The formula (1.4) allows us to write the equality k
k
( / ( * ) x g(y% & (x)
k
Therefore the definition (1.1) of the convolution f*g is equivalent to the following one: (A f)*g^f*g, k
k->comV
.
From the definition (1.1) of the convolution f*g supp (f*g) is contained in the closure of the set l£:i€%i
f = x + y,
(1.8) it follows that
i e supp / , y esupp ] . g
In particular, i f supp / C 5JV and supp 3 C fljv then supp (f*g) C Thus, the set of generalized functions with supports in Bjv forms the convolution algebra (commutative and associative) with a unity where the ^-function plays the role of the unit element.
99
Analyst! on Ike Field of p-A die Numbers
In conclusion we shall indicate one more criterion of the extension of the convolution. Let functions f and g belong to Ll and a function q € Z.* exists suck that OQ
j f{x - y)g(y)dy
—- q(x),
oc
k —
ooinV.
Tften the convolution f*g exists and it is equal to q. • I t follows from the definition of convolution (1.1)
U*9,V) = ^lim^
j f(x) J A (y)g(y)ip(x
Qp = Hm j
Q
p
%
2.
+ y)dydx
k
f(t - y)g(y)dydf
= J
Bk
The Fourier-Transform
of Test
Let tp 6 25. Its Fourier-transform
Functions
F[
= J X ttxMx)dx,
HO
rfOvKK,
QP
P
L e m m a . If
in V.
£ GQ
P
.
(2.1)
i.e. the operation
• Let us prove that &(£) = 0, | £ | > p~'. By performing in (2.1) the change of variable of integration x = x' + a, \a\ = p' we get p
p
= jx (ax'
+ a)Mx'
P
+ a)dx'
QP
= X P ( £ « ) / X QP
P
( £ * W ) ^ ' = X tf°)#0 P
(2.2)
100
p-Adic
Analysis and Mathematical
Physio
As |f-«|p = \f\p\a\ > 1 then for any £, \£\ > p~' there exists a G Qp, |a|p = p' such that x ( £ i ) ji I (see Sec. 3.1). Hence from (2.2) i t follows that (5(f) = 0, | { | , > p " ' i.e. 0 G X>_,. Now we prove that p
p
P
=
e'es_7v,
£ G Q
P
,
i.e. the parameter of constancy of —JV. Indeed
#€ + €0 = / x,(tf + 0*M*)
J ^ W P ^ f #
Q, M |f«fc = K'1,1*1, < 1 and * , ( f *) = 1.
•
Theorem. The Fourier-transform
(2.4)
is valid where
J
=
J-piOW)**.
J (x)dz = j <5(f)V(0*.
f>^£V,
M G f .
(2.5)
(2.5')
• By the Lemma the operation in T>. In order to prove that this mapping is onto and one-to-one it is sufficient to prove the inversion formula (2.4). Let ip G X>jV, then by the Lemma tp G T>~^.
Analysis on the Field of p-Adic Numbers 101 Starting from (2.1) we have / xp(-*om<*£.
J
=
xp(-*o
= J B
j
vix^xpiz'twdt:
xA-Oxpix'tWdx'
N
= j ¥>(*') j Q
MtW
-
*))didx'
B-,
R
=
I
J
p(t(*'-*))df.dx'
X
|r'-i|
+
/
f(x')
l*'-*l>p
l+1
j
j
- z))d(dx'
-i
In the first integral from the right in (2.6) f(x') 1 and i t is equal to (see (2.3) of Sec. 6.2) pf*) j
Xp(i(*'
fl
= f(x)
. (2.6)
and Xp{£( ' — x)) = x
d£dx' = = 9(x) •
B,(*)fl_i The second integral from the right in (2.6) is equal to 0 owing to the formula (3.1) of Sec. 4.3. To prove the equality (2.5) we denote V>(£) = n(f,) € "D, and then vb(x) = fl(x), and it takes the form (2.5'):
j
= J p(0"(0#
which is verified immediately: /
Q,
= j QP
P
Xp
QP
1(0 J v(x)x ((;x)dxdi: QP
^) (fx)dCdx
= J 1}(im&% QP
•
102
p-Adic
Analysis
and Mathematical
Physics
E x a m p l e 1. 6 = A, k
where the functions S and A k
k
k€Z,
(2.7)
are defined in Sees. 6.3 and 7.1:
k
Mo=pWia),A*oo=£i(p-*isi ) P
The formula (2.7) is another way of writing of the formula (3.1) of Sec. 4.3. E x a m p l e 2. 7 e Z.
1
where the function 6(|z| — p ') is defined in Sec. 6.2. The formula (2.8) is another way of writing of the formula (3.2) of Sec. 4.3. p
E x a m p l e 3. 7 g Z, k = 1, 2,... ,p - 1 i ^ A ^ W x p t f c p - ^ K O = p^mip
~ P^HZo ~ *)
(2-9)
• The desired quantity owing to (2.7) is equal to 4 - 7 ( * j r * + f ) = 6i- (kp^+0 k
=
p
l-J
1
t 0
E x a m p l e 4. |4aL >
1
1
= p -<Sl(p -<\kp-r
+{]„)
i f rfk = P ,«0 = * 7
othewise
2
p "^
-
P
p (2.10) The formula (2.10) is another way of writing formula (1.7) of Sec. 5.1.
Analysis
on the Field
of p-Adte
Numbers
103
E x a m p l e 5 . p $ 2, | a | = p * * , 7 € 2 1 -
p
F[
2
X p
( x M| a
7
a ;
| - p ) ] ( 0 = |a| p
Al-7(0-
1 / 2
D
(2.11) The formula (2.11) is another form of formula (1.8) of Sec. 5.1. E x a m p l e 6. p / 2, | a | > p
1 - 3 7
p
, 7 € 2
F[v>x )A (*)](0 = ^(a)K 2
7
_ 1 / 2
Xp ( - £ )
A, ( i ) .
(2.12)
The formula (2.12) is another form of the formula (2.1) of Sec. 5.2. E x a m p l e 7. p = 2, 7 e 2 2
F[x («z )A,(x)](?) 2
= v ^ »
(-£)
1
TO-*"?),
H2 = 2 " ^ ,
A_ (£),
|a|2-2 -
2
7
| | > f l
2
2 7
,
23-^.(2-13)
The formula (2.13) is another way of writing formula (2.3) of Sec. 5.2. E x a m p l e 8. p / 2 , 7 € 2 ,
F[ (x )S(\x\ v
D
p
-
p
£ n(k) = 0 i<*
l
m
= p>- n'(to)6(\i\p
1
1
" P " )
(214)
»'(*) = 0.
(2.15)
where l'(M=
£
u(i)exp(2Ti^),
£
The desired Fourier transform is reduced to the integral
104
p-Adic
Analysis
and Mathematical
Physics
y
If \Hp < P~ ^ e n Xp{Z*) = 1 and
1 =
N
Kir = P
a
>P~
U*b
7
7
V
(xo)dx• = P "
1
£ n(fc) = 0 . i
I * W + 7 - 1 > 1 then
=
+tip+•
• • )(*o+*«»+... )}
and hence like in Example 24 of Sec. 5.1 1 = 0. {€*}, =
P
l
y
If |£| = p~ p
then
and owing to (2.15)
1= /,(« )«p(2«^U s, 0
E x a m p l e 9. Re a > 0
-1 fj(i^ip)] (2-16)
• By changing the variable t = xf, dt = \i\ dx, and using the formula (3.2) of Sec. 4.3 we obtain for the desired Fourier transform the equality p
Analysis
*lp<M,
= Kir
on the Field
|t),<|m(|
of p-Adic
Numbers
105
r
P ' " - ^ / MO* 7
£
fl
f l v
r
/
£
P "
7
if
°
<
o
-N
_p«-i+(l-l)
2
P "
7
A
i f ^ > l
0<7
P~
where we have denoted \m£\if = " p^. p
The following formula F[
( ' - £ f ) F M f|Y,
^eV, ^0
(2.17)
a
is valid, in particular
•
(2-18)
• The formula (2.17) follows immediately from ( 2 . 1 ) F[
aX
+ b) p{(x)dx X
= \a\; j
X
QP
- H?*» H )
QP
/ H O * (!»') * Q
-
P
Every function
* (!) •
"
106
p-Adic
Analysis
and Mathematical
Physics
for some a" G B _ i which do not depend on
N
l
for some a G B _ j . By applying to the last equality the inverse Fourier transform and using the formulas (2.18) and (2.7) we get the representation (2.19). • The set of trigonometrical
|
polynomials
E
fofeS*^
r e A/I
\v—finite
) 2
is dense in C{K), and thus in L (K) for any compact set K C Q ,' Aere Af is any countable everywhere dense set in Q-. p
• It follows from the representation (2.19), because V is dense in C(K), see Sec. 6.2. • 3. The Fourier-Transform,
of Generalized
Functions
In accordance with the formula (4.1) of Sec. 6.4 the Fourier transform / of a generalized function / G T>' we define by the formula (/.v) = ( / . p ) ,
*ez>-
(3.1)
As the operator ip ip is linear from 25 in 25 (see Sec, 7.2) then the functional from the right in (3.1) is linear on f , so / G 25', and the operator / —• / is continuous from 75' in 75' (see Sec. 6.4). The inversion formula f =/, -
/ e 75'
is fa/id , so lAe Fourier transform f —* f is the linear isomorphism V 25'.
(3.2) onto
Analysis
an the Field
0} p-Adic
Numbers
107
• The formula (3.2) follows from the formula (2.4) and from the definition (3.1): (/> 9) = (/> P) = (/, #) = (f,h
= ( / , P)>
V€ U .
•
In the case / = vj £ T> the formula (3.1) is reduced to the equality (2.5'), and therefore the Fourier transform introduced by (3.1) is in fact an extention of the classical Fourier transform (2.1). The formulas (2.16) and (2.17) are carried over generalized functions
r\flu
l
+ 6)1(0 = K
?=/,/(*-6) =
x
F[f]
P
fy,
f € V,
a + 0.
X (&0/(0
(3.4)
P
where the generalized function f(ax + b) is defined in Sec. 6.4. I f / £ i then the formula (3.1) is equivalent to the formula (2.1) 1
/(0
If f £ Lf
= j Xp(tx)f(z)d ,
and there exists q
cc
(. € Q
x
then the Fourier-transform
.
(3.5)
£ Lt_ *ueA fnaf
y X (£*)/(x)dx — i P
P
8
(i),
&
1
co in T}
1
f exists and is equal to q.
• I t follows from the continuity of the Fourier transform in Z>' and from the formula (3.5). In fact, ii(p~* !*!,)/(*) — * / ( * ) ,
^
> co in D'
then
k
F[U{p- \x\ )f\ v
= j
(t:x)f(x)dx
Xp
—
f,
and hence / = a owing to uniqueness of the limit.
k-
oo in V
,
•
108
p-Adic
Analysis
and MathcmaLical
Physics
A n analogy o f the Riemann-Lebesque T h e o r e m If f E. L
1
then f is continuous on Q> and f(f) —» 0, \f\ —* oo. p
p
• The continuity of / ( { ) follows from the representation (3.5) according to an analogy of the Lebesque Theorem on limiting passage under the sign of an integral (see Sec. 6.4) owing to majorization \f(x)X (Z*)\ P
= l/f>)|,
*GQP-
Now we prove that / ( £ ) —^ 0, | £ | —* oo. As V is dense in L Sec. 6.2) then for any e > 0 there exists
1
(see
\f{x) - P P
N
= 0, | £ | >
for
some W g l Therefore
J [f(x) - •p(x)]x (t:x)dx
+J
p(i) p«x)^
p
then
1/(01 =
P
X
QP
< £ .
E x a m p l e 9. « = 1,I = «
(cf. (4.4) of Sec. 4.4).
(3.6)
I t follows from (3.1) and (3.2): &
V
= (X, f G
•
Analysis
on ike
Field
oj p-Adic
Numbers
109
E x a m p l e 10. The formula (3.7) of Sec. 4.3 takes the form
(3.7)
E x a m p l e 1 1 . The formula (2.11) of Sec. 5.2 takes the form
( - £ )
i/2
two*)}®
= K(a)\W; X
P
,
a*
0.
(3.8)
K ^ O i n P ' .
(3.9)
E x a m p l e 12.
1
f
X^aWal; ^
(~ £)
^
6(0.
• I t follows from (3.8), owing to (3.6) we have
T h e o r e m . In order f (E V and supp f C By, it is necessary and sufficient thai f £ £ and the parameter of constancy of f is > —N, in addition / ( O = (/(*),
.
(3.10)
• Necessity. By using the formulas (4.3) of Sec. 6.4 and (6.4) of Sec. 6.6 from (3.1) we derive the representation (3.10):
{f,
= \f[x),A (x) N
\ = j (f(x),
^(*)x i§*))v(m, P
J
ft, V*T>.
110
p-Adic
Analysis
and Mathematical
Phytic!
Further i f £' € -B-JV then | £ ' i | < 1 and XpB'x) p
/«
+ f)
= (f{x),A (x) (U N
= 1 for a: e B , N
+ £ » ) = (/(*), A ( s )
Xp
w
X p
and
( £ x ) ) = /(€),
so that the parameter of constancy of / is > —iV.
"
Sufficiency. Let / € £ and its parameter of constancy is > —JV. By applying the inverse Fourier-transform to the equality / ( » + «') = / ( « ) .
x€Q ,|£'l
w
P
and using the formula (3.4) we get the equality X (xx')f(x) P
= f(x),
xe® .
(3.11)
p
1
xx>
But for any \x\ > p* there exists some x' € S_AT such that Xp( ) From (3.11) it follows that f(x), \x\ > p , i.e. supp / C B . p
N
p
4. The Space L
N
/ 1. •
2
2
2
The space L is defined in Sec. 4.1. We introduce in L the scalaT product (/,
f{xW)dx,
f,g£L
2
,
QP
so that
= (/,/)*
The Caushy-Bunjakovsky inequality is valid:
l(/,ff}|
f.geL
2
In terms of scalar product the Parseval-Steklov equality (2 .5) takes the form (p,^) = (p,^),
eV
Theorem. The Fourier-transform mutually continuous. In addition
maps L
/ ( £ ) = ^fim j f(x) (t:x)dx Xp
2
2
onto L
in L
* In the noim || || we shall omit the index 2, ||/||s = ||/||.
(4.1)
2
;
2
one-to-one and
(.) 4 2
Analysis
ike inversion
on the Field
of p-Adic
Numbers
111
formula
f{x)
=
and the Parseval-Steklov (f.9)
l i m f HOXp(-xOdi 7-co B-, J
it I
2
(4.3)
equality = (f,9)A\f\\
f,gev
=
(4.4)
are valid.
• Let / € i 7 ~ > c o in L :
2
- Then f
y
€ L for all 7 € E and f 1
= Sl(j>-i\x\ )f p
— /,
y
2
J\f,{x)\dx
= J\f(x)\dx
2
<
JdxJ\f(x)\ dx
B, I I / -
A l l
=
2
|fl
/
2
\f(*)\ dx
—
T
< P
T
/
!
J
00.
0,7
Therefore | j / j | — * | | / | | , 7 - * oo, T
AOO = /
/ ( * M « * ) < f e g Z. n£ (see (3.5)), 2
2
and the Parseval-Steclov equality (4.1) is valid (as V is dense in L , see Sec. 6.2):
HAN = NAN — \\f\l liA-/rll = llA-A'H—•o.
(4.5)
T - O O ,
T,T'^00.
(4.6)
The limit relation (4.6) shows that the sequence {A>7 —* 00) converges in itself in L . By the Riesz-Fisher Theorem there exists F £ L such that A —* F, 7 —* 00 in i , . On the other hand f-, —> f, 7 —• 00 in V. Hence / = F £ £ , / — » / , 7 —f OO i n L , and the representation (4.2) is proved. From 11/11, 7 - » 00 and from (4.5) the equalities (4.4) follow. The 2
2
2
2
UAH
2
7
112
p-Adic
Analysis
and
Mathematical
Physics
inversion formula f = f owing to (4.2) takes the form (4.3). Thus the operator / —* / maps L onto L one-to-one continuously. • 2
2
2
C o r o l l a r y . The Fourier-transform
f —* / is a unitary operator in L . 2
A special orthonormal basis in L will be constructed in Sec. 9.4. L e m m a . For any f E L
2
the equality is valid:
lim p->*
j\f(x)\dx
2
o
=0 .
(4.7)
2
Let / G L and e > 0. There exists N € 2 such that 2
dx < v . 4
Supposing f > jV and applying the Cauchy-Bunjakovski inequality we get j\f(x)\dx=p-^
j
2
B-,
\f(x)\dx
+ p-T
j\f(x)\di
2
B-J\BN
Br* -,1/2
j
J
dx
li
<\j+p ^\\f\\<e
2
\f(x)\ dx
-I 1/2
+ P
-7/2
/
dx
ifp^||/||<|.
5. Multiplication
of Generalized
A sequence fw*, k -* oo}, w € jV G 2 such that k
7 1
Functions is called the i-seguence, i f there exists
w ( z ) = **(*) = P * S l ( p * H ) , t
k > N.
The sequence {6k,k —* oo} is called i/te canonical 6-sequence.
Analysis on the Field of p-Adic Numbers 113 As i t was established in Sec. 6.3
—» £>, k
ui
k
—> oo in V' .
I f {n*,fc —* co) is a 1-sequence then {ij ,k —* 00} is a ^-sequence, and vice versa (see (2.7) of Sec. 7.2). Let generalized functions / and g from T>' be given. Their product f • g we call the functional defined by the equality k
(f-g,
= l i m (y, (/*«*)*>)
(5.1)
i f the limit exists for any ip £ V. The right-hand side of (5.1) defines a linear functional on V, so that / • g € V. The equality (5.1) is equivalent to the equality (/ •9,f)=
l i m (9, (f*u )
k
for any 6-sequence {w , k —* 00}. The definition (5.1) can be reformulated in the form k
f
g = J i m (f*6 )g h—* 00 k
if the limit exists in V. The product g • f is defined by the similar way: g - / = l i m f(g*6 ) t —• fx;
(5.1')
k
if the limit exists in T>'. E x a m p l e 1 . Let / and g be continuous on Q . Then their product / • g exists and coincides with the pointwise product f(x)g(x). p
• Owing to (5.1), (1.7) and (3.4') of Sec. 6.3 for any
= li™ (ff. (/***)*») = Jim / g{x)
Q
= J i m / g{x)
QP
y)dydx
QP
R
-
= Jim J M O y « ( « M * ) / ( * QP
-
Wtdx
= / Qp
f(x)g(x)Lp( )dx, x
114
p-Adic Analytit and Mathematical Phyiict
from where we derive ( / • g)(x) = f(x)g(x).
•
R e m a r k . The statement in Example 1 remains valid for those / and g from Ll for which the function OQ
/
ff(*M«)/(«-0««.
vei>
is continuous in 0, e.g. if / £ Lf^ and g £ L ^ , ± + | = 1, p > 1. o c
E x a m p l e 2. I f / £ 2?' and a £ £ then the product a • f exists and it coincides with the product af introduced in Sec. 6.4.
• Owing to (5.1), (1.7) and (3.4') of Sec. 6.3 for any
(«•/,*>) = \^(f,(a'6 )
= if* =
Ur^ ^/(x),^)
k
- y)dy
/M0«(*-fR
& / ^ ( 0 ( / ( ^ U ( ^ ( r - 0 W = ( / , H = (a/,p)
from where we derive that a • f = af. Sec. 6.6.
If f,g £ f
J a(y)S (x
Here we used the equality (6.4) of •
and supp g C Bfj then ike following equality is valid
F[f*g]
=f
g -
(5.3)
Analyst) on the Field of p-Adic Numbers 115
• By using the formulas (3.1), (1.4) and (3.10) for all
= ( / ( * ) x m>*it(M*
+ y))
= ^/W. / tam AN(y)xp(€»))v(Oxp(€*)de
= =
/W,/s(OWOt (W) P
(/,/'I5P]) = ( / , S V )
= (ff/,v)
from where we derive F[f*g] = gf = g~-fasg££ we used the equality (6.5) of Sec. 6.6.
(see example 2). Here •
T h e o r e m . Let f,g€ V. In order that the product f • g exists it is necessary and sufficient that the convolution f*g exists. In addition the following equality is valid F[f-9]
= f*9,
F[f*9]
= f-9
(5-4)
• I t follows from the definition of the convolution, the product, the Fourier-transform and from the equalities F[(f*S )-g] u
= (A f~)*g,F[(Akf)*g] t
= (f*S )-g, k
k 6 %.
•
Corollary. If the product f • g exists then the product g • f exists and they are equal: fg = gf. (5.5)
• I t follows from the commutativity of convolution, see Sec. 7.1.
116
p-Adic Analysis and Mathematical Physics
V I I I . Homogeneous Generalized Functions 1. Homogeneous
Generalized
Functions
Let it(x) be a multiplicative character of the field Q , n-(xy) = ir(x)w(y) (see Sec. 3.2). A generalized function / from T>' is called homogeneous of degree ir(x) if for any and r £ Q>* the equality p
(/W.v(f))=T(OI*l (/,W
(ii)
P
is fulfilled, i.e. owing to (4.5) of Sec. 6.4 f(tx)
teQ;.
= *(t)f(x),
(i.i')
R e m a r k . The equality (1.1') allows to speak about a value of a generalized function / at the point x ~ 1:
Now we shall describe all homogeneous generalized functions. According to Sec. 3.2 all multiplicative characters n(x) can be represented in the form ^(x) = ^ (x) = \ \;~^ ( ) (i.2) 0
x
l x
1 - <
where 7r(p) = p * and iri(x) is a normed multiplicative character of the field Q such that p
wM
=
MP) = * i ( l ) = 1,
| * i ( « ) | = 1;
n"i(x) is a character of the group So. 7 / T ( I ) ^ 1 is a multiplicative character of the field Q
then
p
J n(x)dx = 0,
(1.3)
7 £ 2.
(1.4)
• As ir(x) 1 then there exists a £ Q such that |a| = 1, TT(<J) / 1. Changing the variable x = ax of integration in the integral (1.4) we obtain the equality (1.4): p
p
1
j v(z)dx
=j
n(ax')d(ax')
= w(a) J *(x')dx'
.
•
Analysis on t/te Field of p-Adic Numbers 117
For every character ir (x) by the formula a
(*«,*>)=
we define the generalized function iv from V a
l
/ \z\;- n(zM*)dz,
V€V.
(1.5)
Qp
For Reo > 0 the integral (1.5) converges absolutely and defines a holomorphic function; for the others a we define it by means of analytic continuation. Later on we shall use the following definition: a generalized function f 6 W which depends on a complex parameter a is called holomorphic in a domain O c C 1/ for all
the function a —* (f , 'P) is holomorphic in 0 . a
a
The function v is holomorphic in the domain 9ia > 0. Its analytical continuation is given by the formula a
j\x\°- M*)[?{x)-9(W*+
j
l
So
N r ' ^ N * ) ^
Q,\B„ l
+ fWj\x\;- ^i{x)dx,
(1.6)
The last integral to the right in (1.6) for ffi(x) ^ \by virtue of the formulas (1.3) of Sec. 4.1 and (1.4) is equal to 0: I |*£-ViM
£
7
1
P ' " - ' / *i(x)dx
= 0;
/or wi(z) — 1 (Ais integral by virtue of the formula (2.5) of Sec. 4.2 is equal
Bo
Therefore the representation (1.6) can oe rewritten in the form 1
(* , p)=J\x\ - * (x)[
p
B
1
j
It^nMipWdx
Q,\fi„
a
+ { [0
+
?
(1-8) if
118
p-Adie Analysis and Mathematical Physics
The generalized function jr„ is entire I / T I ( I ) ^ L; for holomorphic everywhere except the points « i = ^ - , Inp
TI(I) =
fre*
where it has simple poles with the residue f^pS{x); generalized function of degree Tr (x).
T
a
1 i( is
(1.9) is the homogeneous
a
• I t follows from the representation (1.8). The first integral in (1.8) is an entire function on a as ip(x) — ^(0) is equal to 0 in a vicinity of 0. The second integral in (1.8) is also an entire function on a as
where ip is an entire function. Therefore (it ,
0
r e s ( 7 r , ^ ) = res ^ ^ r r ^ ( O ) = ^ 7 — ( 6 , ¥>)• a=a„ a=a l—p Pinp a
a
k
Homogeneity of the generalised function ir follows from (1.1) and (1.5): for 9ca > 0 Q
(*•<«>.* (?))
=
j ^ r ' M ^ ^ d x
= J
l
\txX- ^(tx')
QP
J \At ^')WW l
= \t\;-'\t\ ^{t) p
Qp
for other a- by the principle of analytical continuation. S-function is homogeneous of degree \x\
p
1
,
•
Analysis on the Field af p-Adic Numbers 119
• I n fact, (6,v(j))
l
= vl
f e v j e q ;
p
.
•
Let us introduce the generalized function ^ r j p by the formula
( " K - W * ^ - / ft* — <"
0 )
1
Tfte generalized function P r 4 - is not Homogeneous o/ degree IzL . P
•(^(?)W^~/f<* B
® \B
0
r
M«swr
0
i»'u>i*i;
p
IP
\
I*)P
i f O < | t | < l , ^GD, v(0)^0.
•
p
L e m m a . / / / ; , i — 1,2,... ,n, are homogeneous g-neralized functions of degree j r . { z ) respectively where all a.; are different, and they sattsfy the equation £ fi = 0 then fi = f = . . . = / „ = 0. Q
2
l
•
For all ip G V owing to (1.1) we have /
=
£ ( ''*'(T)) E ^<(*)!*U/i.p) = o, w
From here putting t = p , JV £ Z and using (1.3) we get S Ki
N
p " ' ( / ^ ) = o, i
icflg.
120
p-Adic Analysis and Mathematical Physics
and hence (fi,) = • - • = (f ,
n
2
= = / « = 0.
The following Theorem gives a description of all homogeneous generalized functions. T h e o r e m . Every homogeneous generalized functions of degree Tr (x) = | z | p 5 r i ( x ) is Cir„ if iri(x) ^ 1 or ifxi{x) = 1 then a ^ 0; ewery homogeneous generalized function of degree \x\~ is C6,' where C is an arbitrary constant. a
-1
1
I Let / ^ 0 be a homogeneous generalized function of degree 7r and also iTi(x) ^ 1 or i f n~i(x) = 1 then a / 0. At first we shall prove: there exists a number C ^ 0 such that for any ip € V,
(f,
(1.11)
P
The support of / contains points different fromO. (Otherwise we would have f = C6 (see Sec. 6.3) and / would be a homogeneous generalized function of degree ja: |— which is excluded.) Therefore there exists a function u £ D , w(0) = 0 such that ( / , w ) = 1 and (x„,w) / 0. By virtue of (1.1) we have 1
(f(x),w(j))
=^(i)i%,
*€«£,
and hence /(/(*)." ( f ) ) ^ f
= K,^)>
^(0)=0.
(1.12)
As u(x) and ^ ( x ) vanish in a vicinity of 0 so « ( f ) m w ;
1
€f(Q ). 2
Then owing to the formula (6.5) of Sec. 6.6 from the equality (1.12) we derive (1.13)
Analysis on the Field of p-Adic Numbers
121
In the inner integral in (1.13) we change the variable of integration (for any fixed x £ 0) 2
t=^j,dt
= \x\ \x'\; dx', p
(see Sec. 4.2).
As a result we obtain
(1.14)
Here we again used the formula (6.5) of Sec. 6.6 as
By using the property (1.1) (x' ^ 0!) on the right side of the equality (1.14) we get the equality (1.11)
in which C = l/(ir ,u). From the equality (1 11) it follows: either / = CTT or supp ( / — Crr ) — { 0 } . But the last is impossible otherwise we would have / — Cir = C\b for some constant Ci which by the lemma is possible only for C\ = 0 and thus / - Cir = 0. Let now / ^ 0 be a homogeneous generalized function of degree | * | , • Let us suppose that supp / contains points different from 0. Then repeating literally the previous arguments we are convinced that the formula (1.11) is true also for ir (x) = | x j r a
a
Q
a
0
a
(1.15)
Using the definition (1.10) of the generalized function "Pj^r- we rewrite the equality (1.15) in the form
122
p-Adic Analytii and Mathematical Pkytics
Therefore f-CP ± 1
= C6 1
r
where C\ is some constant. The generalized function / — CiS is homogeneous of degree \x\~' but P j ^ p is not. This contradiction proves that supp / does not contain points different from 0, and thus / = CS. • Let us calculate the following integral (see Sec, 4);
1<7
if < 0; for 9to > 0, a £ a k £ 2 the integral (1.16) we define by means of analytical continuation. By virtue of the formulas (1.7) and (1,16) the following equality is valid k
J\x\;- dx=
j
1
Q,
j
1
\x\;- dx+
[*|,
1
\ \;- dx x
= o,
|i| >i P
ajfea ,ifc€S.
(1.17)
t
Therefore the formula (1.8) for T I ( T ) = 1 takes the form
reVtd?
a ,k k
<=Z.
(1.18)
l
We shall call the generalized function \x\°~ homogeneous of degree a — 1, \tx\^
L
= \t\;- \x\;-\
te%,
2. The Fourier-Transform of Homogeneous Functions and T-Function
fc s. £
Generalised
By the Theorem of Sec. 8.1 all homogeneous generalized functions of degree Tr (x) = |a:lp* Ti(z) are Cit i f either n\(x) ^ 1 or o / 0; in case TTI(X) = 1 and a = 1 they are Cb. Here C is an arbitrary constant. -1
a
a
Analyst) on the Field oj p-Adic Numbers 123
The Fourier-transform of a homogeneous generalized function JT„ is a homogeneous generalized function i „ of degree it~ '(ajjjljw = klp""^'^). 1
• I t follows from the formulas (3.3) of Sec. 7.3 and (1.1') that
v
M ' O = \t\; F
[». ( f ) ]
= I V - (j)
MO
Therefore T „ ( | ) is proportional to the homogeneous generalized function 1
M O = r (* )icVX (0P
(2.i)
a
A factor of proportionality r ( x ) is called the T-function of the character p
0
JT (X). Q
Putting i n the formula (2.1) £ = 1 and using (1.3) we obtain the equality T (n ) P
l
= n (l)
a
= J \^- Mx)x (x)dx.
a
(2.2)
P
The integral on the right-hand side of the equality (2.2) is understood as the sum of the analytic continuation on the parameter a of the integrals on the right-hand side of the equality (2.3):
= j\x\ - ^(x) (x)d + a
j
1
p
Xp
x
Bo
a
1
\x\ - r (x) (x)dx. p
J
1
(2.3)
Xp
Qv\Bo
The following functional relation for the T-function is valid 1
1
r K)r ( r- kl - ) = n(-i). P
P
J
(2.4)
p
• I f we apply the inverse Fourier-transform to the equality (2.1) then we obtain (see Sec. 7.3) % = * a
0
= r (* )F[\t\;"^(-{;)] p
a
= ^ F ^
1
^ -
1
] .
(2.5)
124
p-Adic Analysis and Mathematical Physics 1
1
As I T " | f j " is a homogeneous generalized function of degree T ^ t f JKIp (see Sec. 8.1) then its Fourier-transform is a homogeneous generalized function of degree - 1
K^kl-M-H ^*)Hence using again the formula (2.1) we have l
1
1
Fte \£\; ]
1
= w \t\ - )**. r
from here and (2.5) the formula (2.1) follows:
^ =^r <^jei -> . P
p
•
0
Note that the formula (2.4) reminds the relation
r ( t ) r ( i - ( ) = -; 7T
sin iti
for the classical T-function. Let the range K of a character ni(x) ^ 1 be positive.
Then
r (jr ) = P°%,t(Ti) p
(2.6)
a
where the numbers a ,k(*i) P
= J TnfOxptP *')^ -
(2.7)
So
satisfy the relations L
a ,fc(Ti)ap, (7rf ) = p - V i C - I ] P
(2.8)
t
Ka(*i)l
/ 2
= p-* .
(2.9)
• Taking into account the equality (1.4) from (2.2) and (2.3) we get
r (*,)= P
£
1
P ^ -
'
£
-oo<-f<0
=
£ KKoo
c 1
P*"- ' / /
K^
*i(x) (x)dx. Xp
f^ ) (x)dx
P^°-V
x Xp
/
Analysis on the Field of p-Adic Numbers 125 y
Performing in the last integrals change of variable of integration x = p~ t, dx = p dt we get y
r,0«) = E
P
y a
[
Mp-^xpip-^dt
= E
p Mp-'') [ M^xpip-^dt
=
P ° I *i(t)Xp(p-'t)dt
ya
7
E
(2.10)
as owing to (1.3) T ! ( p - > ) - [ ^ ( p ) ] " = 1. 7
Now we prove that J*i(t) (p-'t)dt
= 0,
Xp
7
T*fc,
(2.11)
So
Let 7 > 1. We have 7
Xpfp-^t) = exp[2xi(p- to + p
_ 7 + 1
t i + -.. + P " % _ ! ) ] ,
te5 . 0
As the range of the character T T ^ I ) is equal to it > 1 then ir,(t) depend only on r , ( i , . . . , f _ i - Further for k > 2 the equality is valid (see (2.3) of Sec. 3.2) 0
t
1
E *i(*o + hp + • • • + i t - i P * " ) = 0. °<**-i
[*i(t)x (p- t)dt
=c
>
P
/„
E 0<1,
0
... •
E l
E
E °<<*-i
7
exp[2«(p- ( + P" 0
ff
E i('o o
7 + 1
*i + •• • + P " V l | 1
+ d P + - • • + d - i P * " ) = 0;
126
p-Adic Analysis and Mathematical Physics
for 1 < k < 7 (2.11) follows from f
[Mt)x ( - t)dt 5
= c
P P
0
...
E
* ('t> + * i p + 1
°<'i-a
•
£ l
£ 0
- -Mi-iP* )
1
£ exp|2jri(p-^ + p - ^ ! ! + . . . + p - ^ - i ) ] = 0. 0<(,_,
0
Now the equality (2.6) follows from (2.10) and (2.11). To obtain the equalities (2.8) and (2.9) we calculate the Fourier-transform of 6(|x| — p )iti(x): k
F
m M p - p > i ] = J *i(x)x (t:x)dz
= p* j
P
S„
^(p-'Dxt-ip-'Wdt
So k N
= p" j s
^{t) {p- - f.'t)dt
=V
Xp
(i)
l
w
y (p- - i)* p
So
fl
= / * i (^) /
w
»i(i)Xp(p-'- 0^
So k
_ r P ^\n^,k(^).
^ = o,
i.e. F[6(\x\
l
- p*)*,] = p*6(|£| - l ) 7 r r ( O a p , ( f i ) -
p
p
(2.12)
t
N
In the calculation of equality (2.12) we have denoted \f\ = p and { ' = \f\pH and have used the formulas (2.7) and (2.11). From (2.12) by virtue of the Parseval-Steklov equality (see Sec. 7.4) the equality (2.9) follows: p
J \* (x)?d l
=
X
l
1
p (l- -j
5k 2 t
a
1
2
= P K. ( ri)| /K- «')l^' = P * ( l i
1
M'l)?-
Analysis on the Field of p-Adic Numbers 127
To prove the equality (2.8) we apply to the equality (2.12) the inverse Foil rier-transform k
l
l
6(\x\ -p )^(x)=p Fm\ -l)w: (-0]a , (n ) P
p
P t
i
1
1
1
=p *pA^)*: (-mm\p-i)*i- )In order to calculate the Fourier-transform of the function $ ( | £ ] n — 1) (0 d \ \p = P i ' = \x\pX, and from (2.7) and (2.11) as above we get w
e
e n o t e
x
N
x
So
= J Tr (0xp(r'V£K = / 1
So
=
S
N
* t W]
x (p~ t)dt P
0
J ^\t) (p- t)dt N
l
=
Xp
k
^(x)a (^ )6(\i\ -p ) Ptk
p
So 1
because the rank of the character rrj" is equal to it. From here and (2.13) the equality (2.8) follows l
*i(x)
= p a (7ri)3rf ^ - l ) * ! Pit
l
)
•
•
From the formulas (2.6) and (2.8) we derive M ^ j r ^ O ^ M - i ) .
(2.H)
Comparing the formulas (2.14) and (2.4) we get l
T (*; ) 1
or by changing i r " ^
1
k
=
p
P
l
1
r (*; \x\; ) P
by n
a
T (iv \x\ ) p
a
p
t
=p r„(7r ). a
(2.15)
The formula (2.15) can be interpreted as an analogy of equality r ( i - ) - l ) = tT(t) for the classical T-function.
128
p-Adic Analysis and Maihtmatits.1 Physics
Now we consider the special case it\{x) = 1, i.e. the rank of the character T\{x) is equal to 0. In this case we denote
The formula (2.4) takes the form r ( a ) r ( l - a) = 1. p
(2.16)
p
Now we prove the formula
• To this end we use the representation (2.2)-(2.3) and the formulas (2.5) of Sec. 4.2 and (3.2) of Sec. 4.3 r ( o ) = j\x\^ (x)dx
+ J
1
p
Xp
B
|»|,>1
0
= j\x\°-'dx+ B
[x^x^dx
£
J
a l)
P^ -
1
0
1-P"
Q
F
X (x)dx P
ft.
l-p~
a
From (2.17) i t follows: T (a) is holomorphic in the complex plane a except at the points p
t€2
(2.18)
where i t has simple poles with residue j E j j j ^ r ( a ) has simple zeros at the points 1 + art, £ € S. From the formula (2.1) we derive p
1
r
F N T ] = p(«)Kf. 7"ne generalized function
* € Z.
(2.19)
Analysis on the Field o] p-Adic Numbers 129
is holomorphic on a except the points 1 + at, k € 2 tonere if Aas simple poles with residue — f^\x\p*; in addition f
=6,
ak
feeS.
(2.20)
• It follows from the results of Sees. (8.1)-(8.2). We have Q _ 1
|i| res
p
,
U|a-t l i m (a - 1 - Q ) _ . . p
=
t
=
lim
"
ell —
£
t
plnp
i
1
p
-
1-a
» '— '*) ^
^ •
Further from the formula (1.8) of Sec. 8.1 for ifi e V we have
where tp is an entire function. From here it follows the formula (2.20) 0
i.e. f —*6,a—>at
in V.
a
I
E x a m p l e . Let (see Sec. 3.2) ^c(x)
l
= \x\^- sgn x,
p/2
c
3
where e £ Q* . Then ±jsgn {-\)p°-^
i f e = p,pn,
I±E*-»
i f c = n.
c
r (<*) f » =H { P
2
2
2
( -
2
1
)
• For either e = p or E = pn, x ( x ) = 7 r i ( i ) = 1 and the rank of the character x i ( x ) is equal to 1 (As every i € Q of the form x = 1 + pt, P
130
p-Adic 1
' i ' and Mathematical Phytici
\t\ < 1 \B the square of a p-adic number, see Sec. 1.4). Now using the equality (2.8) for k = 1 p
a ,i(*i)a P
1
P|1
(ir,) = p " ^ - ! )
and from (2.6) we get (2.21). For £ = n, sgn„x = 1 if j(x) is even and sgn„r. = —1 if i(x) is odd. See the Lemma below. Thus T I ( Z ) = 1 and * (x)
=
a
a
1+
\x\ - ^ p
and therefore r , K ) = r
p
|
a
-
+
)
= _
^
=
—
1
L e m m a . Letp-£ 2 antic be a unity, E ^ Q* . A number x € Q belongs to Q if and only 1/7(1) is euen. p
p l
• Let i e Q J belongs to < K - i i e . it is represented in the form 2
2
x = a - eb ,
a,b e Q , P
(a,6) # 0.
(2.22)
If 7(1) would be even then the congruence 2
a -£ f> 0
2
= 0
(modp)
(2.23)
would be valid and thus £0 is a quadratic residue modulo p that contradicts to the fact that e g % ' (see Sec. 1.4). Conversely, let t E Q J and 7(1) be even. By the Chevalle Theorem (see [37] — 1) the congruence a
2
~ £0*0
-
XQCI = 0
(mod p)
has nontrivial solution (an,&o.Co)> Co ^ 0 (otherwise i f en = 0 the congruence (2.23) would have a nontrivial solution that is excluded). Hence the equation a - eb - xc = 0 2
2
2
Analysis on the Fieli oj p-Adic Numbers 131
is solvable by c ^ 0. From here it follows the representation (2.22)
3. Convolution Function
of Homogeneous
Generalized
Functions
and
B-
Let » „ ( * ) = l i j j - ^ i t * ) and
= li|i-Vi£l)
be multiplicative character of the field iQ , and JT„ and are the corresponding generalized functions, and a and 0 ^ a (see Sec. 8.1), p
k
The convolution ir *ir'g exists, and it is holomorphic in the tube domain Re a- > 0, R e ^ > 0, Re(o+/3) < 1, and is homogeneous generalized function of degree a
^(s)^*)!*!, and represented by the formula (***jr£J(*3 = J Tr (y)ir' (x a
0
- y)dy
= B {* y )\x\;+^(w[){x) p
a
(3.i)
g
where fJp(w£) = J ,
}r
l ^ - M l - f l ^ M O ^ l ( 3 - 2 )
s
S ( i a t ^ ) ' called the B-function of the character x ( x ) and P
0
JT^(X).
• By virtue of the results of Sec. 7.1 the convolution Tr *ir' of the locallyintegrable functions Jr (z) and x^(x) exists, it is a locally-integrable function, and i t is expressed by the integral (3.1): a
g
a
(TT *^)(X) = Q
j
1
\y\;- ^ (y)\x-yf -W (x-y)dy. 1
p
l
132
-p-Adic Analysis and Mathematical Physics
The last integral is absolutely convergent and defines a holomorphic function of the parameters [ot,0) in the tube domain Re a > 0, Re0 > 0, Re{a + 0) < 1. I n fact,
B
B*
k
= J
\tx\^ (tx)\ -txf -W (x-tx)d(tx) 1
X
p
1
B„
f
l
= }x\;^- n(xW (x) 1
- tg-ViM^Cl -
\t\;-i\i
W
l
\'\
mixiz+p-^x)^)
j
I
^
I
I
-
^
-
^
K
O
-
^
where K is an arbitrary compact of Q . From the representation (3.1) it follows that the convolution ir *ir' homogeneous generalized function of degree k | p * 7 r i ( x ) f l i ( 3 ; ) . p
a
+,3-1
0
is a •
For the parameters (a,0) lying outside the domain Re a > 0, Re 0 > 0, Re(« + 0) < 1 the convolution T * j r ^ is defined by means of analytical continuation of the right-hand side of the equality (3.1) on (a, 0). 0
Our nearest goal is to express the rJ-function by the T-functions and to define a maximal domain of analyticity of the convolution 7r„*7r^. As T * 7 T ^ is a homogeneous function of the degree q
=
kir^ViTUO)
then applying the formula (2.1) to the equality (3.1) we get o
,
1
FK.7r^-r (x ^|x| )S ,(7r ,,^)|f| - -' K<)- (0. p
o
p
(
t
p
(3.3)
On the other hand by applying the Fourier-transform (see Sec. 7.5) to the convolution v fir'^ and by using again the formula (2.1) we get a
,
1
= r (ir )r,(^)|ei7-' (* xi)- (()f
a
1
(3-4)
Analysis on the Field of p-Adic Numbers 133
By comparing the equalities (3.3) and (3.4) we obtain the formula
In addition the equality (3.1) takes the form p
0 *»,)(x) =
Hp--"* ' O i ' i H * ) •
o
(3.6)
+,J-1
For ( X I T £ ) ( Z ) ^ 1 the generalized function |x|p" (ffi7ri)(a;) is entire (see Sec. 8.1) and r ( j r 7 r ^ | x | p ) is entire non-vanishing function (see Sec. 8.2), and therefore the domain of analyticity of the function (3.6) is defined by singular points of the function r ( 7 r ) r p ( ^ ) : i f JTi(x) ^ 1 and 7Tj(x) ^ 1 i t is an entire function; if = 1 and ir\(z) ^ 1 the singup
a
p
0
lar points are the straight lines a = at, k = 0 , ± 1 , . . . ; i f T I ( X ) ^ 1 and f i ( x ) = 1 the singular points are the straight lines 8 = a t , k = 0, ± 1 , . . . . For 7ri(x) = J T I ( X ) = 1 singular points are the straight lines a = a , k = 0,±l,...,3 = pt,k = 0,±l a+8 = l + ak = Q,±l,.... For T T I ( X ) ^ 1 and TV[{X) = it^ (x) singular points are the straight lines a + 8 = 1 + at, k — 0, ± 1 , In this case k
kl
1
1
t
r K)rp(^)=p<^- ) 7r (-l) P
(3.7)
1
where k is the range of the character 7r . a
• The formula (3.7) follows from the formulas (2.6) and (2.8) 1
t
r (Ta)r (^) = p - V f c f n J p ^ O r r ) = p < P
p
t
+
^ S M ) •
•
Let Ui(x) = T { ( X ) = 1. Like in Sec. 8.2 we denote B (n ,^) p
=
a
B^(\^~ M§- ) X
%
=
B {a,B). p
By this the formulas (3.1) and (3.5) take the classical form a
1
\x\ -'*\xf p
p
a
1
= B (a,0)\x\ ^- , p
p
(3.8)
134
p-Adic Analysis and Mathematical Physics
If we take in account the formula (2.16) we get for B (a,0) rical form B (a,0) = T (a)r (0)T ( )
the symmet-
p
p
p
p
p
where a + 0 + 7 = I. Now we shall prove for Re or, Re 0, Re(a + 0) / equalities
F
F \ H •MP =
(3.10)
y
— 1 + ctki It £ Z the
o +0
* H ) = n\z\ i
(3-i2)
P
• For Rea > - 1 , Re 0 > - 1 , Re(o + 0) > - 1 the product \x\« • \x\? exists in V (in the sense of Sec. 7.5 }, and it is equal to | z | ' so the equalities (3.11) and (3.12) are valid. In fact, the functions \x\° and \x\^ belong to L ^ , and the function (see (5.2) of Sec. 7.5) 1+
3
p
/
\ \^{x)\x-t:\idx x
Q,
is continuous at the point £ = 0 for every owing to the majorization Ixl^l^llT-fl^Cmaxdzl-.lx^),
i t
supp p,
\(\ < 1 p
and the Lebesgue Theorem on limiting passage under the sign of an integral (see Sec. 4.4). For the other a and 0 the equalities (3.11) and (3.12) are obtained by analytical continuation of the right-hand sides on a and 0. • Analogously, i f / e £ and Reo, Re/3, Re(a + 0) / - l + u , t e Z then t
' [\*\$f(*)] a
=
1
F[\x\ -(\x\ f)} p
= (F[\x\;] 4.
Homogeneous
=
p
* F[\xf ]) p
Generalized
=
"[ l * f ft*)],
(3.13)
F[\x\ ']*F[\xf }*f p
p
* f = F[\xf
• r>j*j$.
Functions
of Several
p
(3.14) Variables
The results of Sec. 8.1-8.3 without essential changes are carried over the homogeneous function \x\^of degree a — n depending on n variables X = {xi,X ...
,i„),
2L
\x\ = max(|a;i| , \x \ ,... p
p
2 p
,\x \ ) n p
(see Sec. 1.7).
Analysis on the Field of p-Adic Numbers 135 At first we shall calculate the integral (4.1) Be The formula (4.1) for n = 1 coincides with (1.7). • By induction on n. For n = 1 the formula (4.1) is true. By supposing i t true for n we shall prove i t for n + 1. Denote x = (x,x i), £ = ( c i , X 2 , . . . ,x ). By using the Fuhini Theorem (see Sec. 4.4) we are convinced that the formula (4.1) is valid for n + 1: n+
n
= j
/ \x\^-"-Ux So
^ x ^ l ^ - U x d x ^
BQ
= j
j
J
\ \°-»-ld + xn+1
x
Bo pl,
dx
|S|,>|x„+i|
F
- 1
= y"|z i|p (ix„ i n +
+
B*
+ U
Ix^-'didx^
- J
J B \i\ <\x„ \
BB 0
a
0
r
^-"^didXn
+l
Br, 1
1-P" 1-p-*
I"?
+i
BQ
i-p-"
1-P
1 - p-o+l 1 - p-<* 1 - p-a+1
i_ -»-i p
1 - p"
Let us introduce the generalized function \x\° " from P'fQjJ) by the formula (cf. (1.6)) «-".v) = /
+
|x| °-"Mx)-v,(o)]dx p
J
/
|*|*-»rt«)«fa
+
v
( 0 ) i — V 6 T > ( Q P ) . p i(4.2) 1
136
p-Adic Analysis and Mathematical Physio
The generalized function \x\°~
n
is holomorphic everywhere except the
points (1.9) at =
2hrt , Inp
, k e Z, m
where i t has simple poles with the residue
pMnp
v
The Fourier-transform formula is valid: f l W r " ] = rjtoJKI-", where T
* € I
is the n-dimensional T-function ( r
p
r?(«) = /
k i r
f
t
p
(4.3)
— T) p
4
^ f * ^ = T = ^ - -
4
( - >
For n = 1 the formula (4.3) coincides with the formula (2.19). The formula (4.4) follows from the more general formula (m ^ 0) F[\{x,
n
a
m ) | ; - ] = T;(c)Q(\m£\,)[\£\;
- \pm\;\
(4.5)
where x, £ € QJJ, | ( z , m ) | = m a x ( j i | , | m | ) . In fact,for Re a > n the following limit relations p
n
n
\(x,m)\;- ^\x\;- ,
p
p
n ( K W - i .
Manama) - » .
**-*«»
in D'(Qp) are valid, and the equality (4.3) follows from the continuity of the Fourier-transform operation. For other a one uses an analytical continuation on a.
• To prove the formula (4.5) we use the induction on n. A t first we shall prove it for n = 1, j
a
l
ffe m)\ - xp(xO°x p
= W « ( K I
P
H K i r - H 9 -
4
6
( - )
Analysis on the Field 0} p-Aiic Number! 137 It follows from the formulas (3.1) of Sec. 4.3, (2.16) of Sec. 7.2 and (2.19):
j[max(\ \ ,\m\ W- (zc:)d l
X p
P
Xp
X
l*l»<M»
M*>M*
1
rn
J W^i*®**
1
= \m\;- \rn\M\ £\ )
+ FIM?- }-
p
1*1,
< M ,
a
= M ? 8 < K U + r (or>|€|p- - | T n | f i ( | m £ | ) | ^ ^ p
p
p
-r (a)\i\;"[l-U(\mf\ )} p
P
= r , ( « ) | f f t l f l m f W + M a n a m a ) (1 - | r p ^ ) •
Let the formula (4.5) be true for n, and we prove it for n + 1 i.e. for
# 0 t - , m ) C — ] = i7+ (a)n(|me| )DCt'" - |pm| ] • 1
1
a
F
P
Denoting
C(0
= f[|(^,m)|*-"],m = m a x ( | „ | , | m | ) ,
and using the Fubini Theorem we get
r
+ 1
p
p
(47)
138
p-Adic Analysis and Mathematical Physics
y'{max[|x|p,max(k i|pJm|p)]} -' -Vp{(*,0)' n+i a
,
iirf:,;
n+
- r;(a -1)
/
fi(|m4|p)[|c|p-
0+1
-
teMT^JfrPM*
iffp<Mr
+ r ^ o -1)
J
ociteipjrj^**
1
mt^xMn^dt
-
|«U>|m|,
+ r"(« - i ) | ? L -
0 + 1
/ i - r"(« - i ) - ° P
+ 1
7
2
where / = 2
y
£j(i(£ip)xp(ii i)^ n+
l«lr>Mp = « < K I P )
/
Xp(tU+i)dt
l">l,<[
y .i*i,
= fJ(|mf|p)
xpw
n +
i)^-
y
X
lp<|m|
klp-^d^ipM^+ilp-lmlp +
p(i? i)rfi n +
p
fidm^+Up)
Analysis on the Field of p-Adic Numbers 139
Here we have used the formula (3.1) of Sec. 4.3. By using the formula (2.16) Sec. 7.2 we calculate the integral I?.
• / lM»
J
= fi(|m£| ) p
W T S J O W P
|m|,<W,<|fl7
!
J
l
/
\t\;- x (ti )dtP
n+1
Ll'l
a
l
\t\ - x (tt] )dt p
P
n+i
,<\m\.
P
= n(|mfi )||fi -^(|£|;Me P
-r («)|f p
+
p
a
n + 1
L- [i-fi(K
n + 1
1
fi(K| )mf\- |^ il ) p
+
P
0 +
1 - p -1 il ) ^ + r (a)|f„ | -
| )] p
P
r T
P
+ 1
a
p
i -
l€-+ilr"*i^r
m
By substituting values of the integrals Ii and i"2 desired expression (4.7):
(4-8) we obtain the
140
p-Adic Artalytii and Mathematical Phyaici
i&M
r
+i
= >
- m{\™z\ )(Wp\s\r
1
-
F
\n\p\pm\r )
1
+ r "(a-i)r!(Kl )n(|fX- |u | )||| p
P
1
a+1
a
p
n
- r p ( « - m\rn£\ )\m\ \i\; P
p
- T (a -
p
p
l)ii(\m£\ ) p
1
= -r«+ (a)Ii(K| )|p | p
m
p
n + 1
1
t t
Q
+ r ( )n(Ki )ii(|$| - |f | )(|$| - - |£ ,i; ) + r " (a)fi(|mf| )|f | =i; («)n(Ki,)(i€t -|pm|-). • I f we act like in Sec. 8.2 and use the formula (4.5) then we obtain the following generalization of the formula (3.8) p
a
P
p
I 1 + 1 a
+ 1
p
p
n + 1
+ 1
p
p
n +
p
t t
n
p
\x\- M v
n
a
=
+?
i
K( MA°p -' > a + 8±a
k
where = C(«)r?03)r;( - « - / ? ) . n
(4.9) (4.10)
I f we take into account the property (see (4.4), cf. (2.16)) r;(a)r
n p
(4.11)
(n-a) = l
then the equality (4.10) takes the classical form Note the symmetrical expression for f ^ ( a ^ ) = rp(a)r;(/3)r;{ ) 7
where a + 0 + y = n.
Analysis on the Field of p-Adic Numbers 141
R e m a r k . The formulas of Sec. 8.4 to within notations are obtained in the paper [22] by V.A.Smirnov. They are used in the perturbation theory in the p-adic Euclidean quantum field theory with the propagator of the form
Chapter 2 PSEUDO-DIFFERENTIAL OPERATORS O N T H E F I E L D OF p - A D I C N U M B E R S
Pseudo-differential operator (on the field of p-adic numbers) in an open set O C Qp we call the operator A of the form {A1>)(x) = J a ( £ , * W t f ) > ( - ( f , * M , X l
* e O
which acts on complex-valued functions ip(x) of p-adic arguments i £ Here we suppose that functions ip(x) are extended by zero from the set O on whole space
o The function a(£, z), £ € Qp, i G O is called symbol of the operator J4. We use here some notions and results on the spectral theory of operators from the books by Reed and Simon [175], Dunford and Schwartz [60] and Yosida [237]. I X . The Operator D ° a
The operator D" : ip —* D ip is defined as convolution of generalized functions fand xp (see Sec. 8.3): a
a
D j,
= f_ *ib, a
143
ajt-1.
144
p-Adic Analysis and Mathematical Physics
D" is a pseudo-differential operator with the symbol |£|p owing to the formula of the Fourier transform of convolution (see Sec. 7.5) a
F[D i>]
= |££ • £,
The operator D" has been introduced and studied by V.S.Vladimirov in paper [206], 1. The Operator
a
D,
a /
-1
By virtue of the results of Sec. 8 the generalized function
is holomorphic on or everywhere on the real line except the simple pole at a = 1 with residue — ; in addition /o(x) = 6(x) and (see (3.8) and (3.9) of Sec. 8.3) p
/«*/«
= U+fj,
a,0,a + 0?l,
«,/?£«.
Let a generalized function ip from D'(Qp) be such that the convolution f- **l> exists (a ^ —1). The operator D"il> = f- *ip we call for a > 0 the operator of (fractional) differentiation of order a, and for a < 0 — the operator of (fractional) integration of order —a; for a = 0, D°tp = 6'yj, ip is the identity operator. a
a
1
E x a m p l e 1 . The analogy of the first derivative (a = 1, D = D) :
P +
1
*
I f ip € V then this formula owing to (1.18) of Sec. 8.1 takes the form
<»X-> - - ^ ( M , - . * . or owing to (2.19) of Sec. 8.2 (Dj,)(z)
=
J
|fU(Ox (-£*K. P
Pseudo-Differential Operator! on the Field of p-Adic Numbers 145
E x a m p l e 2. expression
a
The derivative D yj,
a > 0, ^ £ f
is given by the
or equivalently
E x a m p l e 3. The primitive D ° V , a; < 0, a ^ — 1 where vj £ P is given by the expression (see (2.19) of Sec. 8.2)
(g»(z)=
/
|«-»|,-^(ff)dlr
(1-3)
Q or equivalently f / KI?#€)x,(-{*)<*«,
- K a < 0,
a
(D V)(«) = < *! I i m m v x p ( - s * ) - m w ,
(1-4) « < - x .
I f a generalized function ^ belongs to £ ' and a,0,a + 8 ^ —1 then the equalities are valid C i y V = f ' + ' V = D<*D"i>.
(1.5)
• I t follows from the formulas (3.13)-(3.14) of Sec. 8.3 and (2.19) of Sec. 8.2: +
F[D° ^}
= F[f- -t,*1>]
= /_„_„ • i> = \t;$*P4
= | f i ; • mt$)
= \t\f-F\f-fi**]
a
0
= F[D"{D i>)] because y) G £ (see Sec. 7.3).
=
a
F[D^(D u^)]
=
F[f- *D»rb] a
146
p-Adic Analysis and Mathematical Physics
From the formulas (1.5) for 0 — — a it follows the equalities a
a
D D- tl>
tt
a
= f s D- D $,
a t ±1.
(1.6)
R e m a r k . The equalities (1.5) are valid also for those generalized functions 0 G V for which the convolutions f- *(f-0*yj), / _ £ * ( / _ *i/>) and f-a-0*'l> exists in T>'. The existence of these convolutions is essential for the validity of the equalities (1.5) as the following example shows: i f a > 0 and ^> = 1 then the convolution / _ „ * 1 = F [ | f | 6(£)] = 0, the convolution / * l does not exist, and the equalities (1.6) are not valid: a
0
p
a
a
D- (D°l)
a
- D-°0 - 0,
a
D {D~ \)
do not exist.
E x a m p l e 4. a 6 R, a G Q " p
D° (a ) Xp
= \a\; (°-z).
X
(1.7)
Xp
• I t follows from the equalities (2.19) of Sec. 8.2, (2.8) of Sec. 9.2 (for a = — 1) and - «J$ + " ) F[D" (ax)} Xp
(see Sec. 7.3),
= F[f_„* (ax)]
= F[/-*]
Xp
( Iflp- ^
-F{x (ax)] p
+ a),
= ! < * ( £ + «)>
(1.8)
and from the existence of the products a
m - m+«) = i < • *(e+->), P
1
• m+
*(£)*(£ + a ) = 0. By using the inverse Fourier-transform to the equality (1.8) we obtain the formula (1.7). •
Pseudo-Difjerentiat Operators on tke Field of p-Adic Numbers 147 E x a m p l e 5. a £ R, 7 £ Z. Let r
7
• ( * ) = J '[«(|€lp-P )/(€)I,
/€!>'.
Then D"*(i)^p'°$(i).
(1,9)
• Like in Example 4 we have *(o = a
F[D $]
m - p
= f \
=
l
a
7
) / ( - o ,
- $
+
fafc
5
7
p*tf>] - ( i a - p ) / ( - o .
7
7
<*=-*.
7ffl
= P ^ ( l f l - p ) / ( - 0 = p *(0 •
•
P
E x a m p l e 6. a £ 1 , 7 G Z, |2n| > p
2-2
p
t
""
3
y
2
- P ) X P < ° * ) I = p T « | 2 a | ; * ( | r | - p ) (az ). p
(1.10)
Xp
a
2
• I t follows from the formula (1.9) for / ( £ ) = X p ( £ ) owing to the formula (2.10) of Sec. 7.2. • E x a m p l e 7. a £ E, p / 2, 7 £ Z,
£ "(*) = 0 i
a
7
O W^)6(kl -P )]-P P
o ( 1
-
7 ,
'j(^)«(|x| - ^. p
P
• As in Example 6 it follows from the formula (2.14) of Sec. 7.2. 2. Operator
D'
(1.11)
•
1
a
Our goal is to extend the operator D on a = —1, so that on the class of generalized functions i/> £ £', (^,1) — 0 it would be continuous on a at
148
p-Adic Analysis and Mathematical Physics
the point a — — I , and the equalities (1.5) would be valid for all real a and 0 (see [206]). At first we shall prove the Lemma. L e m m a 1 . In order that a generalized function f £ £', supp f C Bw satisfies the condition (/,1)==0,
(2.1)
it is necessary and sufficient that / ( € ) = 0,
£ 5_„.
(2.2)
€
• Necessity of the condition (2.2) follows from results of Sec. 7.3. In fact, let the condition (2.1) be fulfilled. As parameter of constancy of the function / ( £ ) is no less than -N then / ( £ ) = / ( 0 ) = ( / , & ) = ( / , 1) = 0 for all £ € B_N owing to the representation! (3.10) of Sec. 7.3. Conversely, i f the condition (2.2) is fulfilled then by choosing N
ip £ T>,
B-y,
supp
j view = i
we obtain the condition (2.1)
J
0 = (/>) = (/,£)=
= | / ( i ) , & (x) N
I
= (/, A ) N
= ( / , 1) = 0 .
•
On test functions
is valid (/«.P)-*-^/ln|*kK*)
(2.3)
Pa cud o-Differential
Operators on the Field of p-Adic Numberi
149
• The existence of a limit and the formula (2.3) follows from the limit relation
" " " p l n p J ^\x\
a-*
i
I.
Q, The legitimacy of limiting passage under the sign of the integral is guaranteed by the Lebesgue Theorem (see Sec. 4.4) owing to the majorization exp[(a - l ) l n \x\ ] — 1
zesupp
| o - l | < l
for some C = C{p, supp ip). Let us denote by / i the regular generalized function —
ttiitf
= -Ifijfi
J * t^ttyfe
f ^
v
-
In \x\ , p
2 4
(-)
Put fTV-#*rS Tfte limit
(2.5)
relation D~"VJ
is uahd" iftp££'
—>
D '
1
^ ,
satisfies the condition
o -
1 in P '
(2.6)
(2.1).
• Let supp tp C B J V and 0, a / 1 we have a
(D~ ip,
p) =
= ( / « ( * ) , (*(»>. «K* + y)))
= / /«0)«0)dx
(2.7)
ISO
p-Adic Analytii and Mathematical Phyiici
as *(«) = & K * M « + 03 =
Aw(»)«»(* + y)) 6 2
Further owing to the formula (6.5) of Sec. 6.6 (as A (y)
+ y) € 2?(Q ))
N
/ o(z)rfz = JWy),A (yMx
+ y))dx=
N
=
V(y). &N(y) J v(x)dx) \
Q„
U(y),A (y) N
= {tp, l ) j /
J
(E,
So the function g satisfies the condition (2.1). Therefore by virtue of (2.3) from (2.7) and (2.5) we derive (2.6):
(D-°y;,
= j
j
f (x)g(x)dx^ a
= J fi(x)(^y),A (y)p{x
+
N
l
= {fi*V,v)
h(x)g(x)dx
= (D- Jp,
y))dx
tk—tl.
U
Now we shall prove the formula
f
l
{
where the generalized function
0
=
v
+
W p
p
m
( 2
'
8 )
U defined in (1.10) o/Sec. 8.1.
• From the formula (3.6) of Sec. 4.3 it follows that the Fourier-transform of the function fi(x) coincides with I f l " for £ ^ 0. Therefore (see Sec. 6.3) 1
(2.9)
Pseudo-Differeniial Operator! on the Field of p-Adic Numbers 151
For definition of the constant C we apply both sides of the equality (2.9) to the test function A<j = Ao (the function A t are defined in Sec. 7.1). By using the integral (2.6) of Sec. 4.2 we get C = (/i,A )-
(pyl-.Ao)
0
ifl,
ieu>i
I f ip e (ip, 1) = 0 then by Lemma 1 ip(f) vanishes in a vicinity of the point £ = 0. Therefore by virtue of (2.8) fiiOHO
(Vjgr +
=
• 1(1) =
teffifc)-
From here as above it follows that the formulas (1.5) are valid for all real a and 8 if ( t M ) = 0. L e m m a 2. If p € D, supp
p
N
p
(/.<«'),¥>(*-*')>=<
—
plop
m
. *' l lp J
. 0=1.
z
(2.10)
• Let tt 1. For | z | > p " , p ( i ) = 0 and \x - y\ - \x\ , y € supp
(f (x-)Mx 0
-
i*'i
p
x'))
] 'i>i E
p
1S2
p-Adic Analyiii and Mathema'.ical Physici
The case a = 1 is considered analogously. The following equalities are valid /!*/-«
= /!-«. =f-«*fx,
«>0.
(2.H)
• For a = 0, fa = 6, and the equalities (2.11) are valid. Let a > 0. Then by the definition of convolution (see Sec. 7.1) for all
= t—too l i m ( / i ( x ) x f- (v)M*M*
+ v))
= l i m / (x)h{x)(f. {y),
+ y))d
a
nk
0
3
t—OO J
= J h(x)(f- (y).
+ y))dz.
0, The legitimacy of limiting passage under the sign of the integral and existence of the last integral follow from the Lebesgue Theorem (see Sec. 4.4) and the Lemma 2 owing to the majorization
|at|p > R,,
k>N.
Thus the convolution fi*fexists, and hence the convolution f- *fi exists, and they are equal. To calculate it we argue by the following way. The product f\ • / _ „ exists, and the equality is valid a
F\Ji"f-c]
= h
a
/-«
(see Sec. 7.5).
By taking into account the formulas (2.19) of Sec. 8.2 and (2.8) we rewrite the last equality in the form
It ia easy to see that for a > 0
Paeudo-Differential Operators on lis Field of p-Adic Numbers
153
Therefore fx**/-] = Kir
1
from where owing to (2.19) of Sec. 8.2 the equalities (2.11) follow
From (2.11) for a = —1 the relations follow /t*/-i = «=7-l*/l.
(2.12)
Now we shall prove the statement: i f ip £ £' then for a > 0 there exist D- D"tp and D D- vj and they are equal to D ' ^: l
tt
l
0 1
x
a
D~~ D ip
1
= D"- ^
a
1
= D D~ i{>
(2.13)
= f- *(fi*il>)-
(2.14)
or equivalently / i * ( / - W ) = fi-c*iP
a
• We use the convolution theory of Sec. 7.1. Let ip kz D- Then for all x £ Q we have p
((/-WMi/),
v{* + y')) = (/—(*). W)M*+v+y'))) = (f- (y),9(x 0
+ y))
where g(x) = (yj(y'),/') exists and is equal to /i_a*^>: ( / l * ( / - « * * ) , * > ) = / / i ( * ) ( / - . ( l f ) , f f ( x + »))
(fi- *it>,-p) a
owing to the equality (2.11). The similar consideration is valid also for the convolution =
- (/-Wi)** •
•
!Yom the equalities (2.13) for a — 1 the equalities follow 1
D~ Dtp
_ 1
= vj = D 7 J V ,
1
V€
(2.15)
154
p-Adic Analysis and Mathematical Physics
We summarize our results in the following Theorem. T h e o r e m . The formulas (1.5), a
8
D D il>
+S
0
a
= D° *l> = D D tj>,
Ve &,
(1.5)
are valid ifa,0,a + 0 j S - l w a > 0,0 = - 1 . / / V £ £ ' Mfc's/ies iAe contfidon (2.1) (V ,1) = 0 (Aen the formulas (1.5) are va/i
a
R e m a r k . The operator D", a ^ — 1 is defined by the homogeneous generalized function / _ „ of degree —a — 1; for a = — 1 this property is lost: the operator D is defined by the function — j ^ l ; l n | r | p which does not possess property of homogeneity. _
3. Equation
1
a
D i/> = g
Let us consider the equation iri>
= g,
get,
a em
(3.1)
with respect to an unknown generalized function yj e V. A solution of the equation (3.1) is seeked in the class of those ip kz W, for which convolution /_„*V> exists in T>' (and it is equal to g) (see Sec. 7). T h e o r e m 1. For a > Q any solution of the equation (3.1) is expressed by the formula iP = D-°g + C (3.2) where C is an arbitrary constant; for a <0 a solution of the equation (3.1) is unique and it is expressed by the formula (3.2) for C — 0. • The fact that D~"g is a solution of the equation (3.1) follows from the formulas (1.6) and (2.15). I t remains to investigate solutions of the homogeneous equation D il> = 0. (3.3) a
By applying to the equation (3.3) the Fourier-transform (see Sec. 7.5) we get / _ ( 0 • vH£) = 0. As / _ ( £ ) f 0 for f ^ 0 (see (2.19) of Sec. 8.2 for o
0
Pseudo-Differentia! Operators on the Field of p-Adic Numbers
155
a ? - I and (2.8) for a = - 1 ) then = 0, { # 0, and thus ^ = C*(£), V- = C. Now we shall prove that for a < 0, C — 0. For a = 0 it is the case. Let a < 0 and C ^ 0, i.e. / _ * 1 = 0. As Q
J
l
f- €L (B ), o
0
/_ (Odi>-oo a
KM, then i f we choose a function , j
0 = (/_„.l,^) =
= J }- (x)dx a
a
k
J
+ Jim
Bo
J
f_ (x)A (x)
f- {x)dx a
= -oo .
•
1
T h e o r e m t*. In order that there exists a solution ip of the equation (3.1) such that ip(£) vanishes in some vicinity of £ = 0, it is necessary and sufficient that g satisfies the condition (2.1). In addition the solution ip = D~°g is unique, continuous with respect to a (E K in V and the formulas are valid D"ip = DP—g
a
0
= D~ D g.
(3.4)
• Necessity. Let ip be a solution of the equation (3.1) and ip = 0 in a vicinity of £ — 0. Then F[D"iP\
= F[f. *iP\ a
= /_„<£) • fa) = git) .
From where it follows that g(£) = 0 in a vicinity of £ = 0. By the Lemma 1 of Sec. 9.2 g € £' and satisfies the condition (2.1). Sufficiency. Let g € & satisfy the condition (2.1). By the Lemma 1 of Sec. 9.2 g(f ) = 0 on a vicinity of £ = 0. As constant C / 0 does not satisfy
156
p-Adic Analyiii and Mathematical Phytio
the condition (2.1) then by the Theorem 1 there exists a unique solution y) = D~"g of the equation (3.1) which satisfies the condition (2.1); | ( 0 = F[D-«g]
= F[f *g]
= /„(£) • g(Q-
a
Last statements of the Theorem follow at once from the Theorem of Sec. 9.2. • 4. Spectrum
of the Operator
D
a
in Qp, a > 0
The operator D", a > 0 is pseudo-differential with the symbol \f\ Sec. 9.1):
(see
p
DH = j i a X r ( - ^ M £ R a
Q, 2
2
It is defined on those functions tb from the Hilbert space L ( Q ) = L (see Sec. 7.4) for which \£,\^vl e i . This set, we denote it by V(D ), is called the domain of definition of the operator D" in Q . I n addition p
2
a
p
3
(D^, )
f %#%g8ffl&
a t
= {D"/ ^D ' )
V
V
2
\\D°yj\\
= J \i\l \i>(t:)\ d£, a
= (D"tp D"tP) }
2
a
ip,
a
i> € V(D ).
(4.2)
The equalities (4.1) and (4.2) follow directly from the Parseval-Steklov equality: (D 1>,tp) = (F[D yj],F[
a
2
here (•, •) is the scalar product in the Hilbert space L (see Sec. 7.4). The defined operator D" is self-adjoint and positive: a
(D yj,yj)
a,2
3
= \\D i>\\
> 0,
O ^ e
V(D°)
so the spectrum of the operator D" is situated on the semi-axis A > 0. Let us consider the eigen-value problem in Q p
a
D tp = \tl>,
2
Vei (Q ). P
(4.3)
Let A = 0. I t was shown in Sec. 9.3 that for A = 0 the equation (4.3) have the unique linearly independent solution in T>'ip = 1. Thus y) = 1 is
Pi cud o-Differential Operators on ike Field of p-Adic Numbers 157 a
a generalized eigen-function of the operator D which corresponds to the eigen value A = 0. However A = 0 is not an eigen-value of the operator D . Further, ran D (range of values of D°) is dense in L ' . a
a
2
• I t follows from the fact that the equation a
D \b =ip,
J
a
has a solution ip = D~°ip from the domain D[D ) i) = F[D~"ip]
= F[f *
a
as ip G T> and
= | £ f « 0 G t>, 4><=V
(by the Lemma 1 of Sec. 9.3 p{l) = 0 in a vicinity o f f = 0), and the set of functions {
L e m m a . The set of test functions satisfying j ipdx = 0 is dense in L . t,
the condition
2
• By virtue of the Parseval-Steklov equality \\tp\\ = ||^|| (see Sec. 7.4) and the Lemma 1 of Sec. 9.3 it is sufficient to prove that the set Z>(QP\{0}) -
f > 6 T> : HO
= 0,
£ € B
K
i
3N
2
G Z)
is dense i n L . But in Sec. 6.2 i t was proved that V(Q \{0)) P
2
2
L (Q \{Q})~L (Q> ) P
P
2
= L .
is dense in •
Let A > 0. I f we apply to the equation (4.3) the Fourier-transform we get the equivalent equation mi?
-
= °-
From here we conclude that desired points of spectrum A (in this case eigen-values) have the form ria
\ =p ,N€Z, N
(4.4)
* However this statement follows from the facts that self-adjoint operator D" has no remainder spectrum and A = 0 is not an eigen-value of D" •
158
p-Adic Analyiii and Mathematical Phftiet
and the corresponding (normed) eigen-functions have the representation s
$m
a
= H\t\*-p M§J
5. Orthonormai
N«)l d£ = l
Basis of Eigen-Functions
Nez.
1
(4.5)
of the Operator
D" [207]
In (4.5) we choose as functions p the following system of locally constant functions on Sfj: for p ^ 2 „2N-I
P .h,e,(0 N
NI
=
N
l
p- \i^p(£ )-Xp lP
' = 2,3
1=1,2
£i = £a + £ i p + . . . + l
P Nk o ( 0 ^ P ^ * K o - * ) ,
4ci
1
-(€ + V - - )
a
(5.1')
p-1,
£i-2p'
_ 2
i = 1,
£j = 0 , 1 , . . . , p - 1,
,
* = 1,2,... ,p - 1,
£ / °> 0
£, = 0 ; (5-1")
for p = 2 •>2N-l-2 —
^ . ( O ^ ^ A ^ ' j x * i = 2,3,...,
(
k=
P W O
£ i
v
+
q
-
2>- -*f\,
0,l,
€i = 1 + ei2 + . . . + e i - 2 2 ' i
N
£
2
= 2 ^ Y2(*2' - 0,
- 2
,
(5.2')
0,1,
f = l , fc = 0 , l ,
ei=0.
As a result we obtain the following system of normed eigen-functions which the Fourier-transforms owing to (4.5) and (5.1) are equal: for p ^ 2
• * ' [ - V
(
i
+
i
p
,
"
N
"
1
)
(5.3')
2
(5.3") for
p =
2
r
92JV-I-2
X2 [—^— &,»,o«) = a ^ ^ b
+ N
(5.4') ftr
2
" 2 )X2(fc2 - £).
(5-4")
Pseudo-Differential Operators on the Field of p-Adic Numbers 159
By applying the inverse Fourier-transform to the functions (5.3) and (5.4) and using the formulas (2.7)-(2.9) of Sec. 7.2, we get the following eigenfunctions of the operator D corresponding to the eigen-value Ajy = p , J V e S : f o r p ^ 2 I k i n d ( i = 2 , 3 . . . , * = 1,2,... , p - 1, e, = e + £ i p + ...+£,-2p'- ) a
aN
1
0
J
* W > )
i
,
N
= P^ y ^*(klp-P - )x ( p
I I kind (1=1,
P
,
£ (
2
p - ^ + V-
A r
- x ) , (5.5') 1
* = 1,2,... , p - 1, t | = 0) x
$ ,hA ) N
!£fisi
N l
(5.5")
N
= p (p - \ \ )x (kp- xy, x J>
P
for p = 2 I kind (7 = 2 , 3 , . . . , fc = 0,1, = 2*^(kl2
-
e, = 1 + £j2 + . - - + £ i - 2 2
^ - ^ Y ^ ' "
2
" *
+2'- -*x), w
2
1 - 2
) (5.6')
I I kind ( ( = 1 , fc = 0,1, £i = 0) = ^ m
N
\ x
- * 2 - | ) - 6(\x - k2 ~ \, W
2
N
2
2
- #-»)].
(5.6")
The eigen-functions (5.5)-(5.6) belong to the space V and satisfy the condition j ^,
M l
(s)dx = 0
(5.7)
because owing to (5.3)-(5.4) ^ ^ , , ( 0 ) = 0 ( see Lemma 1 of Sec. 9.2). The I kind eigen-functions satisfy also the condition
O r t h o g o n a l i t y of the eigen-functions for distinct N follows from the formulas (5.3)-(5.4), for fixed jV and distinct / it follows from the formulas (5.5)-(5.6). Orthogonality of the I kind functions for fixed N and / and
160
p-Adic Analyst; and Mathematical Physics
distinct e follows from the formula (2.10) of Sec. 7.2: Wit**, •
/ • j
x [( -£;)p'P
E i
\(k-k')p
2 , v
x
2
x]dx
+ (i-fc')p'
t-N - p
=_ 0
as EI ^ s'„ so ki-',l>p-
2
2(e,-e;) '- "
, + 1
p
P
2!l N)
,
\4(t,-e',)p'-™\>P*- -
p^'-lEi-eJip
For p = 2 one considers analogously. For fixed N, I and ei and distinct k the functions (5.6) are orthogonal by virtue of the formula (2,8) of Sec. 7.2: i
^
*
^
)
"
/
X2(2'~ x) 2(-2'- ->x)dz N
N
X
S|+I-N
X2(2 i - j v - i a:)dj! = 0, = S
I = 2,3,...
/ I+I-N
Hence, / o r p = 2 (fte eioen-/anc[ions (5.6) form an orthonormal system in L ( % ) . For p ^ 2 the I I kind eigen-functions (5.5") are orthonormal for distinct jfc (and fixed JV) by virtue of the formulas (5.3"). The I kind eigen-functions (5.5') for distinct k and fixed N, I and e are not orthogonal. To construct an orthonormal system of eigen-functions it is sufficient to orthogonalize the system of functions 2
fc=l,2,...
,p-l}.
Pseudo-Differential Operators on ike Field of p-Adic Numbers 161
We shall prove now that as such orthonormal system one possibly takes the following I kind eigen-functions:
t = l,2,... ,
P
(5.9)
- 1 .
At first we establish the relations
Kj
** ™ '
E
-
7
i
>
i<)
^ . ( * ) ,
(5-11)
where e! = — ^=- V>\-N,v,.l(0 and fc' is defined to be the congruence -2e k 0
= k'
(mod p),
(5-12)
*' = 1,2,... ,p - 1.
(5.13)
The formula (5.10) follows from (5.3') and (5.9) owing to the relations
X p i
kp'-^ ) x
=
k
X p
( -^.) V
P
= E K j
'
*p
£ I<J
( ~ ) *(*o - i ) , >
p
x e s,.^.
'
Further, by using the relation
I<«
I<«
I " ' I P-1,
* t = J
, (5.14)
162
p-Adic Analyiit and Mathematical Phytics
from (5.10) we derive the formula (5.11)
E
»f~)^w V
1<>
**- 1
'
K K f - l
e
v
' l<j
V
* '
I
1
VP-
1<J
P
1
VP-
p V
^
I<;'
owing to the equality
-^=r E y
p
i<;
V
P
I<J
*
i<;
V P /
= E I<J
At last the formula (5.12) follows from (5.11) and (5.3'):
• E
i
tj L
J £
'
'
(5.15)
Pscudo-Differential
Operators on the Field of p-Adic Numbers
By virtue of (5.13) and (5.14) for f e S
we have
N
J
L
ip )
163
Xp
l ( *w)
-
E i<)
=
E f x , ( 7 * ) - i l x , f - ^ ) = ^ l
e
o
= P « «
0
- t ' ) -
From here and (5.15) the formula (5.12) follows. • IVom the representation (5.11) it follows that the functions (5.9) are eigen-functions of the operator D". Thus it was constructed the orthonormal system (5.9), (5.5') of eigen-functions of the operator D° for p ^ 2 : N e Z I kind ( i = 2 , 3 , . . . , fc = 1,2 ^,,0)
p - 1,
= P ^ H H
-P'
_ J V
= E + e
e j
0
2
+ .., + c,_ p'- )
l P
2
l
m
) • *(*o - i)^($tp - z%
(5.16')
I I kind (/ = 1, k = 1,2,... , p - 1, c = 0) t
l
&Mf&
N
N
= ^*0ti>-P ~ )Xp(kp- x)
=
(5.16")
Completeness. Let us prove the completeness of the orthonormal systems (5.6) and (5.16) in L ( Q ) . I t is sufficient to prove the completeness of their Fourier-transform on any circle SN , N £ Z as 2
P
3
L (Qp)=
2
E
®t (5 ).
(5.17)
T
-oo<7
To prove the last one it is sufficient to prove the Parseval-Steklov equality for any functions from the dense set N
i X p ( ^ ) , k i p > P~ ) Let p ^ 2. I f |cr|p — p~ (5.2") 1=
22
N
2
in L (S )
(see Sec. 7.2).
N
17
then Xptf ) = 1 on
«('o-*) = P ^
E
and hence owing to
*&,*,o«).
f e ^ .
164
p-Adic Analytia and Mathematical
Let ](j|p = p
1
PhyMici
. Then for £ € SN we have
v
i
l<*
=
\
E
r
f
*
( ^ j i ( 0 -
P
K K p - l
^
P
'
N
Let |ir|p = p"~ , n = 2 , 3 , . . . . Then for { e S e W . = (xp(^)^V.t, ) =
/
t(
= j
ftwstfM-frW
Hence owing to (5.16) C
N
lc i i
we have
N
=
= 0 if / ^ n and for / = n
3
IGJ.v.t = I ^ . W I
a
=
- *)•
For fixed / = n > 2 and t = 1,2,... , p — 1 a number of eigen-functions (5.16') is equal to ( p - l ) p " . Therefore - 2
\\x,{t*)\\b s ) l
tl
=p
N
( i - l )
n 2 N+1
= (p-i)p - -"
Y,
P
*<*«-*)
1<*
=
E 1
E l
E I ^ . I
2
.
so that the Parseval-Steklov equality is realized. Let p = 2. For \tr\ = 2~ or = 2 ~ we have x (t\o) = 1 on S and hence owing to (5.4") xi(t*) - 1&,o.o(0 JV' kla = 2 " - , n = 3 , 4 , . . . as in the case p ^ 2 we have N
1
N
2
2
O N
Cjr M. = l
(j&(W.ft *)=flr A
S
F
A ( 1
o
W.
N
r
w
Fseudo-Differential
Therefore owing to (5.6) C' l ^
t
. . , l
N
i
2
l i
Operators
on the Field of p-Adic Numbers
165
= 0 i f t+ 1 / n and for / + 1 = n 2
N
= h&,*, ,(*)l = 2 -
n + 1
t
.
For fixed I = n — 1 > 2 a number of eigen-functions (5.6') is equal to 2
. r,-2 2
=
2
r,-i
Therefore
l
d
Finally we note that functions of the form 14-=^) satisfy the equation (4.3) for A = Ajv = p owing to (1.7). Thus we have proved the following
(5.18)
oN
a
T h e o r e m . The spectrum of the operator D is essential' and it consists of a countable number of eigen-values \N = p , N G % each of which is infinite multiplicity, and the point A = 0 {the limit point of the eigen-values). The functions (5.16) for p ^ 2 and (5.6) for p = 2 form an orthonormal basis of eigen-functions of the operator V . These eigen-functions belong to 73(Op) and satisfy the conditions (5.7) and (5.8). The generalized eigenfunction 4>(x) = 1 is the only one corresponding to the point A = 0, and the eigenvalues A^r correspond to generalized eigenfunctions of the form (5.18). aN
a
6. Expansions
on
Eigen-Functions
For uniformity we shall redenote the eigen-functions (5.6) rp' by
Nt
(p = 2)
N k Ci
T h e o r e m on e x p a n s i o n i n eigen-functions
N i: C (
of the operator
a
D, NCZ,
1=1,2,...,
fc
= l,2,..., -l,
* The definition of essential spectrum is found in Sec. 10.1.
P
(6.1)
166
p-Adic Analysis
and Mathematical
NeZ,
Physics
1=1,2,...,
k = 1,2,... , p - l ,
£i.
2
T h e o r e m . Every function f £ £ ( Q ) is expanded in the in eigen-functions {
Nt
(6.1)
Fourier-series
Cj
/ ( « ) = E
E
E
l < ! < »
JVGZ
E / W * W * 3
6
2
<->
1 < * < P - 1
2
TAe series (6.2) coraueiyes in i ( Q ) , and the Parseval-Sleklov equality is valid p
2
\\f\\ = T,
E
E
1
JV€Z
E i / W -
1 < * < P - 1
M
*
2
In the other words the space L ( Q ) is expanded in a direct sum of finite-demensional eigen-subspaces (series) W^, N £ Z, I = 1,2,..., p
z
i (Q ) = E P
Nez
n
E
6
® *-
5
<->
I
For p / 2 the subspace r f ^ , / = 2 , 3 , . . . are spanned on the eigen-functions of I kind (5.16') so that dim H' = (p - 1 ) V " ; 'H are spanned on the eigen-functions of I I kind (5.16") so that dim H - p - 1 . For p = 2, H , I = 2,3,... are spanned on the eigen-functions of I kind (5.6'), dim H* = 2 ' ; Tijf are spanned on (5.6"), dim Wjy = 2. 2
N
1
N
l
!
N
N
N
- 1
R e m a r k . I t is easily seen that just constructed eigen-functions of the operator D", a > 0 (see for example the formulas (5.5) and (5.6)) by complex conjugation convert either in themselves (real) or in other eigenfunctions of the some series (complex). Therefore a complex eigen-function (5.5) or (5.6) gives contribution 1 in dim 7i' rather than 2 as it is the case in the classical mathematical physics. N
Pseudo-Differential
X . p-Adic Schrodinger
Operators on the Field of p-Adic Numbers
167
Operators
A pseudo-differential operator A of the form Ai{> — a*ij> + V • i!> we call the p-adic Schrodinger operator. Its symbol is the function
5(0 + V{z) where a{£) is the Fourier-trans form of a (generalized) function a(x)\ the function V(x) is called a potential. The simplest example of such operator is the operator D " = / _ * , a > 0. Its symbol is \£\ (see Sec. 9.). In presentation of materials in these section we shall follow mainly the work [208], 0
p
1 . Bounded
from
Below
Selfadjoint
Operators
Let A be a selfadjoint operator in the Hilbert space H with (dense) domain of definition T>(A). We denote by p(A) the resolvent set, by c{A) the spectrum and by V{\) the projector-valued measm-e of the operator A,
j
A=
xdp(x),
"{A)
(A) = j
XdiP^M),
• ,i>ev(A). P
°{A)
Let A £ c{A). In accordance with definitions (see [175], v . l ) we say that A £
e3i
is infinite dimensional for alle > 0; if the projector (1.1) is finite dimensional for all sufficiently small e > 0 then A £ o-,j\ -(A)(p,i\ (A) is the discrete spectrum of the operator A). Thus a<
(A),
c (A) esa
si:
N tT (A) disc
= 4>\
A € o~&, {A) if and only if A is an isolated point of the spectrum c{A) and A is the eigen-value of the operator A of finite multiplicity. An operator for which c{A) — c*4\ {A) is called operator with discrete spectrum. sl:
sc
16S
p-Adic Analysis
and Mathematical
Physics
Another classification of points of the spectrum a(A) is based on expansion of the spectral measure dP(X) (precisely measures d(7>(A)^, ip), tp € H) on singular, absolutely continuous, and continuously singular parts. I t gives respectively
M
n
U ff^A) U
pp
o; (A). ins
However these sets may intersect each other (for details see [175], v . l , ch.VII.2). Now we suppose that a selfadjoint operator A is bounded from below, i.e. there exists a constant C such that 2
(A^)>CM\\ ,
^keV(A).
(1.2)
The closed bilinear form generated by the operator A we denote by A{ip, ip); its domain of definition we denote by Q{A) D T^{A) so that
A( ,i>) = (A
(1.3)
V
We denote by Xk(A), k = 1,2,... the mini-max numbers which are defined by the mini-max principle x (A)= k
sup
4rfi?
inf
*e
(1-4)
The following Theorem takes place (see for example [175, v.IV]). Theorem. Let A be a bounded from below selfadjoint operator in the Hilbert space H with domain of definition V(A); let A(f, \p) be a corresponding bilinear form with domain of definition Q(A). The following conditions are equivalent. - 1
(I) The operator (A — A ) is a compact for some A £ p(A); (II) The operator (A — A ) is a compact for all A £ p(A); (III) The set of functions - 1
|>eD(A):
|M|< ,
if compact for all a > 0 and b > 0;
f l
||Aifr||<6]
Ps cud o-Differential
Operators on the Field of p-Adic
Numbers
169
( I V ) The set of functions [ip e Q(A) : H0JJ < a, A(il>,ii>) 0 and b > 0; (V) There exists an orthonormal basts {ip* £ D(A), k = 1 , 2 , . . . } of eigen-functions ipt of the operator A which correspond to eigenvalues \t, A
,
m
k= 1,2,... ,
in addition At is of finite multiplicity, Ai < A < . . . ,
and Xk - * oo,
2
(VI)
X (A)^+oo,
fc-*oo;
(1.6)
k^oc,
k
where numbers \k(A)
(1.5)
(1.7)
are defined by the mtni-max principle (1.4).
In other words by realization one of the conditions ( I ) - ( V I ) the others conditions will be fulfilled, and the spectrum c(A) ~ [A — A*, k = 1,2,...] is discrete, in addition At = A t ( A ) —* oo, k —* oo. E x a m p l e 1. Let a potential V ( | r | ) be bounded from below and locally finite in Q : -C < V(\x\ ) £ L ^ ( Q ) . T h e p-adic Schrodinger operator p
p
p
p
a
Aip = D 4>- V(\x\ )yj, r
a>-\
p
(1.8)
has symbol
and the domain of definition V(A)
2
= [
2
|£| ^£L (Q ), p
p
2
V(\x\ )ip £ L } . P
z
2
It is bounded from below and symmetric in the Hilbert space L ( Q ) = L . It admits the Friedrichs selfadjoint extension (see for example [175], v.2) with the help of the closed bounded from below bilinear form p
A() = j\f\°v{t)~kt)dt Q
a /
+j
= ( D / V . - D V ) + (V>,V0
V(\x\ Mx)ii>(x)dx P
(1-9)
170
p-Adic Analysis and Mathematical
Physics
with the domain of definition 2
2
Q(A) = [
P
+ \C\ +l
p
It is clear that V(A)CV(A)
C Q(A).
E x a m p l e 2. Multidimensional p-adic Schrodinger operators with symbols 2
2
\Si\ + ...U P
+ v(z),
p
KcOlp
+
vi*),
1
1
where a potential V € i j ^ Q p ) and Vfz) > - C , z G Qp , give examples of bounded from below symmetric operators in L ( Q p ) . 2
2. Compactness
2
in
n
L (Q )
As the space Q£ is locally-compact then some criterions of compactness of functions of real arguments are directly carried over to complex-valued functions of p-adic arguments, for example the Ascoli-Arzela Lemma (see [175], v . l , [237]). // M is an infinite set of continuous functions on a compact K C Qp which is bounded in C(K) and it consists of equicontinuous functions on K, then a sequence may be chosen from M which converges in C(K). Let G be an clopen set in QJJ (for instance, a ball B , a sphere S , the exterior of a ball B : Q \B the exterior of a sphere 5 : Q \S , whole space Qp and so on). We shall consider the space L (G) (see Sec. 4.1) as a set of those functions from L (Q ) whose support is contained in G, i.e. every function / from L (G) is defined on the whole space Qp and vanishes almost everywhere outside of G. r
T
p
T
ry
r
p
r
2
2
p
2
2
Definitions. 1. We shall say that a set of functions M C L (G) consists of equicontinuous on the whole in L (G) functions i f for any e > 0 there exists n = n G H such that for any function / from M the inequality is valid 2
c
J\f(x G
2
+ h)-f(x)\ dx , <£
VftG5 . n
Plendo-Differential
Operator! on the Field of p-Adic
Numbers
171
2
2. We shall say that a set M C L {G) consists of functions with equicontinuous L ( G ) - integrable at infinity if for any E > 0 there exists N = N £ 2 such that for any f £ M the inequality is valid 2
c
J
\f{x)\Hx<€.
G\B
N
3. We shall say that a measurable real function F(x) defined on a set G C Qp tends to + oo at infinity (i.e. F{x) - * +oo, \x\ - » oo, x 6 G) i f G is an unbounded set and for any M > 0 there exists N — N £ 7L such that f 0 ) > M for all | x | > , ^ £ G. p
e
N
p
P
A n a l o g y o f t h e R i e s z - K o l m o g o r o v c r i t e r i o n o n compactness: I n order that a set Af C L (G) is compact in £ ( G ) , it is necessary and sufficient that i t satisfies the following conditions: 2
3
7
( I ) is bounded in L {G); ( I I ) consists of equicontinuous on the whole in L (G) functions; ( I I I ) consisis of functions with equicontinuous L (G)-integrals at infinity. 2
2
In terms of the Fourier-transform from the Riesz-Kolmogorov criterion we obtain the following criterion of compactness: In order that a set M c I (G) be compact in L (G), it is necessary and sufficient that it satisfies the conditions ( I ) , ( I I I ) and the condition 2
2
( I I ' ) the set [g = f,f 6 M] consists of functions L (Q )-integrals at infinity.
with
equicontinuous
2
p
From the last criterion i t follows directly t h e analogy o f Rellich's c r i t e r i o n : If a set of functions f £ L {G) is bounded in L {G) and satisfies the conditions 2
2
J «(t)\f(Z)\ dt:
f€M,
jp{x)\f{x)\ dx
} £ M,
3
2
G
(2.1)
(2.2)
172
p-Adic Analysis
and Mathematical
Physics
where
R e m a r k . I f G is bounded then the conditions ( I I I ) and (2.2) naturally are absent. 3. The Operator
a* + V
The operator o* •+• V- = A is pseudo-differential ( f t t i t * ) = j a(OHOXp(-M)di
+ V(z)i>(z).
(3.1)
«; 2
We shall consider the operator A in the Hilbert space L {G) where G is a clopen set in Q". Therefore in the equation (3.1) x belongs to G. The symbol of the operator A is the (generalized) function 5(0 +V(x),
xGG.
We suppose:
V(A)
2
= [il>€L (G)
: aj>€L (®;),
2
V> €
L {G)}.
Under these assumptions the operator A is symmetric and positive. The corresponding bilinear form is:
AO,¥>) = /
m m m d £ + j
Aif ip)
5 ( O W O I
l
= j
2
# + J
n*)^*)^)**,
,
2
V(x)\tp(x)\ dx
G
Q;
with the domain of definition Q(A)
=
[y. €
2
L (G)
:
v/5^I (Q;), J
VK^,
t (G)] 2
G
(3.2)
(3.3)
Pscido-Difjfcrcntial
Operator) on the Field af p-Adic Numbers
173
which is closed and positive. Therefore the operator A admits the (unique) Friedrich selfadjoint extension A with a domain of definition V{A) — V{A*) (~\Q(A) C Q(A), and 4 & $ = $ M ,
€V(A).
The "boundary value" problem for the operator A on a set G is posed in the following way: Let / £ I?{G), find a function (A) which satisfies the equation (A
x£G.
(3.4)
For / = 0 we have the spectrum problem for the operator A. R e m a r k 1. "Boundary" conditions in the "boundary value" problem (3.4) are hidden i n the definition of the quadratic form (3.3) in which it is supposed that functions ip vanish outside G. By virtue of the analogy of Rellich's criterion (see Sec. 10.2) any bounded set M in L ( G ) for which for some b > 0 2
A{vj,iP)
yj £ M
2
is compact in £ ( G ) . From here by using the Theorem of Sec. 10.1 we conclude that the Hilbert-Schmidt theory is valid for the operator A, namely one has the following M a i n T h e o r e m Let locally bounded functions a(£) and V{x) be bounded from below and tend to +oc at infinity in Q™ and G respectively. Then the spectrum of the operator A = a* + V. ' ) is discrete and it consists of real eigen-value Ai < A < . . . , A* —* +00, k —* 00; every eigen-value is finite multiplicity. Corresponding eigen-functions ip^ £ V(A) form an orthonormal basis in L (G), 2
2
Atpk = Afc^jt,
(Atpk:
k,j = 1,2,... .
(3.5)
The following variational principle is valid: \
k
=
min
A
i>i £ Q( )
* More precisely, the spectrum of its selfadjoint extension A.
(3-6)
174
p-Adic Analysis
and Mathematical
Physics
where the min is realized on an eigen-function tp — ip corresponding to the eigen-value A*. f o r A ^ A*, k = 1,2,... the operator {A — A } is compact, so the equation (3.4) is uniquely solvable in T>(A) for any f £ L ( G ) , and its solution is expressed by the formula k
- 1
2
E
(f>
„
7
,
l
For A — At a solution of the equation (3.4) eiisis i / , and only if, f is orthogonal to all eigen-functions corresponding to the eigen-value A , t
i/m+s)=^ where
j = Q,t,...,ni -l,
(3.8)
l
is the multiplicity of A^; (Ais solution is expressed by the formula *=
E
x r # +
E
(3-9)
luftere c,- are arbitrary constants. R e m a r k 2. The Main Theorem is valid also for those bounded from below pseudo-differential operators, whose symbol a ( ( , i ) is locally integrable in Qp x G and every set of functions
tb^L\G):
is compact in
|M|<1,
/
a(£,i)x (-(e,i)WO^K^
2
L (G).
Conversion o f the M a i n T h e o r e m . If the Schrodinger-type operator A = a* + V. in Qp with a symbol of the form ^CKXIP, - - •
AUP) + V(\XI\ ... p>
,\x \ ) n p
where V and V are locally-bounded and bounded from below functions m Qp such that the mini-max numbers \k{A) —* +oo, k —• oo then V — +oo, \£\ —* oo and V —> +oo, \x\ •-* oo. p
p
Pseudo-Differential
Operators
on tke Field of p-Adic Numbers
175
• A t first we consider the case n — 1. Let V ( | i j ) not tending to +00 if \x\ —* 00. Then there exist K > 0 and a sequence pt —• +oo, k —* co. Pi £ 2 such that V(p -)
p
p
On the eigen-functions of the I kind (see (5.16') of Sec. 9.5 for p ^ 2 and (5.6') Sec. 9.5 for p = 2) {4- *,i Mi fi
* = 1,2,-..}
f
2
\&-r>AA(0\
P
=
(3.11)
2
2 p
^P >- 6(\t\p-P - *}
the quadratic form
®*
Q
P
takes the values
p
=
2
2
^P '- Jvm )S(\e\ - -^)d£ P
+ P
p P
-TTP"'"/v(\x\ )H\z\ - ^) p
=
+
W
)
p
P
,
and hence owing to (3.10) is bounded on k by the number 2
s u P ( p - " ) - r K. *>l P
2
By the Theorem in Sec. 10.1 the set of functions (3.11) is compact in L {Q ) which is not possible owing to their orthogonality. The contradiction just obtained proves that V ( [ z | ) —* +00 when | z | —> 00. P
p
p
176
p-Aiic
Analytit
and Mathematical
Phyncs
Similarly i t is proved that Pfjflp) —• +00, \f\ —* 00. By supposing that it is not the case for a corresponding sequence of integers pt —* +00, k —* 00 we choose eigen-functions p
K..i,i(*)>
*= 1,V..}
(3.12)
similar to (3.11). For n > 1 the proof is similar. As an orthonormal system of functions one must take products of functions (3.11) and (3.12) with distinct arguments.
•
a
4. Operator
D, a >0 in B
r
Let us calculate in explicit form all eigen-values and eigen-functions of the operator D in the disc B C Q , r G 2 . By the Main Theorem of Sec. 10.3 the spectrum of the operator D in the disc B is discrete and the eigen-functions form an orthonormal basis in L (B ). According to the definition (see Sec. 10.3) those eigen-functions of the operator D° in Q whose supports are contained in B are eigenfunctions of the operator D" in B . Eigen-functions of the operator D° in Q have been calculated in Sec. 9.5. a
T
p
a
T
2
r
p
r
r
p
Beforehand we prove the following L e m m a . Let a function f(\x\ ),
x G Q be such that
p
£
P "
7
<
p
p - l / ( p - * ) l < 00 T
£
W ) I < ° ° .
0<-f
0<-»
for some a > 0. TAen
+
1 1
kip" -
1
/
/(Mp)<& +
\»\,<\'\,
/
a
1
l»l - - /(l»lp)^ p
\»W>\*\ (4.1) r
/n parricaior, /or / ( | x | ) - Sl(p- \x\ ), p
p
r £ 2 (4.2)
Pseudo-Differential
forf{\x\ )
=
p
Operators on (As Field of p-Adic
Numbers
6{\x\ -f),r£t p
a
(D f){x)
=
P
Q p
P
Q
2
a ( 1
ti _~, p
a
• By the definition of the operator D
|
r,(-a)
r )
" .
* € S.
(4.3)
r
(see Sec. 9.1) we have
/(|y| )rfy+
[Bir-VCIsUdff
/
P
lsl#.
ISIP>1*!,
/(I«IP) p
d
j
r (- ) Q
y+
J
lsl <M,
Iflp""
(4.4)
1
W»>l*lf
P
But denoting | z | by p ^ we have p
1
I
a_1
r.(-«)
I
dy+
ls|«
r (-«)
-a-l AT-l p
1
W '
lwl,>kl. £
+
P
p
-C«+l)v(l-i
JV+1<7<M
1 " M
1
1
^
L PP
V
a
p y i - p p -
= Nw
177
i_p-o-i
Lp ' V
p/
i - p - °
= Fl. 1
P
P
P °
+
From here and (4.4) the equality (4.1) follows. The equality (4.2) follows simply from (4.4): if \x\ < p then T
p
h>l,
-o-l
l-p-°
p ^
1
- !
1
- I
17S
p-Adic Analysis and Mathematical
Physics r
The equality (4.3) follows from (4.1) i f \x\ — p .
•
p
Let p ^ 2. The I kind eigen-functions r vanish in B . The I I kind eigen-functions r are equal to 1 in B . But by the Lemma (see (4.2)) V'o(x) E 1, i £ B , is the eigen-function of the operator D in B corresponding to the eigen-value N
£
r
r
N
0
r
r
a
T
= z£h«*'-* (The equality (4.2) for VJ (X)
im
= 1 and x e B
0
r
takes the form (D°ib )(x) 0
=
As the eigen-functions
c
p
2
p
r
r
Hence we have just obtained the following eigen-values and orthonormal basis of eigen-functions of the operator D", a > 0 in B (p / 2): r
X
1
0
2
° = p^-rTP^ " '
h =P
A < 1
"
R >
.
1 1
K
I
N
= P^ ' 1&-rUfch
D
multiplicity 1; J = 1. 2 , . . . ,p — 1,
multiplicity p— 1; A pO<*-'>, Ikind i =
<&, ,m> riitt
i = 2,3,... I I kind p i _ , j , o ( ^ ) .
2
2
P - 1 -
,k,£i,
J — 1.2,... ,p — 1,
r
e = 0
j» = M
multiplicity
(P-1) + (P-1) P+--- + ( P - 1 ) V "
2
, :
1
+ P - 1 = ( P - 1 ) P - , fc=2,3
Let p = 2. Supports of the I kind eigen-functions r they vanish in B . Supports of the I I kind eigen-functions f , 1 - N < r for j = 0 or N
T
N
j
a
tf
l
Pseudo-Differential
Operators on the Field of p-Adie Numbers
179
2 - N < r for j = 1 (see (5-6") of Sec. 9.5) are also contained in B . (Note that the support of the function r , and f _, {x) = 0 in B i f 2 - N > r. By the Lemma ipQ = 1 is an eigen-function of the operator D" in S corresponding to the eigen-value A (see (4.3)); AQ is a simple eigen-value. Hence we have just obtained the following eigen-values and orthonormal basis of eigen-functions of the operator D , a > 0 in B (p = 2): r
l
N
0
r
N
fi
r
r
0
a
r
X
r
° ~ « + i _ l ' Votx) = 2 > ,
multiplicity 1;
2
A) = 2
o ( 1
a
r )
- ,
3
I I kind
multiplicity l ;
rAQ
r
A = 2 < - >,
I I kind
A = 2«C*-'>,
Ikind^_ - ,(z),
2
2
k
j = 0,1,
rJi0
r j
multiplicity 2;
/=2,3,...,i-l,
E
2
I I kind V j t - r j . o f ) '
J = 0.
1
J=0,i,«,
multiplicity
3
_ l
2(1 + 2 + . . . + 2 * - ) + 2 = 2 * , fc = 2 , 3 , . . . .
5. Operator
a
D,
a > 0 in S
r
Like in the case of the disc B we calculate in explicit form all eigenvalues and eigen-functions of the operator D° on the circle S C Q , r £ l Let p / 2. The I kind eigen-functions ^ • (see (5.16') of Sec. 9.5) with the indices I — N = r have their supports in S ; the others vanish outside S . As for the I I kind eigen-functions (see (5.16") of Sec. 9.5) their non-zero traces on S are the functions r
r
P
(
r
T
r
l,v ( ) j X
r i
= x Up - *), P
3 = 1,2,...,p-1,
x€S .
(5.1)
r
The functions (5.1) are linearly dependent on S as r
1<J
l<j
-
£ exp f 27ri-x J = — 1 , i<j
x £ S. r
P
The remaining functions { t / j , j = 1,2,... , p — 1} are linearly independent
180
p-Adic Analyiit
and Mathematical
Phytic!
on S as (see (3.2) of Sec. 4.3) r
[«/,«*)=
r
/
1
Xp(U-k)p ~ z)d
X
Sr (5.2) 1
l -IT*- ,
i**.
and thus p-1 - „(r-l)(p-l) - 1
det I
-1
-1 p-1
-1 -1
-1
p-1
=
p
r -r-l P
^
0
As a basis of the functions (5.1) we choose the functions & = 1 ,
0J = X p ( j P
r - 1
r
1
»)-Xp((j + l)p - *).
j = l , 2 , . . . , p - 2 . (5.3)
By the Lemma of Sec. 10.4 the function V'o(x) = 1 is an eigen-function of the operator D" corresponding to the eigen-value (see (4.3)) (5.4) The functions i/ij, jf = 1 , 2 , . . . ,p — 2 are proportional on S respectively to the functions T
Pi-r4l») = ^[vi-r,y.o(*)-¥'i-rj+i,o(*)].
J = 1,2,... , p - 2
(5.5)
(see (5.16") of Sec. 9.5) and therefore they are the eigen-functions of the operator D" corresponding to the eigen-value Ai = p " < ) . Now we sum up the results. For p ^ 2 the operator D" in S , r € 7L has the following eigen-values and normed basis of eigen-functions: 1 - r
r
2
\
P°+P- c.<W, n
=P
p-1
, multiplicity 1;
j = 1 , 2 , . . . ,p — 2, multiplicity p — 2; 0
f c
r
A =p ( - ', t
dU^.fr),
J = 1,2,--. . p - l . e * .
Pseudo-Differential
Operators
multiplicity (p - 1) p
on the Field of p-Adic Numberi
,
181
k = 2,3,.. - .
The listed eigen-functions are mutually orthogonal except the functions j = 1,2,... ,p - 2} for which owing to (5.3)
{>f>\- j, T
J
-1/2
if j = * + l , o r * = ; + l ,
For p = 2 the eigen-values and the orthonormal basis of the eigenfunctions of the operator D° in S are: r
Q
A
2
2 < -"' ° ~ 2 + - 1' a
A, - 2
1
0 , ( 2
r
- ',
X = "(*+i-O t
=
2
^_
r l 0
3
(a;),
^_i-r,i.
t
- 1
multiplicity 2 * ,
6. Operator
J-E •
2
D" + V(\x\ ),
m u l t
'P
l l c l t
y !;
multiplicity 1; e t
W,
3=
1.2,^
it = 2 , 3 , . . . .
a > 0 in Q , p £ 1
p
p
a
We consider the Schrodinger-type operator A = D 4- V ( | x | ) (see example 1 Sec. 10.1) by assumptions that the potential V"(|x| ) is a finite bounded from below function and V ( | r | ) —• +00, |a:| —• 00. The operator A is bounded from below and selfadjoint if its domain of definition V{A) consists of those functions i/> from £ ( Q ) for which D ip + V(\x\ )yj £ L (Q ). The operator A satisfies the conditions of the Main Theorem of Sec. 10.3. In particular, its spectrum is discrete. Let Ao, A i , A j , . . . be its eigen-values, AQ < Ai < A < . . . , ipa,
p
p
p
2
a
P
2
p
P
2
M
8 5
s»eP{A),
* =0,1,2,....
!
All I kind eigen-functions ip • (see (5.16') of Sec. 9.5) are the eigenfunctions of the operator A corresponding to the eigen-values N
X'
aN
N
l N
=P
The multiplicity of \ '
c
+ V( - ),
1 = 2,3,... ,
P
N
is no less than
( P - I ) T ^ y
NeZ.
(6.1)
182
p-Adic Analysis and Mathematical
Physics
where (x,y) are (different) solutions of the Diophantine equation 1
p « + Vfp"- ) =
o
N
N
+ V( '- ),
P
2,3
P
x £ 2.
(6.2)
This number is finite by the Main Theorem of Sec. 10.3. Further the I I kind eigen-functions (see (5.16") of Sec. 9.5) and (5.5))
rV- 1
, z
0
j = l,2,...p-2
(6.3)
are the eigen-functions of the operator J4 corresponding to the eigen-values Aj =p
a J V
i f
1
A r
+ V(p - ),
iVeZ.
(6.4)
The problem is how to find deficient eigen-values and eigen-functions of the operator A? As it follows from the results of Sec. 10.5 the subspace spanned on those eigen-functions { y > w '
= 2,3,... ,
j=
1,2,... , p - l , € i ;
j = 1,2,... , p - 2 ;
N
N
£Z)
which supports are contained in the circle S , r £ 7L: r
'-2,3
Wlrj,,,, Vl-rj,
i=l,2,...,p-l,e,;
J = 1.2
p-2} 2
has codimension 1 and it is orthogonal to 1 in L ( S ) . In the other words every function tp from L ( Q ) which is orthogonal to all eigen-functions (6.5) for all r £ Z is a constant on every circles S , i.e. it is expanded in the orthogonal canonical basis { V ? ( j | — p ) , y £ 2 } in Ll(Q ) r
2
p
r
T
p
£ -00<-J<00
V ^ M P - P
p
7
} ,
^
7
= V>(P ).
(6.6)
Pseudo-Differentia!
on iht Field 0/ p-Adic Numbers
Operators
183
2
The set of functions ip from L (Q ) of the form (6.6) forms the Hilbert space of LQ(Q ) which is a (closed) subspace of L (Q ). I t is isomorphic to the Hilbert space of sequences $ = {ip , with the norm P
2
P
P
y
\\nl=
£
P W ,
(6-7)
-oo<7
besides
7. Operator
D
a
+ V(\x\ ),
a > 0 in Z $ ( % ) (p ^ 2)
p
At first we prove the following statement: The space £o(Qp) is invariant with respect to the Fourier-transform eration.
op-
• I t follows from the formula (3.3) Sec. 4.3: The Fourier-transform of functions from V(Q ) depending only on | i | (dense set in £ Q ( Q ) ) is a function of the same class. As the subspace L\{Ql )\& closed in L (Q ) and the Fourier-transform operation is continuous in L ( ) (see Sec. 7.4) then this statement is valid for all functions from L (Q ). • p
p
P
2
P
p
2
P
2
p
From the proved statement it follows that the operator A maps T>{A) D ^o(Up) where V(A) is the domain of definition of the operator A (see Sec. 10.6) into l o ( Q p ) (as its symbol |{ |£ + V(\x\ ) depends only on | £ | and \x\ ). p
p
p
Denote by Ao the restriction of the operator A on subspace LQ(Q ). Its domain of definition is V(A ) ~ V(A)nLl(Q ). By the Lemma in Sec. 10.4 the operator AQ on functions ip from P(AQ) has the form P
D
a
(A iP)(\x\ ) a
p
= (D iP){\*\p)
p
+ V(\x\ )i>(\x\ ) p
p
(7.1)
where D" is the positive integral operator
a
(D iP)(\x\ ) p
= j
fC(\x\ ,\y\rH(\y\p)
(7-2)
184
p-Adic Analysis
and Mathematical
Physics
with the real symmetric kernel r;\- )t-<"-\
T
a
(7.3) ( -1)(1-P — » ) '
In terms of sequences
T
•
P
_
T
-
= { V v 7 6 2 } the operator -4o takes the form (7.4)
-03<7'<«l P
r„(- )
P
'
a
c
a
p (p +p-2) pc+i _ i P
'
(7.5)
y = 7,
P-1 7
'>7 + l.
The matrix K^i is evidently symmetrizable. Our goal is to investigate the spectral properties of the operator Ac. The operator A is bounded from below and selfadjoint with the domain of definition V(A ) = V{A) n L Q ( Q ) . I t satisfies the conditions of the Theorem in Sec. 10.1 with H — Z,jj(Q ). In particular, its spectrum is discrete, i.e. it consists of eigen-values {p^, Jt — 0 , 1 , . . . } : 0
0
p
p
r»0 < P i < r*2 < • • • ,
- * OO,
P-k
k
—
OO,
every p,k is of finite multiplicity «$; corresponding eigen-functions Ao4>k =
0* £ T3 {A ), 0
k = 0,1,...
(|x | ) , p
(7.6)
form an orthonormal basis in i j ^ Q p ) . 8. T/te Lowest Eigenvalue Here we shall prove that Schrodinger operator
\
0
the lowest eigenvalue A
A = D° + V(\x\ ), p
0
of the
p-adic
c>0
is simple, Ao = Po, satisfies the estimates 1
inf V t p ) < A < inf ( 1 - -rez -res V p 0
- oo <7'<7
(8.1)
Pseudo-Differential
and corresponding
Operators on the Field of p-Adic
(unique) eigenfunction ipo(\x\ )
Numbers
185
is positive.
P
• For the lowest eigenvalue Ao of the operator A the following variational principles valid (see Sec. 10.1 and 10.3): ,
a
. ,
X o
=
(D ip,iP)
*$U
+
(ViP,ip) =
p p
(
j
W
o
)
a
= (D
Q
(8.2)
0
where tpo(x) is any real eigenfunction of the operator A belonging to AQ; the kernel K ( | z | , \y\ ) of the integral operator D posses the property: K(t,r) < 0 for r jS r and K(t,t) > 0 (see (7.5)). A t first we prove that 0, x £ Q . a
p
p
p
• Let, conversely, there exist bounded vicinities U(x') and V(x") of points x' and x" such that 0, x £ U(x') and
0
0
0
(ib (x)
=
N
0
N -
- A- (x)],
N
N
+00}
fromX>(Q ) C T)(A)' w i t h supports in Bjv\2?_w, SO that —t (p , N —». 00 in L . From here and from the closure of the quadratic form A(ip, ip) (See Sec. 10.1) and from (8.2) i t follows p
B
2
. A(ip .ipN) 'nf ' c
- (Atp ,i>N) = i n f ...
N
r
| | 2
v
N
Further for sufficiently big iV, V(x')
/
. = (A
| | 2
U U(x")
0>
C B \BN
£ ( k | , l v l p ) [ ^ ( x ) ^ ( y ) - \M*)\
0
0
„, 8.3
and hence
N
rWMft
p
f a
* ^
8 4
(-)
l«l,*l»lp where n > 0 which is equal to i) = - 4
J
J
K(\x\ ,\y\p)<po(\x\ )\
p
0
p
V(x')V(z") * Every eigen-function of the operator A is a locally constant function in Qp\{0) as the kernel of the operator D" depends only on |x| and |y| . p
p
186
p-Adic Analysis
and Mathematical
Physics
From (8.4) we conclude that
j
(AU \,UM)= If
+ j
£(1*U?/l )liM*)l + j
7
KZ(\ \ ,\x\ M (*)\ dx x p
P
N
ft, <
J
® p
p
2
\4>N\)
N
llllMIP
N
(Aji ,^ ) N
-
\x\ )ip (xfdx p
N
(Aii< ,i> )-n. N
Hence, taking into account that | | | ^ « | | | — \\i>tf\\ (A\tb \,
X
+ J K(\x\„
=
p
N
r
N
+ Jv(\x\ )vjx(x) dx-T,
y
V(\x\ )\tp ( )\*dx
£{\x\ \y\ W„(x)j, (y)dxdy pi
\*N(y)\dxd
P
w
e
derive inequality
- v
N
IhMP
'
1
which owing to (8.3) contradicts to the variational principle (8.2).
•
Similarly one can prove that any eigen-function ^ o ( l ' l p ) of the operator Ao corresponding to the lowest eigen-value /i is non-negative in Q . 0
Now we shall prove that ^ o ( | x L ) > 0 , » £
p
Q. p
7
• Let, conversely, there exists ) £ Z such that V'ofp ) = 0. Then by integrating the equation A ipo = D°ip a
+ Vipr, = Hoyj
0
0
on the circumference S and denoting y
7
V- = 6 ( H - p ) € 2 > ( Q ) P
P
we obtain the contradiction a
(AoV-o, 0 ) = ($a,Aaib} = (ii> ,D i>) 0
+ (yp , V$) = P-oW-o, V>) = 0 a
Pseudo-Differential
Operators on the Field of p-Adic
Numbers
187
as
(T/>D,V» =
0,
(ii> ,D°yj)=
(l-p-)p
a
7
7
/
£(l*lp,P )^(k|p)dx<0.
From here it follows that pr, is a simple eigen-value.
• I n fact i f another linear-in dependent eigenfunction tp corresponding to fir, would exist then what we have just proved is ^ ( | i | ) > 0, x € Qp- I t is possible to construct a linear combination of the eigenfunctions Vo and ip which takes both positive and negative values which contradicts to the positiveness of every such eigen-functions, I p
From the variation principle (8.2) But AQ < po is not valid otherwise J / J ( | I | ) > 0 would be orthogonal in A = up and
P
0
it follows evidently that Ao < /iQ. the eigen-functions ipo{x) > 0 and L (Q ) which is impossible. Hence in fact on \x\ . • 2
p
p
The estimates (8.1) follow from the variational principle (8.2). The lower bound follows evidently
ffl
Ao = (D Vo,^o) + ( ^ o ^ ) > 1
/ v ( | * | ) i & ( | x | p ) < f e > inf P
0
ftp
To obtain the upper bound we put in (8.2)
$=
A (x) y
=
Si(p-'*\x\ )eV(Q ). p
p
As 2
||A^|| = \ P
A ( 0 = p^iry K|p) 7
(see Sec. 7.2)
V(\x\ ). p
188
p-Adic Analysis
and Mathematical
Physics
and also ipo ^ A , then 1
Ao < inf p - [ ( D " A , A ) + ( V A , A , ) 7
7
1
7CZ 7
= inf p
= inf
T1
P
•Y€Z
(|e£A £ )
/
7
V(\ \ )d X p
l^+fl-J-V
= inf 1 - iez V P
9. Operator
+ /
7
(
£
3
7
7
£
K(p-')p '
-oo<7'<7
P "
+
1
)
7
'+P-
T
E
-oo<7'<-7
D" + V{\x\ ),
a > 0 in Q , p ^ 2 ( c o n t i n u a t i o n )
p
P
We sum up the results of Sees. 10.6-10.8 in the following T h e o r e m . The spectrum of the operator D" + V ( | x | ) is discrete and it consists of the following eigen-values and eigenfunctions: The lowest eigenvalue Ao of multiplicity 1 satisfies the estimates (8.1); corresponding eigenfunction is V>o(\x\ ) > 0, Z G Q . Eigen-values pt, k = 1,2,..., pt —* +co, k —* oo ore 0/ multiplicity p
p
p
n* + ( p - 2 K + ( p - l )
2
£
p'?"
2
(9.1)
l
where wt is a number of solutions {Nf, H ^ p ^ {{jV?i'*)i
i = 1,2,... , « t } of the equation
+ Vf^-^),
NEZ;
(9.2)
i = 1,2,... ,6*} are all solutions of the equation U k
=p
aN
N
+ V( '- ), P
NtZ,
I = 2,3,... ;
(9.3)
corresponding eigenfunctions are: Tl> (\x\ ), fj( ),N=Nti l+i
Vl
^
J i £ l
p
X
(i),W = < ,
i = 0 , 1 , . , . , m - 1, J = 1.2,... , p - 2 , i = l , 2 I= | , j = l , 2 , . . . , p - l ,
w
t l
1 = 1 , 2 , , . . ,6*.
Pseudo-Differential
Operators on the Field of p-Adic Numbers
189
Eigenvalues X =p°
N
l N
+ V(p - ) Lp. ,
N
r
JVeZ,
k
k(=Z
(9.4)
+
are of multiplicity (p-2)
+ ( -l)
W
2
£
P
l
where rj
N
is a number of solutions {xf, >> =P
AX
1
(9.5)
i = 1, 2 , . . . , njv) of the equation
1
+ V(p -*),
N
{(x^ ,
p*?-> N
xeZ,
(9.6)
y f ) , i = 1,2,... , c/v} are all solutions of the equation X% = p « + V ( p ' " * ) ,
xEZ,
( = 2,3,...;
(9.7)
comsponrfinjj eigenfunctions are: •PNA{Z), x
VN,,, ,{ )> c
s
N =z ,
j = 1,2,... , p - 2 ,
&=
i = yf,
i = 1,...
j = 1,2,... , p - 1 ,
,n , N
i = l
c. N
Eigenvalues *N = P
ON
N
+ V(P'- )?V<,*XN,
^ £ 2 , i = 2,3,...,
*ez+,
Ar' (9.8) e
S
are of multiplicity (p-i)
2
£ p-r-'-' l
0.9)
N
where { ( z f ' ' , J/f''), i = 1,2,... , djvj} are af/ soijifions o/ (fie eguation ^ = P
Q
I
+ W
r
) -
z
y = 2,3,...;
(9.10)
corresponding eigen-functions are: vVi,c».
* =
/ = y?'',
j = 1,2,... , p - l , 2
i = l , . . . ,
The listed eigenfunctions form a normed bases in Z, ((Q ), they are mutually orthogonal except the functions {
190 p-Adic Analysis and Mathematical Physics R e m a r k . The eigenfunction V'o (l^lp) > 0 which corresponds to the lowest eigenvalue Ao defines the ground state ("vacuum") of the physical system; the "vacuum" is unique. Eigenfunctions which correspond to eigenvalues of multiplicity > 1 define "degenerate" states. 10. Example.
Potential
\x\^, a > 0 ( p ^ 2)
For the potential V ( | r | ) = |z|p" the results of Sec. 10.9 admit some refinements. The estimates (8.1) take the form p
i e
zV
P/
1-P""-
1
7 +
a i
^ inf ( l - - ^ P " ° P iez V Pj 1 - p " " ^
-
1
1
2(1-P" ) 1 - p~ ~ a
l
Now we shall prove the following Lemma. All solutions of the Diophantine equation N
p<* + p« 2
M
= p" + p°y,
M,N, ,y€Z X
2
if a > j - - have the form: J
1
— Inp
either x = N, y = M or y = N, x = M.
• Without loss of generality we may suppose that N = max(N, M, x, y). I f we suppose that N ^ x N ^ y then J: + 1 < N, y + 1 < N and t
2p«(N-l)
=
p«(W-l)
+
c(N-l)
p
> ar* p
+
cy _ cN
p
p
+
cM
p
>
p
cN
Paeudo-Differential
Operaion on the Field of p-Adic Numbera
191
which is impossible for a > jSj*. Hence, either JV = x (and then M = y) or N = y (and then M — z ) . • From the above Lemma it follows that the equation I
oi
\< =p'> -rp »-*K
N,x€Z,
N
1 »€S+
(10.2)
)
has only two solutions x = N, y = I and x = I — N, y = I i f / ^ 2JV and has only one solution i = JV, y = I i f i = 2JV. From here it follows that in the Theorem i n Sec. 10.9 the following cases are possible. I f or ^ jj^ then wt may take only values 0 and 2, in the last case for solutions JV* and Wf of the equation (9.2) the relation JV* — 1 — JV* is valid; 6* may take only values 0 and 1 i f / = 2JV, and values 0 and 2 i f / ^ 2JV, in the last case solutions of the equation (9.3) have the form (JVf,({), ( i f - JV*,/*); n = 2 and for solutions i f and arj' of the equation (9.6) the relation x = 1 — x^ is valid; c?j = 0; ttjv,; may take only value 1 if / = 2N and value 2 i f / ^ 2JV, in the last case solutions of (9.10) have the form fc
N
/ N,l
N,i\
i JV,I
N,i
N,l\
In conclusion we note: i / ^> is an eigenfunction of the operator D" + [arjp corresponding to eigenvalue A fften its Fourier-transform ip is also an eigenfunction which corresponds to the same eigenvalue A so that
£
X
10 3
£®P
( ')
l
by the Fourier transform goes to itself (see Sec. 9.4): a
\Z\Z
•
R e m a r k . For the eigen-functions of the / kind fpNj,t (see (5.16') of Sec. 9.5) the formula (10.3) is simplified
P't-ifj'ty
£
' = ~T7>
J" = - 2 e o i
(mod p)
192
p-Adic Analysis and Mathematical
Phytict
(see (5.12) of Sec. 9.5) i.e. eigenfunctions of the series H' (1 > 2) by the Fourier transform go to eigenfunctions of the series Hj_jv and they belong to the same eigenvalue X' = (cf. Sec. 9.6). N
N
1 1 . Operator
D" + V(\x\ ),
a>0
p
Outside
of a Disc
(p ^ 2)
Let us denote by G, = Q , \ B , _ i = [x € Q : \x\ > p ' ] P
P
the exterior of the disc S j _ j , s £ Z , We consider the operator A = D" + V([z|p) in the set G, under the hypotheses of Sec. 11.6 concerning the potential V ( | x | ) . As in Sec. 11.6 we conclude that the basis of eigen-functions of the operator A in £ ( G j ) consists of those eigen-functions ip'pj j (I > 2) and
2
t l
0
A vj 0
= D i>
k
+ V ( | x | ) ^ - fiki>k,
k
p
G V(A ) 0
C L&G )
(11.1)
t
which correspond to eigen-values p, . Let us try to find in an explicit form { u * , 0t(|x}p), k = 0 , 1 , . . . } . Beforehand we note the following equality which follows from the Lemma in Sec. 10.4 (see also Sec. 10.7) k
a
r
D 6(\x\ -p ) p
=
£
K 6(\xi -p->), ry
x € Q p , r € S ( > 0 ) , (11.2)
p
a
— oo <7 < CO
where or
Ky —
7
o r
y = r,
"P" , -pp-^+ih+r,
r
p°+p-2 *-~F+r=T'>
-PP~ ,
p-1 " - - p T ^ )
7
>
r
(11.3)
i
_(p-l)(p«-l) - p(i- —i) • P
(11.4) Let ^(|x[p) be an eigenfunction of the operator Ac, in G which corresponds to the eigenvalue u provided that ip(p') / 0. We shall see i t i n the form of a formal series in the orthogonal canonical bases in L\(G ) (cf. Sec. 11.6) s
a
*{\*\P)
=
£ s
r
d *(|x|p-p ), r
d,?0.
(11.5)
Plcado-Differenlial
Operator! on the Field o] p-Adic Number!
193
Then owing to (11.2) and (11.4) for i e G , we shall have a
D W\*\ )=
r
£
P
d D"6(\x\ -p ) r
p
I
Y
E * E Ki s[\ \ -p^= i
= i(|z| P
a P
)
22 K\*\ -P ) j
x p
E «
P
2 c;av+ * ( H - p )
7
K
£ j
^
= *(|z| -p') l/C„d,+ £ /C d ] \ j+l
+
r j
7
E ^I^IP-P ) s + l<7<™
E >Cryd + tC^d y,< < -l r
= *(ki -p') (c^+p)p" \
a j
P
-
22
P
(
p- °
r
+ 1 ) 7 + r
r
7
d -
P
J
d
£ j
o
r
P
t
r
d | / r
7
r
P
E p~
Q i
7
'
7
7+l
d ]+ 22 * ( w - p ) y j + l<7
+ <rp- « - p
r
+
y
7+l
/
(11.6)
Denote
22
p-
a
r
(11.7)
d =Kd . r
s
Then owing to (11,3} the equality (11.6) takes the form oV(klp) = (
+
22
(
1
P
-^ -
-P')
PK)d b{\*\p a
\ - P K d , + p
i
i
22
a
- ^ d y -
P
•+ 1<7<« \
p-
>
o
r
7
c , d Wi -p ) / (11-8) 7
r
r
P
where i t is denoted 1 +
r
C , = l-p( ")( -''>. 7
r
(11.9)
I f we substitute the expressions (11.8) and (11.5) in the equation (11.1)
194
p-Adic Analysis and Mathematical Physics
for \x\p > p* we obtain {l
,a
d,(p -'
- K)6(\Z\ P
+
- p')
p
a
22
l-P^ +p^ d + s
J2
r
S+l
+
6(\x\ -p ) P
j<-y
J2
r
P'^C^d-A
r
d v(p )H\x\ -P ) r
= P 22
P
j
/
^(kip-p')
*
i.e. ( L P
- M + P
|
,
_
-
! ) A
r
)
-K
+ V(p>) = p.
P
X + P
r
+ d V(p )
22 s
= pd ,
r
(11.10)
P'^^ydy
r = s + 1, s + 2 , . . .
T
(11.11)
The formula (11.11) gives the recursion relation for determination of the coefficients -pttd.+p
ay
E
p- C , d r y
d =
—
r
y
,
;
d, = l, r = s + l,s + 2 . . . . (11.12)
provided (see notation (6.4)) u
# p d - > + fyry
= x[_ ,
r = s +1,* + 2 , . . . .
r
(11.13)
W i t h the help of equality (11.10) we eliminate from the equalities (11.12) and (11.7) an unknown quantity W=
i
(A p
1 1
-.-/i),
and as a result we obtain the recursion relation
d,
„
\
[p-^l-r
x
\
i, - 1 ,
r
+P
22 .<
= s + 1, 6 + 2 , . . .
7
c^p-^dA, /
(H-14)
Pseudo-Differential
Operators
on ike Field c-J p-Adic Numbers
195
and the transcendental equation a
Y
p- *
V-.-/0-P-",
j + l
d (p) = d r
(11.15)
r
"
for the determination of eigen-values p = pt (provided that the inequalities (11.13) are fulfilled). The formula (11.8) takes the form a
B #([4,)=
22
<*(H- ')
(n.16)
P
j
where
\M-vty)]d„
d! = ^ d r
P
r
- p
j2
p-^dy-pp-^y
21 P
r< <™
7
,
V
J<7
7
(11.17)
These arguments have a formal character as we have not considered here the convergence of the series (11.5), (11.15) and (11.16). Now we sum up
our results in the form of the following
Theorem. Let an eigenvalue /i of the operator A in G satisfies the conditions (11.13) and its corresponding eigen-function \p(\x\ ) satisfies $>(p') ^ 0. Then il>(\x\ ) is represented by the series (11.5) in G,, and coefficients d, = d (ji) satisfy the recursion relation (11.14) and p satisfies the transcendental equation (11.15). Conversely, every (reat) solution ii of the equation (11.15) satisfying the condition (11.13) is an eigen-value of the operator Ao in G,, and the function ib(\x\ ) constructed by the formulas (11-5) and (11.14) is a corresponding eigen-functions which satisfies a
3
p
p
s
P
V-(P') * 0.
12. Justification
of the Method
of Sec. 10.11
Now let us investigate the convergence of the series (11.5), (11.15) and (11.16). To this end we estimate coefficients d for r —* +00. From the formulas (11.9) and (11.13) the estimates follow: r
a
i
l-p- -
< 1,
s <
7
< r - l ,
r
A}_ ~V(p ), r
r - * + c o . (12.1)
196
p-Adic
Analysis and Mathematical
Physics
From here and (11.14) (for fixed fi) the recursion estimate follows
= I , r > s+1. lP
_ A
l-rl
\
s
<
/
7
(12.2) Choose now an integer $a > s such that (i2.3)
^ p - ^ ^ u r ^ ^ i . s <7 0
(The series (12.3) converges as A } _ ~ V(p ) the recursion inequality (12.2) in the form y
7
K|<,
-,\C-VP L rl
-
ai
Y
\
—» +oo, 7 —* +00.) Rewrite
p- \dy\),
r>s
0
+ l
(12.4)
J
la
where
c = |p-Ai_,|+p
Y
P ' ^ K I -
J<7<J -1 0
If we introduce the new sequence r =jd (fi-Ai_4i. r
»•=*«.
r
w+i*—
(12.5)
the recursion inequality (12.4) takes the form
T
1
p'^i/'-AUr !^!,
Y
r
j <
7
, 4o
7
= |d, (/i-Aj_. )| 0
0
1
r>a
0
(12.6)
+ l-
From here we derive the estimate M r, < — - .
r > s, 0
Mi = m a x ( C , T ) . ao
(12.7)
• For inequality (12.7) we prove by induction method with respect to r, This inequality is true for r = s . Supposing it is true for SQ + 1 , . . . , r we 0
Pseudo-Differential
Operators on the Field of p-Adic
Numbers
197
shall prove i t for p + l . Then from (12.6) and taking (12.3) into account we have r.
+ 1
£ sa
- , _ , _ , - . ^ - <
P
l
!
l
M
l
(
1
+
r
^ )
=
i
* _ .
^y^r
From the estimate (12.7) and owing to (12.5) and (12.1) the estimate follows (for some M) l
r
\d \<MV- (p ),
r>s.
r
(12.8)
From this estimate i t follows that the series (11.15) converges. Now we suppose that the potential V ( | z | ) satisfies the condition p
[v-*(\x\ )dx=(l- -) 1
Y
p
Q
/ V - V X ^ -
(12-9)
J
p
Under this condition and owing to (12.8) the series (11.5) converges in Ll(G,) as Y, p |dr|
2
a
fl
Prove now the estimate \d' \< M r
2 P
-
( a + 1 / 2 ) r
,
r>
(12.10)
S
where the numbers d' are defined in (11.17). r
• By using the estimates (12.8) and (12.9) for r > s we have (i
22
r,a
K\
P
7
p-° i^i + ^ "
r<7
22
( a + 1 ) r
7
P I^I
J <7
+ M
22
P
P ^ I ^ ' V ) !
r<-»
+
P
- W M
P
22
7
p i^~V)i
s
< Mi
+p-
r
p
i
-° iv- (
( a + I ) p
./
P r
)i+
22
/
p\
22
P -
H
(
2
O
P
+
7
I
^
H
2
22
/
( P
7
)
P
7
^
2
( P
7
)
198
p-Adic Analysis and Mathematical Physics
for some M i and M . 2
Here we have used the inequality r
r
\V-\p )\<M p- '\
r>s
z
which follows from (12.9).
• r
From the estimate (12.10) it follows that the series
2
c o n _
£ P Kr| fl
A ip = D tb + Vtp € a
Ll(G,).
0
Thus the function ^ ( | i | ) belongs to the domain of definition T>(AQ) (see Sees. 10.6 and 10.7) and hence it is an eigen-function of the operator Ao- We note also that from the condition Vip £ Ll(G,) it follows the convergence of the series p
£
r
2
F
p V (p )\dr\
2
(12.11)
We sum up our results in the form of the following T h e o r e m . Let the potential V(\x\ ) satisfy the condition (12.9), the eigenvalue fi of the operator Ac, in G, satisfy the conditions (11.13) and its corresponding eigenfunction ^ ( | x | ) be such that ip[p') ^ 0. Then the coefficients d , calculated by the recursion relation (11,14), define this eigenfunction by the formula p
p
T
#6|*)=
£
d S(\x\ r
p
-
r
p ),
and the eigenvalue u satisfies the transcendental equation (11.15); in addition the series (12.11) converges. Remark. The open question is whether it is possible to obtain all eigen-functions of the operator AQ in G, by the stated method. 13. Further
Results
on the
Spectrum
of the
Operator
a
D
f V'dxlp)
a > 0 in Q ( p / 2) has been obtained by A. N . Kochubey [123] under the hypothesis that the potential V ( | i | ) is real and locally bounded. He p
p
Paeudo-Differential
Operatora on Ihe Field of p-Adic Numbers
used results of Sec. 9.5 by demonstrations. proofs.
199
Let us list his results without
Let us denote by H i and 7t the Hilbert spaces which span the eigenfunctions of the I and I I kind respectively (see (5.16) of Sec. 9.5) so that £ ( Q p ) = H i © ft , and by A\ and A% the restrictions of the operator A on H i and H2 respectively. (It is proved that subspaces H i and Hs reduce the operator A ) 2
2
2
7
1. / / the sequence { V ( p ) , 7 = 1 , 2 , . . . } has no finite limit points then the spectrum of the operator A is pure discrete and the operator A has a complete system of the eigen-functions. 2
We remind reader that the spectrum of the operator A pure discrete and consists of eigen-values (see Sec. 10.6)
t
x' = N
a P
N
N
+ v( '- ),
Nez,
P
1=
is known, it is
2,3,...
Let N(X) be the distribution function of eigen-values of the operator A i.e. a number of eigen-values (with regard to their multiplicity) smaller than X. 2
2. Let V(\x\ ) ~ C\x\ , \x\ -* 00 where C > 0 and P > 0, and there exists N suck that V(p') / V(p ) if I £ m, I > N, m > N. Then 0
p
p
m
N(X) = ( p - 1)
f i + £ j ln,A
+ 0(l),
A-
+00.
3. / / V ( | x | ) - * 0 , |x|p —00 then p
= {0}. 4. / / V(|x|p) —• 0, \x\p —» 0, then
5. / / 7
£
|V(P )l
(13.1)
-oo<-y<0 then the spectrum of the operator A
2
(and thus the spectrum of the operator
A) is pure singular. E x a m p l e , For the potential V ( | x | ) = sin(ajx| ), a 6 ffi the condition (13.1) is fulfilled, and cr (A ) = [-1,1] for almost all a. p
ESS
2
p
200
p-Adic Analyiis
and Mathematical
14. Non-Stationary
p-Adic
Pkytica
Schrodinger
Equation
Non-stationary p-adic Schrodinger equation with the potential V ( | z | ) with respect to a wave function ip[t,x) has the form p
D ip = -±-Dli>
+
t
(14.1)
V(x)tp.
For V(x) = 0 we get the equation for a free particle 1
(14.2)
A general solution ip of the equation (14.2) in the class of the generalized functions V(Q ) is given by the formula p
(14.3) where 9(ki,ki) is an arbitrary generalized function from T>'(Q ) support in the manifold \k \ = \k /2\ . p
s p
2
with a
p
• Passing on to the Fourier-transform in the equation (14.2) we get the equation I i*ilp from where it follows that supp i> C
= 0
In the case when
the formula (14.3) takes the form X
id,z)
= Jp(*)x
P
DK
(Y*-* ) -
(14.4)
Pseudo-DiJJerential
Operators on the Field of p-Adic Numbers
201
The formula (14.4) can be interpreted as the expansion of the solution yj(t,x) in plane waves Xp
(^i - y
(i4.5)
t kx
(In order to be convinced that the plane waves (14.5) satisfy the equation (14.2) it is sufficient to use the formula (1.7) of Sec. 9.1 for differentiation of Xp(ax).) For plane waves the adehc formula (see (1.11) of Sec. 3.1) II
x
-kx)
P
k,t,xe Q
= 1,
(14.6)
2
is valid. I t connects the plane waves (14.5) with classical ones =exp
-2«
.
(14.7)
For p = 1 the formula (14.4) gives the function propagator 1/3 p)
K.\ (x)
= F
A (t)
(14.8)
Xp
p
which is the kernel of the evolution operator for a free particle (see Sec. 11 below). Owing to the formula (3.8) of Sec. 7.3 the function K[ \X) satisfies the boundary condition (see (3.9) of Sec. 7.3) P
fC["\ ) x
_ * (x), 6
t - » 0 in D'(Qp).
(14.9)
We consider now the classical Schrodinger equation for a free particle
The corresponding propagator is (see (3.2) of Sec. 5.3) exp -2JTI I
= j IC^Hx)—>6{x),
x» ( ^ t t^O
dk
—t-kx
dk = X (t) x
in V{
-
t
'
x
« (
y ) * '{14.11) (14.12)
202
p-Adic Analysis
and Mathematical
Physics
Here i t is denoted (see (3.3) of Sec. 5.3) A „ ( ( ) = exp ( - i ^ s g n / ) ". For the propagators the adelic formula n
( p)
fC (x)
= l,
t
t,x G Q, t ?
(14.13)
0
p=2
is valid. I t follows from the adelic formulas (1.4) of Sec. 1.1, (1.11) of Sec. 3.1 and (4.2) of Sec. 5.4. Some analogy between "phases" Acoff} and X (t) takes place. According to the sgn of t the real time R can be represented as the union of three disjoint sets: two sectors fR (the future sector) where A ( t ) = e **/ i.e. t > 0, K~ (the past sector) where A«,(i) - e ' i.e. t < 0, P
+
-
4
M
{ /4
and the point f = 0 (present). Analogously, the p-adic time Q can be represented in the following way (the definition of A (r) see in Sec. 5). For p = 1 (mod 4) it is the union (disjoint) of two sectors: where A ( i ) = 1, Q where A ( i ) = - 1 , and the point i = 0. For p = 3 (mod 4) i t is the union (disjoint) of three sectors: (J+ where A (() = 1, Qp' where A ( ( ) = ± t , and the point t = 0. For p = 2 it is the union (disjoint) of four sectors: Q where A ( t ) = ± f t / 4 Q±i where X (t) = i e * ' " ' , and the point ( = 0. The Cauchy problem for the equation (14.10) in the domain IR x R with an initial (generalized) function ipo(z) from V'(IR) is posed in the following way: Find a solution ij>(t, x) of the equation (14.10) in the domain (f > x) 6 R x IR which satisfies the initial condition P
p
p
P
p
P
p
2
p
4
c
p
+
+
1
i>(t,x) —> yj (x), 0
( — +0 in D'(ffi).
Similarly, the Cauchy problem for the equation (14.2) in a sector, say, Qp x Qp i initial (generalized) function ^o(x) from V(Q ) is posed in the following way: Find a solution tp(t, x) of the equation (14.10) in the sector (i,x) c Q|J~ x Q which satisfies the initial condition w
t
n
a
n
P
P
tf(t.r)—»V*(«),
* — 0,
*GQ+ inO'(Q ). p
2
" sgnl is the Maslov index for the Hamiltonian p /2 (see[149)).
Pseud a-Differential
Operators on the Field of p-Adic
Numbers
203
The solution of the Cauchy problem for the equation (14.10) is given by the formula
— oo
where £(t,x)
is the fundamental solution of the equation (14.10)
(2*4 -
s(t,x)=mKt°H*)=<>(t)\j\™p
f )
(see [205]). Here <J(() is the Heaviside function, 9(t) = 1, ( > 0, ${t) = 0, i < 0. It is interesting to note that there does no exist any fundamental solution of the equation (14.2) in the space X>'(Qp). • Let p ^ 2 for definiteness, and there exist a solution £ € 7J>'(Qp) of the equation D £ - D\£ = 6{t,x) in Qj. (14.14) t
Passing on to the Fourier-transform in the equation (14.14) and using the formula (1.7) of Sec. 9.1 for the generalized function £ ( f i , f j ) we obtain the inconsistent equation
as the left-hand side of this equation vanishes in the open set
Ki,f )eQ?: |fi[ -]f ||=0, 3
p
3
(6,6)^0].
•
Chapter 3 p-ADIC Q U A N T U M THEORIES
X L p - A d i c Q u a n t u m Mechanics This section discusses the quantum mechanics over p-adic number field. The simplest but most important models — free particle and harmonic oscillator are investigated in detail. It turns out that these simplest models have a remarkably rich structure. Investigation of p-adic quantum mechanics is of great interest from the mathematical point of view as well as from the physical one. As possible physical applications we note a consideration of models with nonarchimedean geometry of space-time at very small distances, and also in a spectral theory of processes in complicated media. Furthermore it seems to us that an extension of the formalism of quantum theory to the field of padic numbers is of great interest even independent of possible new physical applications because it can lead to better understanding of the formalism of usual quantum theory. We hope also that the investigation of p-adic quantum mechanics and field theory will be useful in pure mathematical researches in number theory, representation theory and p-adic analysis. Let us recall here that the quantum mechanical Weyl representation (see below) has wide applications in number theory and representation theory. However from the point of view of field theory it corresponds only to the simplest model of the free noninter acting system. No doubt investigation of p-adic nonlinear interacting systems will provide new deep pure mathematical results. 204
p-Adic
Quan*u7n Theories
205
In the above we have considered functions of p-adic argument with values in the p-adic number field Q and also in complex number field C. Correspondingly, different versions of p-adic classical and quantum mechanics are possible. Below we propose various most natural formulations of p-adic classical and quantum mechanics. We begin from the investigation of the formulation, which is denned by a triple (£2(Qp), U(t)) where W(z) is the Weyl representation of commutation relations and U(t) is a unitary representation of an additive subgroup of p-adic number field which defines dynamics. Quantum mechanics will be obtained by means of quantization of classical p-adic mechanics. p
1. Classical
Mechanics
over Q
p
Let us start with the consideration of the classical p-adic Hamiltonian equations
where all variables: coordinates q = q(t), momentum p = p(t), the Hamiltonian H = H(p,q) and time t take values in Q . We shall consider only analytic functions q(t), p{() and H(p,q). We understand the notion of derivative in the sense of Sec. 2.2. We consider first the simplest case of a free particle with the Hamiltonian p
M here m £ Q , m ^ 0. Hamiltonian's equations p
p = 0,
9= -p; m
p(0)= , P
have a unique analytical solution for t £ Q m=P>
q(Q) = q
p
q(t) = q + ^ - t
(1.3)
Let us also present a solution for the harmonic oscillator with the Hamiltonian 2m
+
2
(1-4)
* We use the same symbol p for the notation of a prime number and for a momentum. We hope that it does not lead to misunderstanding.
206
p-Adic Analysis
and Mathematical
Physics
where m,u £ Q , t n ^ O , The equations of motion p
2
p = -mu> q,
9=£i
p(0) = p,
j(0) = q
have an analytical solution which is analogous to the solution over the field of real numbers _ /Voswt+ jLpsinwA _ i cos ut — qmu sin utj
T (
/Y\ \pj
n
5 )
Here /cos ^ ~ smut \—mwsinwi coswt/* Properties of the functions sinuii and coswt were considered in Sec. 2.4, these functions are defined by the series (4.3) and (4.4) of Sec. 2, which converge in the region G defined by the inequalities p
\wt\ < -
for p ^ 2 and |wt| < j
P
2
for p = 2.
Region G is an additive group: i f t, (' g G , then ( - ( - ( ' £ G . For such ( and f' the matrices T, satisfy the group relation p
p
p
T,T,.=T ,
(1.7)
t+t
On the phase space V = Q x Q we define a skew-symmetric form p
p
B(z,z') = 'q-pq',
(1.8)
P
where z = (q,p) € V, z' = (q',p') £ V. The pair (V,.fl) defines a symplectic space. We then have B(T z,T z') t
t
= B(z,z%
t€G , p
(1.9)
i.e. the dynamics of the oscillator defines a one-parametric group of symplectic automorphisms of the space (V, B). I t is also true for the dynamics of a free particle.
p-Adic
2. The Weyl
Quantum Theories
207
Representation
Here we construct the p-adic quantum mechanics in which states are described by complex-valued wave functions of p-adic arguments. The standard quantum mechanics starts with a representation of the well-known Heisenherg commutation relation
[4,P] =« in the space L?(M). I n the Schrodinger representation the operators q and p are realized by multiplication and differentiation respectively. However i n the p-adic quantum mechanics we have r £ ! ) and 4>(x) £ C and therefore the operator 4>(x) —• xip(x) of multiplication by x has no meaning. Fortunately i n this situation there is a possibility to use the Weyl representation. Recall that in the Weyl representation in the space Z-s(lt) a pair of unitary operators is considered p
c*« : 0 f » -
V ( * + 9);
e* :
i
— c *'^(*).
In this form it is possible to construct the following generalization to the p-adic case. We consider in the space L2(Q ) the unitary operators P
U : ifr(x) - » ^ ( x + g), q
s
t
n
V : tf(x) -
x(2px)i/>(x),
p
e
where g,p, x £ Q and \ ' additive character on Q (see Sec. 3). A family of unitary operators p
p
W{z) = (-qp)U V , X
q
p
z = (s,P)€Qp
(2-1)
satisfies the Weyl relation W(z)W(z')
= (B(z, X
z'))W(z
+ *') .
(2.2)
The operator W(z) acts in the following way W(z)tb(x)
= (2px + pg)V(x + q) . X
(2.3)
The expression (2.3) is conveniently written in the form W{Z)VJ{X)=
f W{z;x,y)Tp{y)d . y
(2.4)
208
p-Adic Analysis and Mathematical
Physics
The family of operators W(z) defines a representation of the HeisenbergWeyl group consisting of elements (z> a); z S V, a € Q with the composition law p
(*,«) • (*'V)
= (z + z',a + a' +
(2-5)
A representation of the Heisenberg-Weyl group is defined by the formula {z,a)
^
(")W(z)
X
.
(2.6)
Note that a pair (i2(Qp), W(z)) where operators W(z) are defined by the formula (2.4) is a special case of the Weyl system (see Sec. 12.7). In the standard quantum mechanics the utilization of the Weyl representation is technically convenient. As we saw from the above discussion in p-adic quantum mechanics the use of the Weyl relation is the most appropriate way for constructing canonical commutation relations. We consider now a question on the description of dynamics in the p-adic quantum mechanics. In the standard quantum mechanics one starts with the quantum Hamiltonian and then one constructs an operator of evolution U(t). Prom our discussion it is clear that in the p-adic quantum mechanics one needs to construct directly a unitary group U(t). I t is understood we shall use a classical p-adic Hamiltonian for heuristic arguments. As it is known the usual quantization procedure is the following. For each function f(q,p) from some class, defined on the phase space, one associates a corresponding operator / on Z>2(]R). This quantization map f —* f has to satisfy some natural conditions. In general, the quantization procedure is ambiguous and different quantizations exist. t
I f the function f(p, q) is the Fourier transform of a function
= j t*^'>rta,(3)d d/i a
=
t
(2.7)
then the Weyl quantization is the construction of the operator
where p and q are the momentum and position operators. Such quantization theory is closely connected to the theory of pseu do-differential operators. This quantization procedure can be generalized to the p-adic case. Let
p-Adic
Quantum
Theories
209
f{l,p) be a complex-valued function on the p-adic phase space V = Qp and from V(Q ). I t can be represented as the Fourier transform p
HP,q)
= J X("P + fiq)
V(Ql).
In analogy with (2.4) to any such function one corresponds an operator in f =J
W(a,L3)
where W(a,0) = W(z) is the unitary operator (2.1). This function f(p, q) is called the symbol of the operator / . Note an essential difference of such quantization of the p-adic theory from the standard real theory. I n the p-adic theory we cannot quantize polynomial functions f(p,q) since such functions take values in Q but not in C. p
In standard quantum mechanics usually one starts with the construction of the Hamiltonian operator and then one proves its seifadjointness. Then one constructs the operator of evolution. In the p-adic quantum mechanics we can proceed in the following way. As it is known i n standard quantum mechanics the symbol U(t) can be given i n terms of the Feynman functional integral. I t is natural to suspect that in the p-adic quantum mechanics the corresponding kernel will be expressed as the functional integral
K (x,y) t
j
=jx(ij
11^(0.
(2-8)
where integration is performed over classical p-adic trajectories with the boundary conditions q(Q) = y, q(t) = x. Here L(q,q) is a classical p-adic t
Lagrangian, L(q,q)
6 Q and h € Q . The integral J Ldt — S(t) in the p
p
0
formula (2.8) is understood as a function which is inverse to the operation of differentiation, i.e. £ S ( ( ) = L , S(0) = 0, S(t) e Qp. Here we consider the simplest case of the free particle and harmonic oscillator. In these cases i t will be shown that as in the standard quantum mechanics the kernel Ki(x, y) ~ x(£ci(0)> Sci(') ' action calculated on the classical p-adic trajectory. w
n
e
r
e
s
t
n
e
210
p-Adic Analysis and Mathematical
3. free
Physics
Particle
We construct the dynamics of the free particle which corresponds to the classical Hamiltonian (1.2) by means of the Fourier transformation. Let i/i be from LsfQp) and ^(k) is its Fourier transformation. As it is known (see Sec. 7.4) the Fourier transformation F : ij> —* i> is an unitary operator in Z^tQp)' The evolution operator in momentum representation U (t) is given by the formula
U{t)m
= x ( ^ ' ) $(*),
f € Qp,
(3.1)
and in x-represent at ion by the formula U(t) = F~^H{t)F.
(3.2)
We get a family of unitary operators U(t), U(t), and the relation is fulfilled U(t)U(f)
= U(t + t'),
M'eQp,
(3.2)
U(t)U(t')
= U(t + t ' ) ,
U'eQp-
(3.3)
Let us calculate the kernel K of the evolution operator x-representation. Let us consider the family of regularized operators t
U (t)j(k)
3 4
= XN (|-')#(*).
N
in
(')
where V
4
m
/
I
0,
|*|p> * P
A sequence of operators f/jv(i) strongly converges to U(t) when N —» oo, lim \\U (t)tJ> - U(t)j>\\ = lim N
j
2
\4>(k)\ dk = 0,
IH,>p" f € Correspondingly, a sequence of operators Ujf(t) U(t).
t GQ . p
(3.5)
strongly converges to
p-Adic
Quantum Theories
211
Using the theorem on the Fourier transformation of the convolution (3.5) of Sec. 7 let us perform the inverse Fourier transformation in the relation (3.4): C j v W O ) = F-WNWFMX)] =
= F-
1
\
[ — t } FVJ
x
F
(3.6) where K ^
= F [
X
n
^ ] ( 0 =
J
x x ( ^ t + kt:)dk.
(3.7)
The integral (3.7) have been calculated in Sec. 7 formula (2.3). Note, that in particular the function K f - i f $ ) has a compact support. Going to the limit JV —>• oo in the formula (3.6) in the space £ 2 ( Q ) we have for r ^ 0 P
U(t)yj(x)
= J K (x t
- y)4>(y)dy,
(3.8)
where integral in (3.3) is singular one converging in L 2 ( Q ) . The kernel K (x — y) has the form, see (3.1) of Sec. 7, P
t
* - » > = ^ i c * ( - f ^ # For t = 0 we have K (x
-y)
0
=
6(x -
y).
i
*
(3.9)
(3.10)
It is interesting to note, that the relation (4.1) of Sec. 5 for the function A (o) follows from the relations (3.3) and (3.9) without using explicit form of the function A (a) from the Legendre symbol. The relation for free evolution of the operator W(z) follows from the formulas: p
p
1
WmW(z)U{t)-
= W(z,)
t
t G Qp,
(3.11)
vhere *t = («(*).P(*)) = (? - * . ! » /) V + m is the classical evolution of free particle.
(3.12)
212
p-Adic Analysis and Mathematical
Physics
As a summary, quantum mechanics of free particle over the field Q is defined by a triple (L2(
P
4. Harmonic
Oscillator
Quantum mechanics of the harmonic oscillator over the field Q is defined by a triple (Li(Q ), W(z), U(t)), where W(z) is a unitary representation of the Heisenb erg-Weyl group (2.5), and U(t) is such operator, that the relations p
p
U(t + ?) = U(t)U(t'), V{t)W(z)U{t)-
1
t,t'€G ,
(4.1)
p
= W{T z),
t<=G 2<=V,
t
(4.2)
PI
are fulfilled, where the classical evolution T z is defined by the formulas (1.5) and (1.6). Below we put m = w = 1. Let us define the operator U(t) on test functions tp £ I ? ( Q ) by the formula: t
p
= j K (x,y)$(y)dy,
U(t)tp(x)
t
where the kernel K (x,y)
t £ G„,
(4.3)
has the form
t
* « M j - w » j ! i * ( - T £ £ + £ ) « K (x,y) 0
<«>
= 6(x - y).
The expression (4.4) has no sense when sint = 0. Note here that sin( = 0 vanishes only for ( = 0. I t follows from the equality (4.19) of Sec. 2, |sint| = | i | , i e G . p
p
p
T h e o r e m . The formula (4.3) defines a unitary continuous representation V(t) of the group G in the Hilbert space L2{Q )The operator U(t) satisfies the relations (4.1) and (4.2) and maps 7?(Q ) into itself. P
P
P
• It follows from the explicit form of the kernel (4.4), from which we have for ( ^ 0 MO MS"'
tant
sini J
p-Adic
Quantum Theories
213
It is easy to see from the last formula that the operator U(t) for t ^ 0 is a composition of four unitary operators and maps V(Q ) into itself. For t = 0 the operator U(0) = I. P
Let us prove the group property (4.1) on the functions from V(Q ) for p > 3. Let jp{z) = 0 for \z\ > p and U(t')vj(y) = 0 for \y\ > p ( t ' ^ 0). Then p
N
M
p
U(t)U(t')vj(x) =
j
K (x,y)
j
t
\y\
tfi(z)
J
K (x,y)K .(y,z)dydz t
/ s
/ K" (ti P
y2
M
t
M,
\*\,
'i
t
N
r
=
K ,{y,z)${z)dydz
1*1,
f
I*I,
+
t i ? )
+
2
y
+
( s n T 7 s i n 7 ) ) ^ -
M
(4.5)
In order to calculate the internal integral in (4.5) we use the formula (2.3) of Sec. 5. Let us introduce the notations
tant
b=—
tant'
2x sin (
+
2z sin f
(4.6)
Then, assuming that t + t' / 0, tan(t + i ' )
t + t'
(1 — tant • t a n ( ' ) t a n t • t a n V
W
(4.7)
As the left hand part in (4.5) does not depend on M for sufficiantly large M (and fixed x, t, (') the number Af can be chosen as large as is wished and hence we use the upper line in the formula (2.3) of Sec. 5. We have also * 2a
M P
Therefore
p
-M = P
x s i n t ' -+- y s i n i t + t'
214
p-Adic Analysis
arid Mathematical
Pkyiici
Thus we have y 2
/ ^(- (ti
+
+
ti?) ^(s^
+
s]nV))
!
= A„(V tant p
X
^(sin(
+
tant'
(+1'
sin(')
(tan*
+
tan(')
)
'
2
Taking into account the relations A (ac ) = A (a) and (4.1) of Sec. 5 we transform the expression for A : p
p
p
» „ ( - i tani
< W tani'/
p
(-. '•»('+'') V (1 - tanr - tant') tan* tan ('
= Ap [ -
1
"
1
= A ( ( ) - A ( t ' ) - A ( ( + C). / (4.9) p
V
p
p
Let us take into account the relation ,2
,11
Z tant' tan (
x tan t
2
1 x Vsint
1
_J_\ sint'/ 2
( \tant
2
r + z ~ ~tan(i-rt')
+
1
tanC
2xz sin(( + V) '
*
'
Assembling together the relations (4.5), (4.8), (4.9) and (4.10) we get
u(t)u{t'Mx) _ A (t-H') / * ("la^+T) If + ^ p
+
s n ^ )
* * * = ^
+
Wit-
Thus the property (4.1) is proved for t, i ' , t + 1 ' ^ 0. The case when one of these parameters vanishes can be proved analogously and simpler because of the relation K (x,y) = 6(x - y). For the case p = 2 one considers analogously. As it can be seen from the above discussion in p-adic quantum mechanics, in contrary to that of the usual one, a wave function can remain finite under 0
p-Adic Quantum Thcoriea 215 evolution, for example a wave function from T>(Q ). One can estimate a diffusion region. In order to check relation (4.2) one calculates the integral p
/^ j n w M . f p W
= W(T z-,x,y), t
(4.11)
where W(z;u,i>) is the kernel (2.4). 5. Lagrangian
Formalism
In Sec. 11.1 the Hamiltonian formalism of p-adic classical mechanics has been discussed. I n this section we consider the Lagrangian formalism. We begin our consideration from the definition of the integral of analytic function and from the discussion of its properties. Let function / : Q —• Q be analytic in the disk B (see Sec. 2) and points a and b belong to the disk B', which is strictly contained in B. The integral of / from a to b is, by the definition, the following p-adic number: t p
p
jf(x)dx
= f^(b)
(5.1)
a where the antiderivative of the function / was defined in Sec. 2.2. Note, that / ( > is defined in a and b by virtue of the condition a, b E B' C B, see Sec. 2.2. The formula (5.1) defines the p-adic valued functional on the set of analytic functions in the disk B. The properties of this functional are given by the following Lemma. - 1
L e m m a . The integral (5.1) has the properties: b
b
1. J(Xf
+ pg)dx = X J fdx + u J gdz,
a c
a b
b
2. J fdx + j a
fdx = j
c b
Z.j
= fg\
A.ueQp-
a
fdx.
a it
f'gdx a
b
* - J
fg'dx.
o
4. If for any analytic function h in the disk B, which satisfies the conditions
216
p-Adic Analytit
and Mathematical
Phytict
h(a) = h(b) = 0 ike following condition is valid:
J fkdx
= 0,
a
then f = 0. • Properties 1 and 2 are direct consequence of the definition (5.1). Property 3 follows directly from the definition (5.1) and from the Leibnitz formula for the derivative of product. Let us prove property 4. Representing / and h as power series we have: /l =
22
M * - « r .
0
I
y/Adz = E C « . 2 = ^ .
M
0
a
where
ft 0
h(b) = {fh)
n
h(a) = 0, Y l
h (b-ar=0, n
= 22 0
Changing summation order in (5.2) and denote d = b — a ^ 0, we have
J t m .
y j
A
/
f
^
i
-
5
< -
3 )
We shall define « „ , n > 2 arbitrary (taking into account the condition | / i „ d | —» 0, n —- oo only), and the coefficient hi by the formula: n
p
h = -
22 2
ft
_1
»^ -
p-Adic
Quanfum The&ritt
217
Substituting the last formula into (5.3) we get:
y
F=f hdx= f
pi-
y
+1
(5.4)
Taking into account the last formula, we shall reformulate the statement 4 of the lemma in the following way. I f for any p-adic number h , n > 2, which satisfy the condition | / i d " | —* 0, n —* oo the relation F = 0 is valid, then ft = 0 for all it = 0 , 1 , . . . . Let us suppose that lemma is not true. Then there exists such m , that / = 0. Let us define h„, n > 0 by the formula: _1 f 1, n = p - m - l , n
n
p
m
M
l
(l~n)d"+
\ 0,
n ^ p
M
- m - l ,
M
where M E N , p > m + 3. Substituting the last expression in (5.4) and taking into account the notation fc = 0 , l , . . . ,
k + 2' we have F = F(M)=
I
Y
C
J
^ t = 0.
(5.5)
It is easy to see by analogy with the discussion of the radius of convergence of antiderivative (see Sec. 2.2), that
\ r(/) = r 0<*
(* + 2)(Jfc-m)
therefore, the series cj
D<J[< OO
converges. As l i m p
M
(5.6)
t -m
= 0, then we have
Af —.oo
lim Af-co
E a
k — m + pM
E 0
Ck
k-
218
p-Adic .Analgia and Mathematical
Physici
for any N € PJ, N > m. By virtue of convergence of the series (5.6) there exists a constant C , which does not depend on M and JV, such that
lim
E
M—co
k- m
0<
< c.
M
+p
Consequently, beginning from some M we have the inequality:
y
(5.7)
0<*<M
M
a
for some C" > C. Choosing M such that p > r ^ | ~ " d taking into account non-Achimedeanness of the norm and inequality (5.7), we have
0
k — m + pM
E
k-m- p
M
+
M
P~ C
n
M
=
p \c \ . m P
r
If we go over to the limit N —* oo and take into account (5.5) we get from the last formula: M
\F(M)\ =p \c \ F
therefore /
m
m p
= 0,
= 0. The contradiction obtained proves the lemma.
Lagrangian formalism for Q can be constructed by analogy to that of for i t . As in Sec. 1 1 . 1 q and ( - coordinate and time variables - take values in Q and q(t) is an analytic function in some disk fl, then (see Sec. 2.2) its derivative q(t) is analytic in B too. We shall consider only the case, when Lagrangian L(q,q) is an analytic function (p-adic-valued) on Q x Q . In this case the value of L{q,q) on the trajectory 9(1) is an analytic function L{i) - L(q(t), q(t)) in the disk B. p
p
p
p
We shall define an action S as a p-adic-valued functional on the set of trajectories by the formula:
1 S[q] = J
L(q(t),q(t))dt,
(5.8)
p-Adic
Quantum Theories
219
where the integral is understood in the sense of (5.1). Let us define the variational derivative of the action (5.8) by analogy to that of the real number case: SS[q] 8q
dS[q + eh] de
t=0
where e € Qp, h(t) is an arbitrary analytic function in B, that satisfies the conditions h(a) = k(b) = 0. The action (5.8) is stationary at the trajectory q(t) i f the following condition is valid: ssi-,}
fi
7
= 0. 7=1
T h e o r e m . If the action (5.8) is stationary at the trajectory q(t), then the Euler-Lagrange equation is satisfied on this trajectory:
dt\dq)
dq
(5.9)
By the condition of the Theorem, we have:
Te
S[q
+ eh\ t=0
Taking into account statement 3 of the lemma, we get:
Substituting the last equation in (5.10), we have: t J {
id)-fx-
By virtue of the statement 4 of the lemma we get the proof of the theorem.
220
p-Adic Analysis and Mathematical
Physics
We shall call the solution of the equation (5.9) with boundary conditions q(ti) = qi, ^((2) = 92 the classical trajectory passing through the points <ji and qi and denote by g i(t). The action Self* 1,^2] on this trajectory can be calculated by the formula: c
& i p i , * 3 ] = / i(gci(£).?d(0)d(. t
6. Feynman
Path
Integral
One of the possible way to construct Feynman path integral has been suggested in Sec. 11 (formula 2.8). In this section a rigorous construction is given by means of finite approximation method and kernel of evolution operator is calculated for the case of p-adic harmonic oscillator. Let us consider p-adic system with Lagrangian L(q, q), which is an analytic function on Q x Q and let the trajectory q(t) be an analytic function in the disk Bp{. We shall choose an integer n < N and construct the covering of Bpj by disks B ( a j ) , j = 0 , 1 , . . . , p ~ — 1 of radius p without common points (see Sec. 1.3, example 2). In every disk of this covering we shall choose one point tj,j = 0 , 1 , . . . ,p ~" — 1, and also we suppose that to = 0 and t N-*_ — (. For any pair (tj, ij+i), j = 0 , 1 , . . . ,p ~ — 2 let us construct the classical trajectory g i (see Sec. 11.5), which satisfies the conditions p
p
N
n
n
n
N
N
p
n
l
c
Qci(tj)
9 d ( i j + i ) = gj+i-
= qj,
The value of the action on every such trajectory is
S c l M j + i] -
J
£(g (i),
(6.1)
el
i=0,l,... ,p"-"-2. 1
Symbol / L ( g ( r ) , q{r))dT
in the formula (2.8) we shall interpret as follows:
0 [L(r)dr=
lim
= Ji*»
O T
T.
22
[ HQCI(T),
5 e 1
q (r))dr ci
(6.2)
p-Adic
Quantum Theoriei
221
The formula (6.2) can be considered as an approximation of an arbitrary trajectory by segments of classical trajectories. Taking into account the formula (6.2) finite approximation Kj"\x, y) of the kernel of evolution operator can be written as follows: *<">(*,«,)
= c
n
f... f p (
£
X
s [ts.i,tj ] d
I dqi
+i
...dq N-*._ , p
2
where q = x, 9 N - » _ I = y, n is the order of approximation, C„ is some normalization factor. If there exists the limit of Kf \x, y) when n - * — oo, then it defines the kernel of evolution operator: 0
p
n
K (x,y)=
lim
t
l n)
K , (x,y).
n—*—oo
In the case of harmonic oscillator Lagrangian has the form:
and Euler-Lag range equation is q + q = 0.
(6.3)
The solution of the equation (6.3) with boundary conditions «(*<) = *.
q(U+i) =
qi i+
as in the case of real numbers is given by the following formula: - g isin(i gjcosfj+i - f l i + i cosU . W = gjsint,-+i :—; ; COS t ;—; ; Sin f, sin(i - U) sin(r - t.) where U,U+i,t € G (see Sec. 11.1). By means of simple, but time consuming calculations, which are analogous to those in the real numbers case one can prove the following equality: i +
—
i + 1
i + 1
p
2feua(t i-it) (+
sin(t
i + 1
-%)
222
p-Adic
Analysis
and Mathematical
Physics
W i t h the help of integrals which have been calculated in Sec. 5.3 we shall prove the following Lemma. L e m m a . The following formula is valid:
n ,0
K(-^i)\ ,1/2
2 tan a,
*P
-2
£
\
dzi...dxt_i
sin
at
0
""i/l
Xp
2 tan V
0
£ a; \o<«"
sin
£ a; \0
wAere it €%> k > 2, Of € Q |a, | < ±, i = 0 , 1 , . . . , k - 1 an
F
p
• The proof will be carried out by means of induction. When k = 2 the formula (6.5) can be reduced to that of (3.1) of Sec. 5 by means of elementary transformations. Let us suppose that the formula (6.5) is valid for k — n and prove its validity for k = n + 1. Denoting the expression in the right-hand part of (6.5) by F(a ,... ,
0
F(a ,ai,...
,a )
0
k
3
_A (-2a ) ' '- - ~ , .T/T" / ^ ( a , - - - , flt-i )Xp - T T p
2
t
:
0
dx . k
By the assumption of the induction we have: F(a ,.. Q
A (-2a ,)A (-2yl„_ ) _ "p .,««) = p
T
P
1
1
|lat i A ^ n -Ji |p 0
/
2
sin j 4 - i n
«0
/
X p
(t
sinAn-i [tan^-i
+
tan<.„
+
+ 2 tan a„ s-i+i
sina
dz„. n
p-Adic
Quantum Theories
223
Taking into account the formula (3.1) of Sec. 5 one can rewrite the last formula in the following form: A (-2« )A (-2A _ )A p
F(a ,...
n
p
l
1
p
( j u ^
+
,a„) =
0
a 4 - l ( t a n i , , - , + tana.) j n J
2 ^ )
1/2
n
P
•x
2tan 4 _i
P
J
n
( *fl |_ ) ^ain.4„_i sin a„ j •> ( 1 , 1 \ I
+ 2tana„
(6.6)
r
ytan Jt„_, ~ tan 9, J J Using the properties of the function A (a) (see Sec. 5.4) we have: p
^
p
\ 2 t a n A „ _ i ~*~ 2 t a n a „ J
= K ( ^ y )
^
An-i) \ ,sin(a„ 2 s i n a „+ sin-4„_i
p
= A (2a„)A (2 4 _ )A (-2A ). p
p
J
n
1
p
n
Analogously, taking into account properties of the functions sin x and cosz (see Sec. 2.4), the expression under the norm symbol can be reduced to the following: 1
+
tan
1 tana
— \A„ |p. n
The further proof can be carried out by means of elementary transformations under the character symbol on the right-hand part of the formula (6.6). • For the case of harmonic oscillator let us choose the factor C following form:
n
V'V"
n fJ
1
°<j
t
n 0<.-
/-
W+l «ltp
£
(67)
JIP
Then the approximation of order n of the kernel K (x,
*?><*,»>=
in the
y) has the form:
224
p-Adic Analysis
and Mathematical
Physio
By virtue of the Lemma we have: »M,
v
M-2t)
/r' + y
3
xy \
.
The formula obtained coincides up to nonessential factors with the kernel of evolution operator of harmonic oscillator, which have been constructed in Sec. 11.4 (see (4.4) of Sec. 11.4). 7. Quantum
Mechanics
with p-Adic
Valued
Functions
In the previous subsections we considered the formalism for p-adic quantum mechanics with complex valued functions. Here we discuss an approach to p-adic quantum mechanics with p-adic valued functions. Recall the second quantization formulation of the usual quantum mechanics in terms of creation and annihilation operators. Let t be the Hilbert space of sequences / = (/o, / i , • • -) of complex numbers with the inner product 2
(/•») =
(7-1)
The creation and annihilation operators a' and a act by rules a'U
=/
n
+
i,
afh = » / f t - i ,
B = 0,1,...
(7.2)
and satisfy the canonical commutation relations ,
m
[a,a ] = aa -a*a
= 1
(7.3)
for some domain in l . Hamiltonian of harmonic oscillator has the form 2
Ho = ua'a
(7.4)
where u is a real number. One can consider a more general Hamiltonian H = H +V 0
(7.5)
where V is for example a polynomial expression with respect to operators a' and o. The time evolution is governed by the Schrodinger equation =
(7.6)
p-Adic
where ip = tp(t) is a vector froin
Quantum
Theories
225
For a self-adjoint operator H one has iiH
t!>(t) =
e- m-
(7.7)
Now note that Eqs. (7.1)—(7.5) can be immediately extended to the case of p-adic quantum mechanics, i f we consider a space of sequences / = ( / o i / i f •) of p-adic numbers with the inner product
=
£
f n
9
(7.8)
n
n=0
where the series converges in Q . A problem here is that the theory of operators in such p-adic Hilbert space is not developed yet to compare i t with the theory of operators in complex Hilbert space. p
We describe now briefly a p-adic integral calculus and its application in quantum mechanics with p-adic valued functions which has been developed by A . Y u . Khrennikov. Let Q p ( \ / r ) be a quadratic extension of the field Qp. Take p > 0 and let A be the space of analytic functions on p
u = P
taking values in Q {^/r) norm P
{xe® -.\x\ < } p
p
P
and endowed with the topology defined by the ||/||p = m a x | / „ | p , n
if
The projective limit of the spaces A„ A = l i m proj A. p—.co
is a non-archimedean Frechet cpace. We denote the dual space A' and will use a notation j
226
p-Adic
Analysis
and Mathematical
Phytico
analytic in a neighborhood of 0. The Gaussian distribution on Q distribution v £ A' having the Laplace transform
p
is a
1 4
L(v)(x) = e ' *''. Let us define an inner product on the space A, (ft 9) = j
f(x)g(x)u(dx).
The completion of A with respect to a corresponding norm is a p-adic Hilbert space L (Qp, v(dx}). A theory of pseu do differential operators in this space has been developed. 2
X I I . Spectral Theory in p-Adic Quantum Mechanics Let us discuss the spectral problem for a p-adic harmonic oscillator. In standard quantum mechanics over real numbers field one studies spectral properties of the Hamiltonian operator. In p-adic quantum mechanics there is no Hamiltonian operator, therefore spectral properties should be expressed in terms of the group U{t). A t first let us consider a harmonic oscillator in standard quantum mechanics and let U(t) be the corresponding operator of evolution, which defines unitary representation of the additive group of real numbers IR. The decomposition of the representation U(t) into irreducible representation has the form £ (K)= a
e
H
(0.1)
m
where invariant subspaces are strained on the Hermite polynomials. The corresponding eigen-function equation has the form r/(i)V =
iw
e -V,
rpeH . a
(0.2)
Here ui„ are known eigen-values for the harmonic oscillator, which one interprets as the energy levels. Analogously, the study of spectral properties of p-adic harmonic oscillator is connected with the problem of decomposition into irreducible representations a unitary representation of a group G. The solution of this problem is divided into the following steps: p
p-Adic Quantum Tkeoriti
describe the characters of the group G
p
227
(see Sec. 3);
calculate the dimension of the invariant subspaces H \ find explicit formulas for the eigen-functions of the evolution operator U(t). ' We shall determine a decomposition, which is analogous to (0.1), (0,2), a
£ (Q ) = 2
P
e H,
(0.3)
a
r7(*)#=x(trf)&
tfei7 . Q
(0.4)
It is well known that in standard quantum mechanics for the harmonic oscillator the invariant subspaces H are one-dimensional, that is there is no degeneration. However, the spectral properties of the p-adic harmonic oscillator are considerably more complex. In particular, for p = 1 (mod 4) both the invariant vector (vacuum) and the excited states exhibit an infinite degeneration. For p = 3 (mod 4) there is a unique vacuum vector, and the excited states are degenerate with multiplicity p + l . For p = 2 there are two vacuum vectors, and the excited states are degenerate with multiplicity 2 or 4. n
In this section we investigate the eigen-functions by making a unitary transformation to a new representation. The case p = 1 (mod 4) admits the most complete analysis. First we list the prerequisites from the harmonic analysis and the theory of operators. 1. Harmonic
Analysis
Let G be a locally compact commutative group. Every irreducible unitary representation of G is one-dimensional, hence the description of the representations reduces to describing the characters of G. A character of G is a complex-valued continuous function \ '• G —<• C with the properties x(ff + 9 ) = x(9)x(g'). = 1. where g and g' are arbitrary elements 1
of G. Equipped with the operation of pointwise multiplication and the topology of uniform convergence on compact subsets, the set of characters becomes a locally compact commutative group, which we will denote by G. The group G is called the dual of G, or the Pontryagin dual. The group G is compact if and only if G is discrete. We have the Pontryagin duality theorem: G = G.
228
p-Adic Analysis and Mathematical Physics
Let U(g) be a continuous unitary representation of G in a Hilbert space H. Then we have the representation U(g) = Jx(9)dE( ),
(1-1)
x
a
where dE(x) is spectral measure on G. Let the group G be compact. On the group G there exists an invariant measure dg (Haar measure). In this case we will normalize the Haar measure by the condition fdg=\.
(1.2)
c The set of characters G = {Xor(
I
d
Xc,(g) 0(9) 9
= $c,p,
X
(1-3)
a where 6 p is the Kronecker symbol and L,2(G) is the Hilbert space of complex-valued functions square-integrable with respect to Haar measure. I n this case, the Hilbert space H can be decomposed in an orthogonal direct sum a
m
H= H, (1.4) where H is the largest subspace on ail which the representation acts as a multiple of Xa(g)- The Hermitian projection operator on H is given by a
a
a
P
a
=j
Xc,(g)U(g)dg,
(1.5)
G
By (1.4) and (1.5), (1.1) takes the form
(1.6)
U(9) = Y^(9)Pc We have U(g)P
a
=
x (9)P . a
a
(1.7)
p-Adic Quantum Theories 229
2. Operator
Theory
Let A be a bounded operator on a Hilbert space H, and orthonormal basis for H. The (race of A is defined by TrA---
{ipr.}™
Y
a
n
(2.1)
l
The trace is not defined for all operators, and, moreover, it may depend on the choice of orthonormal basis. We have the following facts [175,60,237]: 1) Let A be a positive bounded operator on H. Then the sum on the right in (2.1) converges (to a finite or infinite limit) and is independent of the choice of basis. 2) Let A be positive and bounded in H, and let T be a sequence of positive bounded operators converging to the identity operator in the strong topology. Then n
lim
Tr (T AT ) n
= Tr A.
n
3) Let K be compact, and let dx be a positive measure on K. Let A be an integral operator on L (K) with kernel A(x,y) continuous on the compact set K x K. Then the trace of A is well-defined (independent of the choice of basis and finite), and 2
Tr A=
JA{x,x)dx. K
3. The Theorem
about Dimensions
of Invariant
Subspaces
Characters of the group fl have been studied in Sec. 3. Remember that the group G coincides with B-\ for p > 3 and with 5_2 for p = 2. I t follows from Sec. 3.1 that the characters of G have the form x ( a i ) , where a € I . The set I for p > 3 consists of the elements a of the form: 7
p
p
p
p
,
a = 0 or a =p~" (a
()
T
+ a+
where y = 2 , 3 , 4 0 < Otj < p - 1, a p = 2 the set Ii consists of the elements l
T
2
/ 0, j = 0 , 1 , . .. , 7 - 2. For
0
a = 0 or a = 2~' (l + 2 + a 2 ttl
a
... + a _ a p ~ ) ,
lP
2
+ ... + a _ 3 2 7
7 _ 3
),
230
p-Adic Analysis and A/a f A emit leaf Physics
where 7 = 3 , 4 , . . . , 0 < a; < 1, j = 1,2,... ,7 - 3. As noted in Sec. 11.1, the operator P
a
j
1
= \G \P
x(-*t)U(t)dt,
1/4
(3.1) '
v
for p = 2,
is the projection on the subspace H — P H, H — L^iQp), H„ is invariant under U(t), which acts as a multiple of x{ai) on i t . We have the following theorem concerning the dimension of H . a
a
a
T h e o r e m . The invariant subspaces have the following
dimensions:
For p = 1 (mod 4), d.mH = 00 /or all a G I . For p = 3 (mod 4), if a = 0, then dim/fa = 1; if\a\ = p^ with an even 7 > 2, then dim H = p + 1. F o r p = 2, if a = 0 or | a | = 2 , Men d i m / f „ = 2; if\a\ > 2 and Qi - 1, /Aen dim H = 4. For ali other a e 7 , d i m / / = 0. 0
p
p
a
3
4
z
2
a
P
e
The following result will be used in the proof. P r o p o s i t i o n 1. The dimension of the subspace H ,a terms of the trace of the projection operator P by a
£ I
p
is given in
a
dimH
= Tr P „ .
a
(3.2)
It is easy to prove this proposition by choosing a suitable basis in the space H. In calculating the trace of P we will use statements 2) and 3) of Sec. 12.2. Let us define the bounded operator ui in L (Q ) by the formula a
n
2
u„ii>{x) = f l ( p - " | z | ) V ( x ) , p
p
i> € L ( Q ) ,
where „, ,
f 1,
0
2
p
(3.3)
p-Adic Quantum Tkeorici 231 -n
Since fi(p |xlp) — ' *• 1 uniformly on every compact set as n —• oo, we have w„ —* E in the strong topology (E is the identity operator). By statement 2) of Sec. 12.2 we have
Tr P = l i m Tr K P w „ ) . a
By the definition of the operator P
n
0
n
(3.1) we have
a
1
Tr(w F w ) = |G |-
/
p
(3.5)
a
n—-co
* ( - < r t ) Tr ( w f / ( < K ) * -
(3-6)
n
l*l
Tr (u> U(t)u> )= j n ( p - " | x | ) i r ( z , x ) f i ( p - | : | ) d i ,
n
p
n
=
1
t
/
/
iV -
i iv -
1*1,
I*U
p
2
^l 2x tan^)dx. x (
(3.7) V
'
By the formula (2.3) of Sec. 5 we have
1*1,51 We rewrite the last integral in (3.7) in the form (3.8): j
X
U?
tan ~\ih
= r
j
2
x(2p- Vtan0dj/
1 3
=
7n
hp{W\p- '
for
\t\ >p- ,
\ Pn
for |t|„
P
2
Here we have used the relations 3
A ( 2 p " V t a n 0 = *P (^) P
= A (i), p
| sin i |
p
= |t | , | cos t | = 1. p
P
232
p-Adic
Analysis
and
Mathematical
Physics
Therefore, by (3.7) and (3.9) we have
™
1
>
K
=
^ W
(
f
^ \ r
° " ' ^ " < for|t|
(3,0,
2
p
Let us next consider the case p > 3. Substituting (3.10) into (3.6), we obtain for n = 1,2,,,, Tr
(w P u ) n
a
n
\t\ "
J
p
\t\
p
•
'
m
n
We denote the first term in (3.11) by J< and rewrite it in the form j
Jl=P
2
^v(-ap "r)dr.
(3.12)
Since we are interested in the limit as n —* oo, for any fixed a we choose n such that j a p | = p - " | a j < 1. Then 2 n
2
p
p
2
v_(-«p V)=l,
h=P
I
^ d r .
(3.13)
We evaluate J\ as follows:
=* E
/
E
E (|) / *
-oo<7<0
since
(3.14) -0=*
p-Adic Quantum Theories 233 Here e = 1 for p = 1 (mod 4) and e = i for p = 3 (mod 4). Hence p
p
(3.15) * - ' J0
1
- ; ) - *
We denote the second term in (3.11) by h = P
j ,
+I
p-"
m ^x(~a*)dt. •\P
(3.16)
"
Proposition 2. Let n > 1. Then
j
2
=
( p - l)(2n - 1)
for a = 0,p = 1 (mod 4),
(p_l)(2n-AT) - 1
/or ja| = ^ , 2
2 n , p = 1 (mod 4),
p
1—p
for a = 0,p — 3 (mod 4),
(_1)WE*J. _ E ^ i
| j = p , 2 < Af < 2n,p = 3 (mod 4). a
f o r
N
p
2
• We note that the behavior of the function X (t) depends on p. For p = 1 (mod 4) we have X (t) = 1 for all t for p = 3 (mod 4) we have p
xi(t) = (-iy if \t\ = \ In the case of a = 0 and p = 1 (mod 4) we have P
h
P
I
= V
$r
--»
j
-«is*s->
P
= P
£
= P
p
M
*
,V
f l - ; ) = 0>-l)<2n-l).
E
(3-17)
If |a| = p \ 2 < iV < 2n and p = 1 (mod 4) we obtain, using (4.2) of Sec. 4, p
h =P
j
x(-«t)
Wf
l
p->'+ <\*\,
=p
„
E
/i-i/p,
+ 0, 7 > -Af + 2
-Sn + 1<7<-1 \
=p
y<-*f,
-VP. T=
fi
E f i - i V i = o - i ) ( 2 » - i v ) - i . -a»+l
P
J
(3.18)
234
p-Adic Analysis and Mathematical
Physics
I f a — 0 and p = 3 (mod 4), we have
J = P
£
Z
N
Finally, i f | a | = p ,
(-If f l -i ) = l - > ;
(3.19)
2 < N < 2n, p = 3 (mod 4) we have
p
/a = P
/1-1/p, 7<-JV, \ -1/P, 7 = " ^ + l . \ 0, 7 > -N + 2J
£
_ +i< <-i 2 n
=P
7
E
( - i r f i - ^ - t - i ) ^
1
-
(3 20)
The last expression is equal to 1 for even JV and —p for odd JV, Equations (3.17)-(3.20) prove Proposition 2. • Since Tr ( w „ P w „ ) = Ji + 7 a
2l
the next result follows from (3,15) and P r o p o s i t i o n 3. Let a natural number n and a p-adic number a be given such thatp " > |a|p. TAen 2
Tr
(w P u> ) n
a
n
2 n ( p - 1) + 1
fora = D,p = 1 (mod 4),
( p - l ) ( 2 n - J V + 1)
for \a\ = p ,2
1
for a = 0, p = 3 (mod 4),
= • 0
N
p
1
for \a\ — p ",N p
< N < 2n,p=
1 (mod 4),
odd, 2 < N < 2n,
p = 3 (mod 4), p+l
N
for |a|p = p ,N
even ,2 < N < 2n,
p = 3 (mod 4). Taking the limit as n —> oo in Proposition 3 and recalling (3.2) and (3.5), we obtain the proof of Theorem for p £ 2.
p-Adic Quantum Theories 235 We now consider the case p = 2. In place of (4.11) (for p > 3), we have f or p = 2 Tr
{u P w ) n
a
n
l«|a<2-»-
2-'~+><|l|j<4-
1
(3.21)
= Jl + J 2
As above, one proves that Ji = 2. Next, for ghba - Owe have
2-an+i<|,| < -i 3
2
where, recalling (0.2) of Sec. 5 for ( = 2 ' ( 1 +2*1 + . , . ) we have l
\ l ( t ) = ^
+ (-lY i)
2
,1
=
(-l) i-
Therefore,
2-a-+i<| j < -2 1
= 4i
2
2
£ 2^ -2n+l<7<-2
= 0. L
Assume now that |or| = 2
N
2
h = Ai
and
E -2n+l<>J-2
Hla=a'> i i 1 =
< 2n. Then V
Yl
= 4.'
M]=ai i,=o
U
j
i-l) x(-at)dt
m 2i '
(3.23)
where
fh) =
j
l
Hf X(-m)dt
= f (y) B
/,( ),
|*|a=3* fi=
ll
J (-i) x(-*t)dt, (l|j = 3-> '1=1
(3.22)
J = 0,1.
7
236
p-Adic Analysts and Mathematical Physics N
For |tja < 2~ we have | a ( | < 1. Then /<,(?) = fi(j) and f(j) = 0. For |f| = 2 we have {at} = 1/2. Then again f (j) = / I ( T ) and / ( T ) = 0For | ( | = 2~ we have {at} = 1/4+ ( « i + i i ) / 2 . Then 2
w + 1
2
0
N+2
2
+
/o(7)
=
|^(| + f l3]
« P
/
^ ' " ( - i ) "
1
2
7
"
3
-
[Mill
Similarly, = - ( ( - l ) ^ " . Hence for | f | = 2~ , i.e. for 7 = -JV + 2 we have / ( - i V + 2) - 2 - + i ( - l ) " . Finally, for |r| > 2 " , we have / o ( 7 ) = / I ( T ) and f(j) = 0. We have thus proved the following result. 0
2
N+2
2
J V
1
<
i V + 3
2
Proposition 4 . For p = 2
2
~
3
f 0,
a = 0 or \a\ = 2 ,
\ -2(-l)%
|o| = 2 , A ' > 4 .
2
w
2
TAis Proposition completes ike proof of Theorem 3. R e m a r k . By Theorem, the x(at) are eigen-values of the evolution operator U(t) i f and only i f the number a is of the form: _ 1
7
2
a = 0 or a = p ( a ( j + a*p + . . . + a _ 2 p " ) , 7
0
j = 0 , 1 , . . . , 7 - 2,
a ^Q; 0
where 7 = 2, 3 , 4 , 5 , . . . for p = 1 (mod 4), while 7 = 2 , 4 , 6 , . . . for p = 3 (mod 4); for p = 2 we have a = 0 or a = 2 7 = 4,5,...;
_ 7
2
( 1 + 2 + o 2 + ... +
0^.3^-%
2
0,-0,1;
j = 2 , 3 , . . . , 7 - 3.
We denote this set of indices by J \ the numbers in J are analogous to the "energy levels" in standard quantum mechanics. p
4 . Study
of the
p
Eigenfunctions
Up to now we have worked in the space Z^tQp), however, the rather complicated action (4.4) of Sec. 11 of the evolution operator U(t) in this space makes the analysis of the spectrum difficult.
p-Adic Quantum Theories 237
We first consider the case p = 1 (mod 4). The field Q then contains the square root of —1, i.e., there exists an element r £ Q such that r = —I. We will make a unitary transformation to a new representation (called the Cl-representation), i n which the evolution operator acts in a very simple way. This representation will be used to derive explicit expression for the eigen-functions of the evolution operator. For functions / € L (Qp) e introduce the integral operator 3 with Gaussian-type kernel by the formula p
2
p
w
2
3lflt«J = jx
2
(™ - \z
2
+ 2 » ) f(z)dz.
(4.1)
P r o p o s i t i o n 1 . The operator 3 given by (4.1) is unitary in / ^ ( Q p ) , taies T?(Q ) onto itself, and the inversion formula is valid: p
M
= jx
(p-TX
- 2xz) 3[f](x)dx.
(4.2)
Qp
•
This result follows from the expression 2
= X(T* )F [f(z)
X
£2*),
(4.3)
which is a composition of four unitary operators.
•
We say that (4.1) defines the transition to the 3-representation. In the 3-re presentation, the dynamics is described by the theorem.
T h e o r e m 1. Let p = 1 (mod 4). Then, for any function have
U®3[fm
f ,
= ai/(e- )](z), 2
in i ( Q p ) we 2
\t\p < VP-
(4.4)
• First assume that / G 2J(Q ), so that, by Proposition 1, € P(Qp) and suppose that / ( z ) = 0 for | z | > and 3[/](j/) = 0 for j y | > p . P
N
p
p
M
p
238
p-Adic Analysis and Mathematical Physics
By (3.8) and (3.9) of Sec. 11 after interchanging the order of integration we have
/ *((--^*+G&+*)»)**
(4.5)
The formula (2.1) of Sec. 5 can be used to evaluate the inner integral in (4.5). We set ' tan t
'
' sin t
(4.6)
Then 1
T sin t — cos ( sin
t
(4.7)
> 1
for | t | < 1/p. Note that since the left-hand side in (4.5) is independent of M as M —* oo, we can take M arbitrarily large; the top formula in (2.1)of Sec. 5 can then be used to evaluate the integral. We have, furthermore, p
2~a
—
P
P
x + z sin t
M
= p
P
M
(4.8)
\x + Jsinr|p.
rsin t — cos ( N
Since the variable z lies in the disk \z\ < p , we have p ^ ^ s m i L , < 1 for sufficiently large M and \t | < 1/p. Also, the variable x lies in a bounded p
disk, since it appears in the argument (4.8) of the function f i ( — , V I M
which is nonzero only for p~ \x parameters,
2
"
+ z s i n i | < 1. Hence, for these values of p
-
(4.9) 2
We now observe that cosi — r s i n i = c for some c € Qp. Therefore, using (0.3) of Sec. 5 we have
-% H r = ¥ - 4
(4.10)
•p-Adic Quantum Theories 239 Using (4.7)-(4.10)> for these parameter values we thus obtain
/ M
\y\
MzO (j* llpIp '
=
f
x
1
(4.11)
2
+ &i) ™*)
~~
r s i n i
cos(
1
Consequently, (4.5) takes the form
V, tant
sin((cosr — r s i n f ) 2
z sin( + 2zz: cos t — rsin /
•/»k(-l
(4.12)
We now exploit the relation 1
1 sint(cost —
tant
t 2
T (cost + i s i n t ) cosf —
Tsinf)
sin( cosf — r s i n t e
T!
Tsint
— T cost + sint 2(cos( — r s i n t )
r 2
2r
t
= c o s t + r s i n t.
Then (4.5) takes the form U(t)3[f]( ) x
2
T
= ( r x ) J dz(e~ 'z)x X
(-|*
a
+ 2«) ,
(4.13)
Q,
which coincides with (4.4) on functions / 6 P ( Q ) . By continuity, (4.4) extends to the entire space Lj((Jp). • P
By (4.4), the dynamics in the 3-rep resent at ion is thus given by the simple formula / < * } — / € L
2
( Q p ) .
(4.14)
The 3-rep resent at ion may be regarded as a distinctive p-adic analog of the second quantization representation known i n standard quantum mechanics.
240
p-Adic Analysis
and Mathematical
Physics
We construct explicit formulas for all eigen-functions of the evolution operator in 3-re presentation. Let us find, for example, an invariant vector ("vacuum"), i.e., the element \l> £ f-s(Qp) satisfying U(t)i> = i>,
\t\
(4.15)
p
To do this we find the invariant vectors in the 3-re present at ion, i.e., the functions / E / ^ ( Q p ) satisfying T
f(e- 'z)
= f(z),
\t\
(4.16)
p
We note that every nonzero p-adic number z £ (JJ can be uniquely represented in the canonical form z = p c * e , where |a| < 1/p. The dynamics, given by (4.14), reduces to the substitution 7
a
p
z
_ Ve
a
p
r
0
— e-"z
=pVe -",
(4.17)
i.e., the numbers j and k in the canonical expression do not change. This means that any function f(z) in i ( Q p ) depending only on 7 and k is a general solution of (4.16), i.e. an invariant vector. Equivalently, i f z £ Q has the canonical representation z = p (zo + z\p + . . . ) then any function / ( z ) — f[\z\ ,zo) in i 2 ( Q p ) <s an invariant vector, and, conversely, any such vector is of this form. (
2
p
7
p
Let us give the explicit form of the invariant vectors in the original representation. By (4.1), we have *t*)
= j x (™ - I * a
2
+ 2*z) / ( | z | , z ) d z P
0
Qp
=
£
£
-oo<7
fW>W*M>
(4-18)
l<*
where *M*) =
J
2 T
dz {T - -z* X
+
X
Zzz).
\'\,=r-I t can be seen by computation that lM«)
2
= X(r*
+ 2p-'kx)p-'- Q{p'- \x\ ), 1
1
p
7 < 0.
p-Adic Quantum Theories 241
I n particular, i f f{\z\ , z ) = fi(|z| ), 6{p< - \z\„), 7 = 1,2,..., from (4.18) we obtain the vacuum vectors: P
0
lfro(*) = n(l*lp),
P
= xiT^W
ftftj
- kip),
7 = 1,2,...
The dimension of the vacuum subspace is thus seen to be infinite, i n agreement with the theorem of Sec. 12.3. For the excited states, i.e., for vectors tp i n H the equation a
V{t)i>
a
= x{at)^ ,
a € J ,
a
p
a
c ? 0,
\t\ < 1/p, p
(4.19)
reduces to fo(e-"z)
= (c*t)f (z), X
|i|p < 1/p,
a
(4.20)
in the 3-representation tp = Taking into account (4.17), we find that the general solution of (4.20) is a
f (z)
=
a
p
0
(4.21)
a t
where
2
T
7
<M«) = J X [TX - ~z -\-1xz-
7
araj
0
a
where z = p c * e . 5. Weyl
Systems
and Coherent
States
The Weyl representation of commutation relations plays an important role in the formalism of Sees. 11.2-11.4. Here we study the general properties of this representation. In particular, one of the questions is to describe such representations up to unitary equivalence. Let us start with a discussion of geometry of symplectic space. Let V = Op", « > 1 and B is a nondegenerate symplectic form on V, then the pair (V, B) is called the symplectic space over Q . The subspace of V is said to be nondegenerate i f the restriction of the form B to this subspace p
242
p-Adic Analysis and Mathematical Physics
is nondegenerate. I f (V, B) is some symplectic space, then V can be represented as the direct orthogonal sum of its two-dimensional subspaces hi, i = 1,2,... , n (so-called hyperbolic planes): N
V=
®hi.
i=l
In every such plane hi we shall choose a basis (e,-, fi), i — 1,2,... , n with the property: B(e ,f ) = l. i
i
The basis {(ef , / ( ) , i — 1,2,... , n} of the space V satisfies the conditions (e.-,«i) = ( / i , / j ) = 0, = -(fi.e/)
=
i,j = 1 . 2
n
and is called the symplectic basis of V. The matrix of B in this basis has the canonical form: 0 E' \-K 0 Later on we shall suppose that in V some symplectic basis is chosen and the form B has the canonical form. The following inequality is valid:
z, z' £ V and the norm || - || was defined in Sec. 1.7. Let now (V, B) be a two-dimensional symplectic space over Q. A Weyl system over (V, B) is a pair (H, W), where H is a Hilbert space and W is a map from V to the set of unitary operators on H which satisfies the relation (so-called Weyl relation): W{x)W{y) s
= (B(x, Xp
t
n
y))W(x + y),
(5.2)
e
where x, y £ V and Xp(0' additive character of Q that satisfies the condition X ( 0 = 1 i f and only i f \£\ < 1 (see « 3.1) p
P
p
and the map W is continuous in strong topology on the set of unitary operators on H. One can use operations of direct sum and tensor product to construct new Weyl systems from existing ones.
p-Adic Quantum Theories 243
E x a m p l e 1. Weyl system ( L ( Q ) . W) over the space ( V = Q xQ> ,B), where operators W(z), z € V are defined by the formula (2.3) of Sec. 11. 2
P
p
p
ti
E x a m p l e 2. Tensor product ® ( ^ ( Q p ) , W) of n Weyl system from Example 1 is the Weyl system (L (Q ),WW) over the space (Q ,B), where Wl»\z) = &W(z.), z = (z ... ,*„)eQ 2 n . n
2
p
p
lt
Let us denote by VQ the following compact subgroup of the additive group of the space V : Vo = {z€V
: \\x\\ < 1}.
(5.3)
E x a m p l e 3. Let (V, B) be an arbitrary finite-dimensional symplectic space over Q . We shall define the Hilbert space L \ as the following closed subspace of £ ( V ) p
2
L \ ={<(,£ L {V) 2
: 4,(x + x') = p{B{x,x')){x), X
x' € V }, 0
(5.4)
and the set of operators W f z ) , J € V by the formula: W(z)
x
z€V,
i-EL*.
(5.5)
then the pair (L , W) is Weyl system over (V, B). 2
• Unitarity of operators W{z), z £ V is obvious. I t is sufficient to check Weyl relation (5.2). One has W{z)W{z')
= W(z)\x (B(z',x))4(x
- z')}
= (B(z,
x)) (B(z',x
- z))4>(x
= (B(z,
z')) (B(z
P
Xp
Xp
Xp
Xp
= (B(z,z'))W(z Xp
+ z\ x))(x) .
-z'-z) + z'» M
The investigation of Weyl systems over p-adic symplectic space is essentially based on the notion of vacuum vector which does not have apparently a natural analog in real number case. Let us prove the following important theorem.
244
p-Adic Analysis and Mathematical Physics
T h e o r e m 1. For any Weyl system (H, W) over p-adic symplectic space (V,B) there exists a vector £ H such that the following relation Q
W(x)d,
=
0
(5-6}
0
is valid for all x £ Vb- TAis vector a we shall call the vacuum vector of (H,W). • First we note that Vb (as well as V) is a commutative additive group. Let x = (xi,... ,x ), y= ... ,y„) £ V. By using scalar product n
[*»?)=
22 l
X i y i
and notation . . . , -X2n Xi
* = {~X i, n+
t
x) n
the value of the form B on the vectors x, y £ V can be represented in the form: B{x,y)
=
{x,y).
Let now x,y £ Vb, then we have (see (5.1)) \B(x,y)\ <\\x\\-\\y\\
Taking into account the property of the character xpX (t) P
= l,
|£|„<1,
(5.7)
we get that the restriction (H, W ) of the Weyl system (H, W) to VQ (W = W\v ) is a unitary representation of the group VQ in H. Naturally, Weyl relation (5.2) under the restriction on (H, Wo) has the form 0
0
a
W (x)W (y) 0
0
= W {x + j/), 0
x, y £ VQ.
Thus the restriction of Weyl system (H, W ) on Vb is a unitary representation of the compact commutative group V . By the well-known theorem from representation theory any irreducible representation of compact commutative group is one-dimensional (that is be some character of this group). The group of character of Vb (Pontryagin 0
p-Adic Quantum Theories
245
dual, denoted by VQ) is isomorphic to quotient group V/VQ, Vb — V/VQ (see 3.1) and any character of VQ has the form = Xp((c,T)),
^ (x) a
xeV , 0
where a is an arbitrary element in co-set a G V/VQ. By virtue of (5.7) A 3 does not depend on the choice of element 0: in co-set a . By Peter-Weyl theorem the space of representation H can be represented as the direct orthogonal sum H=
H ,
(5.8)
a
where H is a maximal subspace in which the representation is divisible to Aci(i). By virtue of (5.8) we can find a E V/VQ such that H is nontrivial. Let us denote by || • | | H the norm in H and choose tb G H such that \\XI>\\H = 1, then the vector a
a
a
<Pa = W Q « ) # ,
a
6 a G V/V .
(5.9)
0
can be chosen as required vacuum vector. In fact by means of Weyl relation (5.2) and the condition tp E H$ for z G VQ we have W(x)
Qcr)
P
=
Q « J v ( ( a , x ) V > = IV Q a J ^ =
0
•
The next important notion is the notion of the system of coherent states. Let a>o € be a vacuum vector of the Weyl system ( / / , PV). Let us choose an element a from any co-set d £ V/Vb and denote the family of such elements by JQ. Let us construct the following set of vectors $ C H: $=
{
QEJQ).
a
(5.10)
Let ai and a belong to the same co-set a G V/VQ- Then from (5.2), (5.6) and (5.7) we get 2
4> = Witter,
= W(a
at
7
+ (ai - a ))0o 2
= X (-B(Q , a! - a2))W(Q2)Vv (a r
p
2
= Xp(B(a ,a ))W(a )o 1
2
2
=
2
-
ai)
0
(B(c< a ))
Xp
u
2
a
246
p-Adic Analysis and Mathematical Physics
Hence \ip \ does not depend on the choice of the element a in co-set a £ V/Vo. The set of vectors (5.10) we call the system of coherent states of Weyl system (H, W). The main property of coherent states is given by the following theorem. a
T h e o r e m 2. / / Weyl system (H, W) has unique (up to multiplication to a constant) vacuum vector tpc, then the system of coherent states of(H, W) forms an orthonormal basis in H. • Let us prove that the subspace H 2 W(a)ib . In fact, i f x £ Vb we have
a
is strained on the vector 4>a =
0
W{x)
0
= W{x)W{a)r, = x ( 2 B ( x , a))W(a)
= x ( B ( - 2 « , x))<j> , P
a
thus \\H = 1. Then by virtue of the formula (5.9) and uniqueness of the vacuum vector we get: 0
2 a
= W{-a)<j>, therefore
if the following condition is valid: H = ®Hi
and subspaces Hi are invariant under the action of operators W(x), x S.V.
p-Adic Quantum Theories 247
The set of vacuum vectors of the Weyl system (H, W) forms the subspace Ho which is called the vacuum subspace of (H, W). In this subspace HQ we shall choose some orthonormal basis {d>' , i £ I } . The following theorem is valid. 0
T h e o r e m 3. Weyl system ( H , W) is irreducible if and only if the vacuum subspace Ho of this system is one-dimensional. Otherwise (H,W) can be represented as direct sum of the following type: (H,W)=
*m,w%
where subspace Hi is the span of the basis i ^ = W(a)4,
i€l).
• Let Ho be one-dimensional and <j>r, £ HQ is vacuum vector. Let us suppose that (H,W) is reducible. Then there exists a nontrivial subspace H' of the space H which is invariant under the action of the operators W(x), x € V. Let us consider the Weyl system (H\ W). By virtue of Theorem I there exists a vacuum vector <j>o of this Weyl system. By virtue of uniqueness of vacuum vector we can choose such that 0o =
According to the Theorem 2 the set of vectors { = W(a)
D
a 6 Jo)
forms orthonormal basis in H, but d> £ H', hence H' = H. The contradiction obtained proves the irreducibility of the Weyl system (H,W). Let now Ho is not one-dimensional. We shall prove that subspaces Hi and Hj are orthogonal when i / j , i,j £ 7. From definition of these subspaces it follows that it is sufficient to prove the relation: 0
= Q,
(w(x)
*,yev,
M€i,
(5.n)
Let z £ VQ. By virtue of unitarity of W(z) and formulas (5.2) and (5.6) we have: (W(x)<j> ,W(y)4>i) 0
=
(W(z)W(x)d>lW(z)W(y)4)
= ( (2B(z,x))W(x)4lxp(2B(z,y))W(y)dri) Xp
=
X p
(2B(z,x -
K))(JFfc)4,
(5.12)
248
p-Adic Analyst! and Mathematical
Physics
For any x,y € V , x — y $ Vb we can always find 2 € Vb such that X (2B(z x — y)) / 1 and in this case we get (5.11) from (5.12). I n the case of x — y £ Vb and i ^ j we have P
}
(W(x)
= {W[-y)W(x)^,4)
0
= Xp(B(-y,x))(<j> ,4)
= 0.
0
The formula (5.11) is proved. Let us consider now the space H = ©
and prove that H = H. Let
us suppose that it is not true and consider the orthogonal complement H' of H in H. Because of unitarity of operators W(x), x 6 V the space H* is invariant under the action of all these operators. Let us consider the Weyl system ( H ' , W). According to Theorem 1 there exists a vacuum vector
therefore £0 £ B~ which is impossible by virtue of the condition H C H. The contradiction obtained proves the relation 0
0
H = H=
®H . t
i€T
Invariantness of subspaces Hj, i £ I under the action of operators x £ V follows directly from the definition of these subspaces.
W(x), •
As an application of the Theorem 3 we shall prove the irreducibility of Weyl systems from Examples 1-3. For this, according to this Theorem it is sufficient to prove that vacuum subspaces of these systems are onedimensional. • Let tpr, € LiCQp) be the vacuum vector of the Weyl system (L {Q ),W) from Example I . Then it satisfies the relation (see (2.3) of Sec. 11) 2
P
X {2px + pq)M* P
+ 9) = &>(*),
(5-13)
where 2 = (q,p) £ VQ = B x B , x £ Q. I f we put q = 0 in (5.13) we get the formula 0
0
Xp(2px)4> (x) - o(x), 0
p-Adic Quantum. Theories 249 from which it follows that supp d, c {x € Qp : Q
Xp(2pi) = 1. p € So} = Bo.
I f we put p = 0 in (5.13) we get <&a(x) = <J> (x + q),
x,q 6 Bo-
0
Therefore, ^o(^) = Cfi(jz|p), where Q ( | i | ) is the characteristic functions of Bo (see Sec. 6.2), C is an arbitrary nonzero constant and thus the vacuum subspace H of the Weyl system ( L ( Q p ) , W) is one-dimensional. Irreducibility of the Weyl system (Z-2(Qp), W^) follows directly from the irreducibility of ( ^ ( Q p ) , W) and the definition of tensor product of Weyl systems. Vacuum vector of this Weyl system has the form: p
B
n
d> \x) 0
2
= n(\\x\\)
I I fltl*4.
=
* = (*i,•.•,zn)eQp".
l
•
The following Corollary follows directly from the last Theorem. Corollary 1. Any irreducible
1
Weyl system can be represented as ike direct sum of
Weyl systems.
Two Weyl systems (H, W) and (H, W) over space (V, B(-, •)) are equivalent by the definition i f there exists a unitary operator U : H —• H that satisfies the condition UW(x)
= W(x)U,
x e V.
(5.14)
Let us prove one more Corollary from Theorem 3. Corollary 2. Any two irreducible are equivalent.
Weyl systems over Ike space (V, B)
250
p-Adic Analysis and Mathematical Physics
• Let (H, W), {H,W) be irreducible Weyl systems over the space (V,B). By virtue of Theorems 1-3 for any of these systems there exist unique vacuum vector 4>a € H and o £ H and spaces H and H are strained on the basis of coherent states
* =
= W(a)io,
a e
J ). 0
Let us construct a unitary operator U : H —• H by the formula: U , a
aeJo-
a
(5.16)
I t is easy to see that the operator (5.13) satisfies the condition (5.14). In fact, it is sufficient to check (5.14) for basis vectors d\ and
UW(x)d>
a
= VW{x)W{a)
= x (B(x,
a))W(x
P
P
6. Symplectic
+ a)^
P
= x (B(x,
= x (B{x,
x+a
P
a
a))i
0
I+0
+ «)^o = W(x)W(a)}o
=
W{x)U
Group
The symplectic group Sp(2n,Q ) is the group of linear automorphisms of the space V = Q which preserve the symplectic form. Any element g € Sp(2n,Q ) can be represented as a matrix (gij), 1 < i , j < 2n in some basis. Norm of matrix g means the following p
n
p
p
™ « JffoV
(61)
I t is known that deto = 1 for any g € Sp(2n,
p
L e m m a . The set of matrices G = {g € Sp(2n,Q ) : [\g\\ = 1} forms a subgroup o / S p ( 2 n , Q ) . p
p
• Let us prove that for any element g £ Sp(2n,Q ) the following inequality is valid IMl>t (6.2) p
p-Adic Quantum Theories
251
In fact, taking into account the definition of det g and properties of the norm (6.1) we get: 1 = | det g|p <
max
|jf
Ul
... g
2 n i l n
2n
\ <
\\g\\ .
P
Let now g,h £ G . Let us prove that gh 6 G. Taking into account formulas (6.1) and (6.2) we have:
i < M
<
By analogy we get that i f g £ G, then < j 1
< IMI \M = i¬
£ Sit ft JWS, l
F
1 < Us" !! <
max
_ 1
p
6 G:
K - l f ^ J ,
2
1
< H a l l " - = 1,
l
where iV/jj is the complement minor of element gjj in g.
•
R e m a r k 1. The constructed group G is a symplectic group over ring S of p-adic integers, i t is not commutative and is maximal compact subgroup of the group 5 p ( 2 n , S ) . p
p
R e m a r k 2. I n the case of n = 1 the group of matrices {T , t £ G ) constructed in Sec. 11 and which defines the evolution of harmonic oscillator is a subgroup of the group G. Let g £ 5p(2n,Qp), i f V and gx denotes the following element of the space V: t
p
2n
{ax)i
= £gi>Kj. (6.3) ;=i Formula (6.3) defines the action of Sp(2n,Q ) on V. Let us consider an arbitrary irreducible Weyl system (H,W) over the space (V,B). Because 5 p ( 2 n , Q ) acts on V transitively and preserves the symplectic form we get that ( i f , W } W (x) — W(gx) is an irreducible Weyl system over (V, B) too. Therefore, according to Corollary 2 from Theorem 3 of Sec. 12.5 there exists a unitary operator U(g) . H —* H that satisfies the condition p
p
g
V(g)W(x)
= W(gx)U(g),
}
3
g£Sp(2n,%),
x £ V.
(6.4)
252
p-Adic Analytit and Mathematical Physici
It is known (see [225]) that the constructed set of operators {(7(g), 9 € S p ( 2 n , Q ) } forms the projective representation of Sp(2n,Q ) in H, which is a unitary representation on two-fold covering of Sp(2n,Q ) (so-called metaplectic group). We shall be interested in the restriction of this representation to the subgroup G. Remark also that by virtue of irreducibility of (H,W), formula (6.4) defines U(g) uniquely up to a factor. I f n = 1, then the evolution operator U(t), t 6 G of quantum p-adic harmonic oscillator constructed in Sec. 11.4 is the representation of subgroup {7J, ( £ G ] of the group G defined by the formula (6.4) by means of the Weyl system (L (fy ),W) from Example 1 of Sec. 12.5. p
p
B
p
p
2
P
If n > 1 the similar operator can be constructed by means of tensor product of n operators U(t) of one-dimensional oscillator using the Weyl system {L (%), W ) from Example 2 of Sec. 12.5. I n both cases representations defined by the formula (6.4) are unitary (not projective). This is true for the whole group G. In fact, the following Theorem is valid. ( n )
2
T h e o r e m 1. The set of operators {U(g),g 6 G} which satisfies the relation (6.4) for some irreducible Weyl system (H W) over the space {V, B) gives a unitary representation ofG in H. t
• By virtue of equivalence of irreducible Weyl systems it is sufficient to prove the Theorem for arbitrary chosen irreducible Weyl system. We choose the Weyl system (L , W) from Example 3 of Sec. 12.5 as such Weyl system. 2
In the space
we define the set of operators {U(g),g
€ G} by the
formula: 1
u(g)f(z) = fM
6
= /Or *)*
5
< - )
It is easy to see that U(g) is the operator from L / € i j , z' £ Vr,, g £ G the following relation is valid
2
to L .
In fact, i f
2
/ (z + ' ) = / ( r 2 + , r V ) 1
J
2
J
l
l
,
= Xp{B{g- z,g^z'))f(g- z) = {B(z,z ))f {z). Taking into account the definitions of W(z) and U(g) i t is easy to check the relation (6.4) for Weyl system ( £ £ , W) and operators U(g), g 6 G. In fact, i f / e Z-2 then we have: Xp
U(g)W(z)f(x)
= U(g)\x (BU,x))f(x
- z)} = {B{gz,x))f(g-\x
P
l
= W(gz)f(g- x)
Xp
=
W(gz)U(g)f(x).
3
- gz))
p-Adic Quantum Theories 253
Obviously, the set of operators {U(g),g tion of the group G in L \ .
£ G) forms unitary representa•
Study of the Weyl system gives us an opportunity to get some information about the properties of the representation {U(g),g £ G) defined by (6.4). We shall say that vector <pa € H is an eigen-vector of representation W(g),9 G G) i f it satisfies the condition; U(g) , 0
0
where X{g) is a complex number, |A(ij)| — 1. The following theorem is valid. T h e o r e m 2. Let (H, W) he an irreducible Weyl system over (V, B) and [V{g), j £ G ) he unitary representation of group G in H defined by (6.4). Then vacuum vector ipr, of the Weyl system (H, W) is an eigen-vector of representation {U(g), g £ G).
• By the condition, vector 4>o satisfies the relation: lV(z)d> =
z £ V. 0
The following equality is valid because of invariantness VQ under the action of G: W {z)
a
and hence fa is the vacuum vector of irreducible Weyl system (H, W ). On the other hand from (6.4) it follows that U(g)fa, g £ G is vacuum vector of (H, Wg) too. Taking into account formula (6.6) and Theorem 3 of Sec. 12.5 we get the required statement. • g
7. Investigation
of Eigen-Functions
for p = 3 (mod 4)
As it has been noted in Sec. 12, the investigation of spectral properties of harmonic oscillator constructed in Sec. 11.4 is equivalent to that of representation of the group T of matrices of the type r
'=(-sin
(
2\)>
(see (1.6) of Sec. 11),
254
p-Adic Analysis and Mathematical Physics
which is defined by formulas (4.3)-(4.4) of Sec. 11. This problem has been solved completely for the case of p = 1 (mod 4) (see Sec. 12.4), but for the case of p = 3 (mod 4) only the dimensions of eigen-subspaces have been calculated. Analysis of the representation of the group T is closely connected with that of larger group S of matrices of the type: (-
b
«)•
« ^ Q
P
2
,
2
« + * = l.
In the case of p = 3 (mod 4) the group S is compact (see Sec. 1.5) and is the subgroup of 5 p ( 2 , 2 ) . In order to study eigen-functions it is convenient to consider the phase space V = Qp x Q of classical system as quadratic extension of Q : V = Qp(v'--i) (see Sec. 1.4). Using the notation i — any element z £ Q ( V 1) can be uniquely represented in the form z = x + iy, z denotes the conjugate element from Q (\/—1) 2 = i - *!/, t , j £ Q - The group 5 is isomorphic to the subgroup of the multiplicative group Q ( \ / — l ) of Q p ( V 1) of the following type: p
p
p
—
P
P
P
P
—
S=t{zeQ;(V=l):zz
= l}.
Let us also define the function e" : G —* QpfV—T) by the formula: p
e*' — cos t + t sin £,
t £ G . p
This function satisfies the relation: i e
V ' = < '>. [
i
,+,
e
The group T is isomorphic to a subgroup of Qp(\/—T) of the following type: r = { y ' , f eGp}. T is the subgroup of 5 and the following lemma is valid; L e m m a 1. Group S is isomorphic (for p = 3 (mod 4) to the direct product of the cyclic group Z +\ of order p + l and the group T: p
5~Z (see [82]).
p + 1
xT,
(7.1)
p-Adic Quantum Tkcorici 2SS
I Let us give the sketch of the proof. I t follows from the equation a + b = 1, a,b £ Q , that for p = 3 (mod 4) either \a\ = 1, \b\ < 1 or \a\ < 1 |6|p = 1 (see Sec. 1.4). Therefore, taking into account that sin t maps Gp to G in a one-to-one manner, we get that any a + ib £ S can be represented in the form: 2
2
p
p
p
p
p
a + ib = (a + i&o) ' (cost + i s i n t ) = ( a + i d ) e " 0
0
0
for some t £ G and ao and bo which satisfy the relation: p
2
al + b = 1 (mod p).
(7.2)
0
The set of pairs (ao,6o), which satisfy (7.2), is isomorphic to some subgroup of multiplicative group IF* ( \ / ~ 1 ) of quadratic extension of the finite field IFp which is cyclic (as £.ny subgroup of the multiplicative group of the finite field). The order of this group is equal to the number of solutions of (7.2) which can be calculated by means of Gauss sums and equal p + l . • Before the study of eigen-functions let us prove the following lemma, e denotes some element from Q ( > / - l ) with the property ei = —1. p
L e m m a 2. Any element z £ Q ' { ^ — 1 ) (for p = 3 (mod 4)) can be represented in the following form: k
n
z = re c e",
(7.3)
where r = r(z) £ Q* ; k = k(z) = 0 , 1 ; c is the generator of Z i 2
p+
n = n ( ) = 0 , l , . . . ,p; r = 2
=
S/T;
r(z)£G . p
/
2
• Let z £ Qp(v —T)- Only two cases are possible: either zz = a or zz = —a , a G Qp (see Sec. 1.4). In the first case we shall choose r as the square root of zz which belongs to Qp : r = \fz~i. Then $z £ S, k = 0 and (7.3) follows from (7.1). In the second case r = y/~zz £ Q ' and ±z £ S, k = 1. Further proof is evident. • 2
3
2
The representation (7.3) we shall call the polar decomposition of z £ Q ' ( > / ^ T ) . As i t follows from Lemma 2 this polar decomposition defines on
256
p-Adic Analyiii and Mathematical Phyiict
Q p f y ^ T ) four functions: r(z) : %(V=l)
2
—• Q ;
*(z) : QJCv^T) —
P
n&j : Q J t V ^ T ) — • { 0 , 1
{0,1};
p} and r(z) : Q ; ( V = 1 ) — G„.
By the definition these functions have the properties: r ( e " z ) = riz), n ^ ' z ) = «(*),
i
k{e 'z) = k(z), r(e z) = r ( ) + i, it
(7.4)
z
!e«j;(A
ceG . p
Less obvious properties of these functions are given by the following lemma. L e m m a 3. Let Z, Z* € OJfV^-I) the following relations are valid:
||z|| > P, ||z'|| < 1 (z' € V ) . TAen 0
1) |r(z + z ' ) - r ( z ) | < l , 2) k{z + z') = k(z), 3) n{z + z') = n(z). p
• Let z = x + and z' = x' + iy' satisfy the conditions of the Lemma. 1) By means of elementary calculations we get the formula:
T(z + 2') + 7(*) 2
z
Since r e Q 3 then (r{z + z'))o + (r( ))o
m
- 0 (
p
2
K » + ' ) + K*>IP = niax{|r(z + z%, = mzx{\\z + z'nz\\}
o
d
P) and hence \r(z)\ ] p
= \r(z)\ .
(7.5)
p
From equation ||z|| = Mz)|p = max{|x|p, | y | } it follows p
\x\p<\r(z)\ > From (7.5) and (7.6) we have: P
\r(z + z') - r(z)\
p
\y\ <\r(z)\ . P
(7.6)
p
2
< r ^ L - m a x { | r ( z ' ) i , WUmtW
< L
p-Adic Quantum Theories 257 2) From the definition of the function k{z) it follows that i t depends only on the first term in canonical decomposition of p-adic number zz: k{z) =
k((zz ). 0
z
The relation ((z + z ' ) ( + z ' ) ) = (zz) 0
|(- + z'){7T?)
- zz\
p
follows from the equality:
0
= \z'z' + 2(zz' + z'z)\
< \zz\ .
p
3) By virtue of relations ||z|| = | r ( z ) [
p
= m a x { | x | , \y\ } we have that
p
p
p
either | z | = | r ( z ) | , | y | < | r ( z ) | , or | y | = | r ( z ) | , \x] < \r(z)\ . p
p
p
p
p
p
p
p
Let us
consider the first case. By the definition of n(z) and Lemma 1, it follows that i f \y\ < \r(z)\„ then n(z) = h ( ( * ) J , i f |y| = | r ( z ) | , then n(z) = p
n
p
i [y^J
P
Let us consider the second case (the first one can be
considered analogously). By virtue of statement 1) of this lemma, relations \ \P — \y\p = l (*)lp > P X
r
a n
d inequality \x'\
p
< \r(z')\
< 1 we have:
P
u
Denoting by 6t,
n
the Kronel;er symbol on Q ( \ / — T ) we define: P
= *«,*(..)•
n = 0 , 1 , . . . ,p.
From the definition of these functions and properties of functions n(z) and k(z) we have: c
supp 6^ (z)
n supp Sjf = 0,
c
c
6< (z + z') = 6 (z), m
m
if
6^{c z)^6^{z),
i f n j f m;
||z||>p,
||z'||
teGp.
(7.7) (7.8) (7.9)
As it has been noted in Sec. 11.7 each of them is connected with some irreducible Weyl system over the space (V = Q x Q , f l ) . For example in Sees. 11.4 and 12.3 oscillator has been considered in representation which corresponds to the Weyl system from Example 1 of Sec. 12.5. In the case of p = 3 (mod 4) the investigation of eigen-functions is carried out easily in representation which is connected to the Weyl system (L^W) from P
p
258
p-Adic Analytic and Mathematical Phyiict
Example 3 of Sec. 12.5. Remember that in this case the representation space has the form: =
* Qp) • f(* + A = x (B(z,x'))f(x),\\x'\\ P
< 1},
representation U(t) of the group T is denned by the formula: v(t)f(x)
it
= f( - z),
teGpjeif,
e
and representation V(z) of S is denned by the formula: v(z)f{x)
= f(zx),
zeSjeLf.
In this representation the problem of determination of eigen-functions of the oscillator is equivalent to the finding of any a 6 I (see Sec. 12.3) a complete system of linear independent solutions of the following system of equations in Li(y)\ p
f/(e-'M
=v(at)/( ),
I /(* + *')
= (B{z,z'))f{z),
2
1
WW < 1.
X
'
Taking into account the theorem about dimension of invariant subspaces (Sec. 12.3) for the case of p = 3 (mod 4), we have that, i f a G I , \a\ = p , k — 0 , 1 , . . . the system (7.10) has no solutions; i f a = 0, then there exists a unique linear independent solution of (7.10) and in the case of or 6 J there are p + l linear independent solutions. The explicit formulas for eigen-functions are given by the following theorem. p
p
2 i + !
p
T h e o r e m Let p = 3 (mod 4). If a — 0, then any solution of the system (7.10) is proportional to the function
M*) = HMI> If a G J , « / 0, then p + 1 functions p
are linear independent and give us the solutions of (7.10), where m=
1+
\( {j))>
A= V(-l)
m + 1
«.
" = 0,1,...,p.
p-Adic Quantum Theoriei
2S9
• As it can be checked by direct substitution, tj>o{z) is the solution of ( 7 . 1 0 ) . Let now a € J ,
a ^ 0. From (7.7) we have that the functions ( 7 . 1 1 )
p
are orthogonal for different n and hence are linear independent. Besides that, from ( 7 . 9 ) it follows that the functions ( 7 . 1 1 ) satisfy the first equation of the system (7.10). For a G J , a / 0 we have |a| > p and therefore l lp £ P- Taking into account this inequality, substituting functions ( 7 . 1 1 ) to the second equation of the system (7.10) and using the property 1 ) of the function r(z) (see Lemma 3 ) and formula ( 7 . 8 ) we get: 2
p
p
a
K,(m&MH*)
- a\ ) (-c,r(z
+ **))
p Xp
c
= (B(z,z'))6' (z)6 (z)il(\r{z) Xp
m
-
n
a\ )xp(-or(z)). p
Taking into account ( 7 . 7 ) , the last formula is equivalent to the following one: X (aAT
+ B(z,z'))
P
= l,
(7.12) m
where A T = T(Z + z') - r(z), m
n
T
;
z = r(z)£ cV < >,
iT
g> = r(z')e c e ^'\
z + z' = r(z + z')e
m
r
cV <
i + i
'>.
By virtue of the last equations, the expression for B(z, z') can be transformed to the following form: B(z,z')
= B(z,z
+ z')
= r(z)r(z
+
z')(e£) B(e <*\e (* ''))
= r(z)r(z
+ z')(-l)
m
m
iT
iT
+
sin A t .
(7.13)
For further proof the following equality is required: (for p = 3 (mod 4 ) ) ib
\\e"-e \\
= \a-b\ , p
a,i€G .
(7.14)
p
In fact: f l
t
|
| | ' - ' | | = || ''»"- '>-l|| e
e
e
= maxfj cos(a — b) — l | , |sin(a — b)\ ) p
p
= |sin(a - b)\ = \a - b\ . p
p
Taking into account ( 7 . 1 4 ) we can prove the following inequality: I"AT-|
P
< 1.
(7.15)
260
p-Adic Analyiii and Mathematical Phytics
In fact: \aAr\
p
i
=
i
\a\ \\e ^+*->-e ^\\ p
2
fl
r(,+
= IKK* + ') + ( = ||r(z + z ' ) e
i T t
'
+ I , )
l)
+ **)K * - W + ( « - K * J i ) ^ | | iT
- r(z)e <'>|| = ||»'|| < 1.
Taking into account formulas (7.13) and (7.15) and inequalities | r ( j ) — a | < 1, \r(z + z') — a\ < 1, let us rewrite the expression (7.12) in equivalent form: X ( o A T + ( - l ) a s i n A r ) = 1. (7.16) p
p
m
!
P
From the properties of function sins, which have been pointed out in Sec. 2.4, the following inequality (for p = 3 (mod 4)) follows easily: IsinAr-Ar]
p
(7.17)
p
By means of (7.15) and (7.17) the expression (7.17) can be transformed to the form: X ((« + ( - i r V ) A r ) = l . (7.18) p
Thus, the functions (7.11) satisfy the second equation of the system (7.10), if a, a and m satisfy the relation (7.18) for any A r g G . It is easy to see that a and m from the condition of the Theorem satisfy the equality: p
« + ( - l ) " V = 0, from which formula (7.18) follows.
•
The theorem gives us the opportunity to construct eigen-functions of oscillator for p = 3 (mod 4) in an arbitrary representation. Namely, the following corollary is valid. C o r o l l a r y . Let (H, W) be some irreducible Weyl system over the space (Qp x Qp.fl) with vacuum vector^. Then eigen-functions of oscillator in representation (H, W) are given by the formula: V£ = j
m
Y (-«0^(^ c'V<)Vv»\ P
where a G J , n = 0 , 1 , . . . , p, a and m are as in the Theorem. p
(7.19)
p-Adic Quantum Theories 261 I Let us consider the Wey] system {L^tW) with vacuum vector <po = f2(||z||) and corresponding representation for harmonic oscillator. I t is easy to see that the operator S : L \ —* H defined by the formula $4=
J(,W(z)) , W{z)tp dz, L (V)
<&€LX,
n
(7.20)
v is a unitary operator (see Sec. 11.7). Substituting in (7.20) eigen-functions of oscillator in representation (L, W)d>(z), we get: K
= J dzW(z)^
j
0
v
dyr (y)Xp{B(-z,y))il(\\y-z\\) a
v
= j
dzW{z)i,
j
a
V
d 6< (y)6< ( )Sl(\r(y)- \ ) y
m
n y
a p
V
•XP(-<*T(V))XP(B(
z,y))U(
y
*.\).
Since integration i n the last formula is actually carried out on bounded domain ||y|| = ||z|| — |a| , then we can change the integration order. Taking into account the formula p
J dz (B(-z, Xp
y))Si(\\y - z\\)W{z)fa
=
W(y)vj , 0
v we get K
= J
dy6UiyK(yMMy)-^\p)x (-«r( ))W(y)vj . P
y
0
v r
Using the properties of functions b {y), from the last formula we have: m
C = / Q;
\r\ U{\r P
5
- a\ )dr p
j
^(y)
a
n
a
the polar decomposition,
m
{-at)W{ae
Xp
f e ^ d t .
|ilr
After rejection of unessential nonzero coefficient we get the required expression for eigen-functions. R e m a r k 1. By analogy to representation t / ( t ) , f S G of the group T we can construct unitary representation V(g), g £ S of the group S. I n p
262
p-Adic Analysis and Mathematical
Physics
this case, taking into account the theorem and formula (7.1) it is easy to prove, that eigen-subspaces of V(g) are one-dimensional and eigen-functions coincide with eigen-functions $J of the representation U(t) (but in this case the functions d>a different n correspond to different subspaces). R e m a r k 2. Investigation of eigen-functions for p = 3 (mod 4) (that is the solution of the system (7.10)) have been carried out without the theorem about dimensions of invariant =ubspaces by means of direct analysis of the system (7.10). X I I I . W e y l Systems. I n f i n i t e D i m e n s i o n a l Case Let (V, B) be an infinite dimensional symplectic space over
W(x)W(y)
= piB( ,y))W( X
X
x
+ y)
for all x,y £ V and the restriction of W to any U £ F is continuous in strong topology. Let us be reminded that in the case of dim V < oo (see Sec. 12.4) all irreducible Weyl systems are unitary equivalent (Stone-von Neumann uniqueness theorem) and any Weyl system can be represented as a direct orthogonal sum of irreducible ones. But it is not the case if dim V — oo, 1. Weyl
Algebras
Let (H,W) be a Weyl system over (V,B), U £ F and let tm (H,W) denote the fv"*-algebra generated by the set of operators {W(x),x £ U}. As dim U < oo using the Stone-von Neumann uniqueness theorem (see Sec. 12.4) i t is easy to prove the following lemma. u
Lemma 1.
For any two Weyl systems ( f f j , W j ) and (Jfa/Wj) over
(V, B) and any V £ F there exists a unique *-tsomorphism a of the algebras M (Hi,Wi) u
andW. (H ,Wi) u
2
which mapsW,(x) a(W (x)) 1
=
W (x). 2
into W (x) for all x £ U: 2
p-Adic Quantum Theories
263
The Weyl algebra of the Weyl system (H, W) over (V, B) is defined as a C*-algebra ii(H, W) which is the uniform closure of the union of algebras JDlufif, W), when U runs all subspaces from F: \Jm (H,W).
U(H,W)=
u
U£F
As i t was mentioned, we do not have the Stone-von Neumann uniqueness theorem for the case of dim V = oo. But in this case the so-called C*algebraic uniqueness theorem is valid: T h e o r e m 1. For any two Weyl systems ( J Y W i ) and (H W ) over {V, B) there exists a unique ^-isomorphism a of the algebras iX{H\, W\) and ii{H ,W ) which maps W^x) into W (x) for all x £ V: l p
2
2
2l
2
2
c,{Wi(x))
=
W {x). 2
• Algebras VJlulH, W), U £ F form the partially ordered set under imbedding (this ordering is induced by the natural ordering on F) and by virtue of the Lemma 1 and the Zorn Lemma there exists a unique •-isomorphism 6; of the *-algebras IJ VH^(H , W ). The isomorphism a 2
2
is continuous in the uniform topology (as a *-morphism of *-algebras) and thus can be uniquely extended to the needed *-isomorphism a • 2. Positive
Functionals
A complex valued functional /J : V —• C on the symplectic p-adic space (V, B) is positive, i f
(i) Mo) = l, (II) for any finite sets A j , . . . ,X
n
£€. and » i , . . . ,x
n
€ V we have
l
( I I I ) u is continuous on any U £ F. Positive functionals on (V, B) give us an opportunity to study cyclic Weyl systems. A cyclic Weyl system over (V, B) is a triple (H, W,ip), where
264
f-Aiic Analytit and Mathematical Pkyiio
[H, W) is a Weyl system over (V, B),
2
1
2
On the other hand, positive functionals on (V,B) and cyclic Weyl systems over (V,B) are closely connected. I n fact, i t is easy to see that if (H, W,
^x\^,WW)
(131)
defines a positive functional on ( V , f l ) . The inverse statement is less obvious, but it is true. T h e o r e m 2. For any positive functional u on (V, B) there exists a cyclic Weyl system (H,W,
{f,9)=
£
f(u)g{vMv-u) (B(u v)) Xp
}
t,g€K,
:
2
il/ll = define nonnegative Hermitian form and seminorm on K respectively. Let N be the closed subspace of K consist of / with zero seminorm. Then on K/N the form (•,-) naturally induces the positive Hermitian form and K/N is provided by the prehilbertian structure. The required space H is the closure of K/N with respect to the scalar product mentioned above. The map W. On the space K we shall define the following set of operators W(x),xf=V: W(x)f{u)
= ((Bx,u))f(u Xp
- i),
/ € K.
p-Adic Quantum Theories 265
These operators satisfy the Weyl relation (this fact has been proved in Sec. 12.4, Example 3). I t is easy to see that W(x), x £ V are isometric. Hence, we have correctly defined isometric operators W(x), x 6 V on K/N: W(x)[f
+ N] = W(x)f
+ N,
x£V,
f£K,
which are uniquely extended to unitary operators on H with needed properties. The cyclic vector ip can be chosen as follows: *P =
2
2
2
2
2
2
VWi{x)
x£V
2
defines an operator U : Hi —* H . operator: 2
I t is easy to see that U is an isometric
(VWi{x)
9
= (W (x)
3
2
= p(B(x,y))p.(y X
2
= x (B{x,y)) 2,W {y P
9
- x) =
2
- x)
(W^x^W^y)^).
As Hi and H are dense in Hi and H respectively, then U is uniquely extended to the unitary operator Hi —> H with needed properties. • 2
2
2
R e m a r k . The statements of Sec. 13.1 and 13.2 and their proofs coincide w i t h that of for the real number case. 3. Foch
Representation
Representations of commutation relations (or Weyl systems) are proved to be closely connected with the notion of a lattice in p-adic vector space.
266
-p-Adic Analysis and Mathematical Physics
Let V be a p-adic vector space (finite- or infinite-dimensional). A lattice £ in V is a Z -submodule of V which does not contain any nonzero subspace of V and absorbs V (that is for any i f V there exists A £ Q \ { 0 } such that Ax £ L). In the case of dim V < oo this definition coincides with the ordinary one (that is L is a finitely generated Z -submodule of V which contains a basis of V ) . On the other hand, this notion coincides with that of absolutely convex absorbing set (a nonempty subset A of V is absolutely convex it x,y £ A, A,p. 6 Z implies Ax + /ty £ A) without nonzero subspaces. p
P
p
p
Let £ be a lattice in V. We shall define the following fi+-valued functional pi on V, x g V: 1
int
Pi(x) =
kip •
(This is an analog of the Minkowski functional.) I t is not hard to prove, that for any lattice L the functional p\ is a non-Archimedian norm on V, the topology on V generated by pi we call L-topology. Let now (V, B) be a p-adic symplectic space and L be a lattice in V. The subset L * of V defined by the formula L' = {xeV:S(x,j/)e2p,
Vj/€L}
is a lattice in V and is called a dual lattice. I f L = L " , then L is selfdual. Properties of this duality are rather similar to that of orthogonal complement and therefore are given without proof. L e m m a 2. Let L,L\,Li
be lattices in (V, B). Then we have: (I) ( £ • ) " = £ , (ii) ( ^ - r i ^ - ^ L t n i ; ,
(III) ( L , n L ) - = 2
+ LI
A connection of selfdual lattices and Weyl systems can be easily seen from the following lemma. L e m m a 3. Let L be a selfdual lattice in (V, B).
m i ¥ -»c
, ,
Mx)
fi>
= <
10, is positive.
xeL,
. x £ L
Then the functional
p-Adic
• I t is sufficient to prove, that for any Xj,... the following inequality is valid: Y,
Quantum
, A„ g C and
XiXjMxj-xMBiziiXj))
Theories
Z\,...
,z
n
267
£ V
> 0.
l<»,J
As Ui(x) = 0 for all x L the above inequality should be proved only for the case when all Xj, i = 1 , . . . , rt have the form Xi = a + Ui,
Oi £ L
for some a £ V. Thus we have: £
XiXjHi(uj
- Ui)x (B(a F
+ «*;« + tij))
l
=
£
A A p(BK',"))Xp(S(t j,o)) i
i X
l
1
> 0 KKn
Let I be a selfdual lattice in (V, B ) . By the definition Fock (or L-Fock) representation of commutation relations is described by the cyclic Weyl system (H, W, = {W(a)
« £ T)
is the set of coherent states in H. The Weyl system (H, W, ) be the corresponding L-Weyl
system.
Then
(I) W(z)
268
p-Adic Analysis and Mathematical Physics
(III) (Ac Weyl system (H,W,ip) is irreducible; (IV) the map W is continuous in L-topology on V and strong topology on the set of unitary operators on H. • (I) Because of the relation
n(*)
= (
= {
J] l\LL\
(13-2)
i t is sufficient to prove that the closure k of the linear span of the set of vectors {W(a:)^, x G L) in H is one-dimensional. Let us suppose that i t is not true. Then there exists a nonzero vector i/> from K of the form
which is orthogonal to ip. Thus we have
(^) = £ c
a
= o,
but in this case
The contradiction obtained completes the proof of (I). (II) From (13.2) we can see that the set d> = {W(a)ip, a € T ) of coherent states forms an orthonormal of (H, W,tp), H is the closure of the linear span K of the set {W(x)
For any j3 € V there exist a G T and u G L , such that /? = or + u. Hence, taking into account the property (I) from this theorem and the Weyl relation we have
where C = C a + u X p t - B f a , " ) ) and therefore tpa is a basis in i / . a
p-Adic Quantum Theories 269
(III) From ( I I ) we easily get that any vector from H is cyclic, then (H, W, f) is irreducible. (IV) Let %l> be an arbitrary vector from H. Then i> = £
C W(a)
£ | C
a
0
p < c o
and for i £ L (that is pi(x) < 1) we have WWfr
- n
= II £ ( 1 - X ( 2 B ( « , * ) ) C r V ( a ) | | aer P
2
= £|CJ |l-
t t
P
2
X p
(2B(o,z))[ .
(13.3)
|Ct» | for any e > 0 there exists a
By virtue of convergence of the series aer such A > 0, such that
2
£
Ca| <|.
(13.4)
a€2~,f>,(a)>ll
For all p t ( r ) < ^ and pi(a) < A we have \B(x,a)\
and hence x>(2B(o, / ) ) = 1. Therefore for all pi(x) < m i n { l , ^ } = S we have from (13.3) and (13.4)
||H/(*)^-#= [
£
\o€T,p,(o)
< 2
£
+
£
) -|C | |l-v,(2B( ,z))p 2
a
a
a£T,Pi(tt)>a/ 2
| C | < 6. a
o€T,pi(o)>i
Thus the map W is continuous in a neighborhood of x = 0, the continuity at any other point follows from the continuity at x = 0 and the Weyl relations.
The theorem proved shows that the properties of the L-Weyl systems are similar to that of Weyl systems over a finite-dimensional symplectic
270
p-Adic Analysis and Mathematical
Physici
space. This justifies the name L-Fock for the corresponding representation of commutation relations. 4. Equivalence
of L-Fock
Representations
Let us be reminded, that in the case of dim V < oo all irreducible Weyl systems are unitary equivalent but it is not true i f dim V — oo. Therefore a question about an equivalence of two Fock representations for different lattices L is very natural in the last case. Let Li and L be selfdual lattices in (V, B). We shall say that L \ and L coincide almost everywhere i f there exists a nongenerate subspace U of the space V of finite condimension such that L \ D U = L DU. The following theorem gives us a solution of the problem mentioned above. 2
2
2
T h e o r e m 4. Let L \ and L be selfdual lattices in (V, B). Then L,-and L -Fock representation of commutation relations are unitary equivalent if and only if Li and L coincide almost everywhere. 2
2
2
• Necessity. Let (Hi,Wi,p\) and (H , W ,
2
2
2
2
2
UWi(x)Uis valid for all z EV.
1
-
W (x) 2
Let v : V —> IE denote the map defined by the formula v{z) = \(U W (z)-p )\. 9u
2
(13.5)
2
Let us prove the relation f,(0),
z
I 0,
z £L - L
€
L
l
1 r
+
2
L , 2
v
.
'
In fact, by virtue of the Weyl relation and the Theorem 3(i) of Sec. 13.3, for xi £ Li and x € L we have 2
2
+ x) = |{l/vi, HM*! + 2
*2)^2)| =
= \{UWi{-xi)
2
2
\W (-zi)U-pi,W (x )
2
2
= \(U
2
p-Adic Quantum Thtorio
271
For x £ L1 + L2 by virtue of the relation (L1 + L2)' = LiC\L (see Sec. 13.3) there exists y £ L\C\L such that B(x, y) £ Z and \ ( 2 S ( z , y ) ) ^ 1. Thus we have 2
2
p
(U
=
V
i
p
{W2{y)U
=
X
9
2
p{1B{x,y)){U ,W2{x)>p2), Vl
therefore v(x) = 0. Let now d(Li, £ ) denote the order of the group ( L j ,L )/L . Taking into account Theorem 3(h) of Sec. 13.3, the formula (13.6) and the ParsevalSteklov equality we have 2
2
i H M f l ,
=l|t^ill«
£
=
3
2
3
£
\{U ,W (a) )\ Vl
2
V3
2
v'(0) =
» (0)d(L L ), u
2
and from the last formula we get d(Li,L ) < 00. I t is easy to see that d(L\, L2) < 00 i f and only i f L \ and L coincide almost everywhere. This finishes the proof of necessity. Sufficiency. Let Li and L coincide almost everywhere and so d(L\,L ) < 00. We shall construct the vector yj £ H by the formula 2
2
2
2
2
1
4>2 = d- (L ,L ) 1
2
£ W (<*)V2oeti/(lini )
(13.7)
2
a
Note that the expression under the sum symbol in the formula (13.7) does not depend on the choice of representative in the coset a £ Lij(L\ f l L ). From the Weyl relations and the last formula we get for all x £ L \ \ 2
W ( )yj 2
Xl
= d-HM, L )
2
£
2
X (B(x, P
a))W (x 2
+ c)
2
oCt-i/fl-inia) £
= d-%LxM)
^
#
J
= ^ '
(
1
3
- ) 8
Besides that, by virtue of the Theorem 3(ii) of Sec. 13.3 and the ParsevalSteklov equation we have I I ^ H ^
=
d-\L L ) it
2
£
oeti/(tinii)
I I I W H I H ,
= 1,
(13.9)
272
f-Adic
Analysis and Mathematical
Physio
because groups ^ / ( L i n i j ) and ( L i + L j J / L j are isomorphic and d ( i i , L ) is equal to the order of the group L\/(L\ f l L ). From the formulas (13.8) and (13.9) i t follows that (#2,1^2.^2) is the Z-i-Weyl system. Therefore {Hi,Wi,z) are equivalent by the Theorem 2 of Sec. 13.2 and the Weyl systems (Hi, W>) and (H ,W ) are unitary equivalent. I 2
2
2
2
2
2
X I V . p - A d i c Strings In this section elements of p-adic string theory are given. First, expressions for so-called dual amplitudes are introduced. After that p-adic analogues of these expressions are considered and their properties are investigated. 1. Dual
Amplitides
The origins of modern string theory, as i t is known, originate from dual theory of strong interaction of elementary particles. Strong interactions of hadrons must be described by functions (scattering amplitudes), which satisfy some general requirements, such as Lorentz invariance, unitarity, duality and others. I t is a rather nontrivial problem to construct functions with such properties. In 1968 G. Veneziano proposed the following function describing the interaction of four particles 1
A(s,t)
0
=J dxx-^-^l-x)- ^-
1
.
(1.1)
0
Let k\, k , £3, k$ be relativistic n-dimensional momenta of scattering particles. Their squares must be negative (so called tachyons) 2
*
2
M * n
2
- { * . ) i
2
- - - ( * r ) 1
2
= -2.
(1.2)
The variables s and t have the form 2
* = ( * i + *2) ,
2
t = (k + k ) , 2
3
(1.3)
s
one introduces also the variable u — (itj + fc ) , and besides we have s + t + u = —8. The energy-momentum conservation law is valid, 3
* i + k + k -r Jfc = 0. 2
3
4
(1.4)
jt-Adic Quantum Theories 273
At last, the function ot(s) in (1.1) is linear *(*)=1 + | * .
(1.5)
The beta function (1.1) can be expressed in terms of the Euler gamma function
r(- (,))r(-«(t)) a
A
M
~
(
T(-a(s)- (t))
L
6
)
a
where
J
oo
I »
=
(1-7)
o The Veneziano amplitude (1.1) has the symmetry property A(s,t)
= A(t,s)
(1.8)
(so called crossing-symmetry). I t can be represented in the form 4.,
0
=
£ ; M
±
l
«
^
.
_
^
_
,
(i.9j
n=0
The equality (1.8) and the representation (1.9) ensure the duality property: one and the same amplitude can be represented either as a sum over poles in s-channel (formula (1.9)), or over poles in r-channel. The asymptotic of A(s,t) at large s and fixed f has the form Q
A( ,()~s C), S
(1.10)
so called Regge behavior. I t was proved that for validity of the unitary condition the dimension of space-time must be n = 26. The generalization of the Veneziano amplitude to the case of scattering of N particles w i t h momenta (k\,... , jfcjv) has the form A
N
j
=
dx dx . 2
(1.11)
3
Q
ni'ii- -*'-u-«ii^ si
j=2
n 2
274
p-Adic Analysis
and Mathematical
Physics
Or in more symmetric form
A
, w Yldz^dV*)-
J ]
1
=
K
•* 1 = 1
(1.12)
l
where 1
l
-t
dV = d i o d z i d M * - -*J~ (*» - * i s ) ( * « - * < . )
_1
3
and z i * t . * c are arbitrarily chosen from different variables. Division on dV in (1.12) corresponds to a separation of the volume of the group SL(2, R) from the expression 0
3
r A=
N
Hd •
n
Xi
i=l
k
k
&~ \- < -
(i.i3)
Xm
l
which is invariant under SL(2,R) fractional-linear transformation of variables Xf. There exists another collection of dual amplitudes (Virasoro-Shapiro), which corresponds to integration over the complex plain
B=
ff[d\
N
J
TT
1=4
\
Z
i
-
Z
j
\ W >
(l.H)
4
The formula (1.14) is obtained from the expression
"
i=l
l
after the separation of the volume of the group SL(2,C). i=l,...N. For four particles we have B
4
2
1
k
1
= J d z\z\ ^ <\l-z\ '^'-**.
Here kf = —8,
(1.16)
c In string theory the amplitudes (1.1) correspond to open strings and (1.16) correspond to closed strings. Note that these expressions for scattering amplitudes are only the first terms in the so-called decomposition of the
p-Adic Quantum Tkeorica
275
number of loops for string amplitudes. By analogy to (1.16) one can consider instead of (1.1) the expression A = j dx\x\
kl
k t
4
\l-x\
l l
k t
(1.17)
• where the integral is over all real axis. amplitude A = A(s,t)+A(t,s)
Then we obtain the symmetric + A(u,s).
A
2. p-Adic
(1.18)
Amplitudes
As i t was shown by the development of mathematical physics during the last twenty years, a rather substantial modern string theory stems from a development of the simple formula (1.1). Therefore i t is very interesting to find a generalization of the expressions (1.16) and (1.17) using only pure mathematical resources. According to two representations (1.1) and (1.6) two possible generalizations exist. One can use either the integral representation or the T-function representation. Here we shall consider the integral representation. The function \x\ is a character, so one can interpret the integral in (1.1) or in (1.17) as a convolution of two multiplicative characters on the real axis. Therefore the following generalization is suggested. a
Let i t be a field, j (x) and Tp(x) be multiplicative characters on K, where a and 0 are some parameters on which these characters depend. Let a
My ,yp) a
= J y (x)j (l-x}dx, a
0
(2.1)
K
where dx is a measure on K. For different fields K and different characters f this formula gives us the known as well as new amplitudes. In particular, for 7 o ( i ) = \x\ and K = IR we obtain the expression (1.17), and for K — C we obtain the expression (1.16). Let us consider the case of K = Q , 7 „ ( x ) = 7D(X) — \x\P~ , where a and 0 are complex parameters. The formula (2.1) then gives the B-function considered in Sec. 8.3, a
a
1
p
BP (c,0)
= J
[x\;-'\l-xf -'dx. p
(2.2)
276 p-Adic Analysis and Mathematical Physics The integral (2.2) absolutely converges in the region where Re a > 0, Re 0 > 0, Re(a + 0)<1. In Sec. 8, see (3.9), the expression for B-function (2.2) in terms of r -function was given, p
r
P
( a ) W
TOT where
™
i - p"-
W
1
M
= T^cT'
or in more symmetric form B (a,^) = r ( -)r (^)r (T),
(2.5)
a+p + j = l .
(2.6)
P
p
tt
p
p
where If we put in (2.3) a = —n(s), 0 = — a(t), where » ( s ) is denned by (1.5), and s and t are defined by the formulas (1.3), then we get the simple p-adic dual amplitude B (—at(s),—a(t)). By virtue of the relation p
a(s) + a(t) + a{u)
=-I,
and using (2.5) and (2.6), it can be written in the symmetric form 1_ -«0)-i p
l c=s,t
,u
The formula (1.13) admits a direct extension to p-adic case
Xi
J J
i=l
l<«m<JV
A = [l[d
|*,-, |-*.-*-. M
( 2
.
9 )
After the separation of volume of the group S£{2, Q ) from (2.9) we obtain p
N-2
4»*
=
k k
I
dx2...dx . Yl\ \; ' -\l- \;^ N 2
II 2
|x -i |;*'*». l
m
Xj
Xj
(2.io)
p-Adic Quantum Thcoritt 277 The integral (2.10) can be calculated explicitly giving an answer similar to (2.8). It is interesting to note that these p-adic dual amplitudes can be obtained by the rules of quantum field theory from the Lagrangian corresponding to the following nonlinear equation U
- ' ^ = 4f.
P
(2.11)
Here tb = d>(x) is a real field in n-dimensional real space-time with coordinates x = (XQ, • • • , x„-i), • is the d'Alembert operator
dxl
dx\
dxl_
Y
The equation (2.12) has the static solitonic solution of very simple form n - I ^ =
p
l (
P
l p - 1 .
T T )
e
x
P
f
(2.13)
2pln7
where x = ( # i , „ . , i _ i ) . We shall use the notations n
1(*,0)
M
= J & M * - H l - * |
,
(2.14)
a +0 +7 = 1.
(2.15)
L e m m a 2.1. The following formula for the function (2.14) is valid
where ( is the Riemann zeia function, and arguments a and 0 belong to the region «r > 0, 0 > D, « + 0 < 1, f M defined by the formula (2.15). • Dividing the integration region in the integral (2.14) on three parts we transform it in the form Bfa 0)
r
(
Q
)
r
(
/
?
)
I
r ( /
*
) r ( 7 )
I
r
(
T
)
r
(
"
}
(2 17)
278
p-Adic Analysis and Mathematical Physics
Using the relation for the gamma function T(x)T(l
sin wx and doing simple trigonometrical calculations we reduce (2.17) to the form
B(<*,p) = -
B
cos
CO
s ^
i
IT
cos
&
(2.18)
^r(«)r(/?)r( ). 7
i
Taking into account the functional relation for the zeta function
( 2 * ) \ ( l - . 0 = 2coG^r(«K(>) and the relation (2.15), one get from (2.18) B{cc,0)
=
<(l-o)C(l-/?)C(l-7) c(<*) <m c(7)
R e m a r k . Let us note the following relation between the zeta and the gamma functions
i»r(/?) ! > + /?)
r(/?)r( ) r(/s + ) 7
+
7
c(i- )qi-j3)c(i
r( )r(a) r( + «) T
-7)
a
CO)
7
CO?)
C(7)
,„
1 0
,
'
1
a + /? + 7 = I. 3. Adelic
Products
The following theorem is valid. Theorem region
3.1.
Let real parameters a and 0 take their values in the a<-l,0<-l
(3.1)
Then we have the following relations for the functions (2.3) and (2.16) IK
c
(
1
_
a
)
<
(
1
_
w
(
(
a
+
/ J )
,
(3.2)
p-Adic
Quantum Theories
279
(3 3) with the products in (3.2) and (3.3) on a// prime p teina absolutely convergent.
• Let us rewrite the expression (2.3) in the form
or taking into account (2.15) in the form
M°,?o
=- \ !
r
\ ^
\ J^ r
(3.4)
In the range of values of the parameters (3.16) we have 7 > 2 and B (ct,0) p
<0.
(3.5)
Using the Enler formula (0.1) from Sec. 7.7, we find from (3.4)
nv **
c
wn < m- <-°> Q-B «-y) " C0-«)C(l-/J)C(l-7) C(-«) C(-/J) C(l-«-/?) < ( l - a ) C ( l - 0 ) C(<* + 0 ) a
(3.6)
that is the relation (3.2). The product in (3.6) absolutely converges. In fact, as it is known for the absolute convergence of a product n( *i>) i +
P
it is necessary and sufficient to have the absolute convergence of a series ^ x . We have p
and for or < — 1 < 00. irp-_i p-
T——
280
p-Adic Analysis and Mathematical Physics
Analogously one proves convergence of the products on p in other terms in (3.4). The formula (3.3) follows from (3.2) taking into account the Lemma 2.1. • R e m a r k . In a paper by Fieund and Witten the following remarkable formula was derived B( ,p)l[B {a,0)=l p a
(3.7)
p
There are some subtleties with a convergence of the product and a result depends on the choice of a regularization. We show that there exists a connection between the formulas (3.2), (3.3), (3.7) and the Tate formula. From the formula (2.2) of Sec. 7.7 for 0(A) = 1 it follows the relation
n / vp{xp)\xp%d =n / &a^)}-**fc Xp
?
j
p
j
<-) 38
here the case p = oo is allowed, and 0 is a real parameter. The function tpp{ p) here must belong to the Bruhat-Schwartz space. However we formally put x
ft,t%)=|l-*jf,
(3.9)
where a is a real parameter and then the relation (3.8) yields the formula (3.7). Indeed, one has 1
a
•Pp(y) = c (<*)x(y)\yp\p- - , P / O O , P
where
W
= i l p - C ' P * °°:
C
X
a
» » = -W~ -
sin y r ( l + c) .
Therefore according to (3.8) 1 - x \«\x\fo p
P
= Hc ( )C (-c p p
a
p
- b- 1) = 1,
p-Adic
Y[C {a) p
281
+ a) [ J y ^ T ^ T p
= ~ 2 ( 2 i r ) - ' - sin ^ r ( l
B
Quantum Theories
v
= -
2
(
2„—»=r(i
.
+
)
^ = i.
I f we put
^,(#=-11-^ instead of (3.3), then we obtain the relation (3.7). 4. String
Action
We have discussed the Veneziano amplitude and its generalizations. In fact these amplitudes are the only first terms in the so-called loop expansion. Full answer has the form oo
Ah(ki,
...kf/)
j exp ^ £ f e > * ^ & & ; - ) j JJffcjf.
= jwr)
(4.1)
Here is a compact oriented surface of genus h, £ j , j = 1,... ,N are coordinates of N points on i t , dSj = \fg(£})d £j, g(E) = d e t g p ( 0 , where g 0 is a metric on J2h> >@ ^>^- ^ ' tfiodnli space of Riemann surfaces of genus h,r are local coordinates on Afj,, G ( £ , C ; T ) is a Green function of the scalar Laplacian on £ , d[i(r) is a measure on A/h, & i , . . . are momenta of particles (tachyons), kj G R " , j = 1,...... Jtf". One has i f = 2 2
a
a
=
h
s
t
n
e
a
f t
and ifci - f . . . + i j y — 0. We will discuss this expression in more details in the next subsection. Here we note that this formula is obtained from a formal functional integral
6(k)A (k ...k )^ h
ll
1
f[Dg )
N
o0
J
3
[[DX^Hvitye- , j=i
(4.2)
282
p-Adic Analytit and Mathematical Physics
where 8(k) = M £ * i J is the Dirac 6-function, Here the classical action is j
1=1
2
3
', g
a
(4.3)
0
X = X»{(,),a = 1 , . . . , d is a map from Zh to ffi", d X" vertex operator V(k) has the form a
j
V(k)=
2
d Z^g-e p{ik-X(t:)) X
=
^X»(0,
•
E„ 11
Symbols [Dg g] and [ D X ] mean that one should integrate over all metrics on Z and over all maps X *, a
1
h
5. Moduli
Space and
Theta-Functiona
The action I possesses two infinite symmetry groups. One of them is the group of Weyl rescalings which acts by g p —* e"g p. I t is called conformal or Weyl group. The other group of symmetries is the group of diffeomorphisms Diff, or reparametrizations of ^ » . A diffeomorphism that is continuously connected to the identity is called a local diffeomorphism. The group of these diffeomorphisms is denoted by Diffo. There are also global diffeomorphisms that are not in the connected component of the identity. a
a
The space M
n
= i Metrics on £
I /Diff-
Weyl
is called moduli space. The moduli space A//, is a complex variety of complex dimension 0 for ft = 0, I for ft — 1 and 3ft — 3 for ft > 2. The bigger space
T = | Metrics on £ h
J /Diffo • Wtyl
is called Teichmuller space. The group obtained by taking all of the diffeomorphism modulo the local diffeomorphism is the mapping class group (or modular group). Let us choose a slice S through the space of metrics which is transversal to the actions of Diffo and Weyl. A subdomain of S in
p-Aiic QiHtarn Theories 2S3
which no iwo points are related by a global diffeomorphism is a fundamental domain for the mapping class group. The conformal class of a metric amounts t o a complex structure on . The first homology- group H,{^ .Z) has 2h generators. Let us choose a symplectic basis { o i ,ai-ti &»} of 1-cycles on Ylk satisfying the following conditions on intersection numbers: k
a, - flj =
-
bj = 0,
bj = f i ,
a;
Any two sjTnplectic bases are related by a transformation from the symplectic modular group $ (2h,Z). The space of holomorphic 1-forms on ^ has the basis { ~ \ . . . . - . * } uniquely determined by the conditions r
k
y_
V
= V
ij=
1.... A
*. A matrix f =
(TV,-).
is called the period matrix of and is a complex symmetric h x h matrix with positive denned imaginary part. Given a base point Q: £ one associates to every point Q on a complex /t-component vector / i Q ) by the Jacobi map defined as Q
i=i....a.
This vector is unique up to the periods - So / is an element of the complex torus
called the Jacobian variety of
• D
The theta function is defined for ; 6 ^ ( E k ) >' ff{z,r)
=
, r j e
s
u
£ e x p ( i > n ' r n + 2arin*;). sre»
m
284
p-Adic Analytia and Mathematical
Phyiics
A theta function with characteristics a,0 € 1/22'' is defined by 9
(z, r) — exp(iVn
The parity of 4a'/? is called the parity of characteristic {a,0}. One needs the prime form E(w,tl) which is defined as the holomorphic differential form on £ of weight ( - 1 / 2 , 0 ) x ( - 1 / 2 , 0 ) : A
Jm,T (fi(«i)V3(A(jJ)i/i
(5.1)
where (0,r)- (u;) W l
t=i
E'fw.f) is independent of the choice of the odd characteristic {or,/3}, i t is only zero for w = £. There exists a measure on M reflecting the holomorphic structure of moduli space. According to the Mumford theorem the line bundle n
if® A"
1 3
(5.2)
over moduli space is holomorphically trivial. Here K is the canonical bundle of moduli space, i.e. the highest wedge power of its cotangent bundle. A is the highest wedge power of the bundle of holomorphic 1-forms on the surface. There is a unique holomorphic nowhere vanishing section F of i f ® A ~ . Let vi,...V3A_a be analytic coordinates on A/j,(ft > 2) then 1 3
2
13
dp = |F(u)| (det I m T)~ dv
A dv
(5.3)
is a measure on M^, here dv = dv\ A . . . A dv3„-s.
6. Multiloop
Amplitudes
To define the functional integral (4.2) one uses the procedure of quantization of gauge theories. One can argue that the amplitude (4.2) has the form A (k ,...k ) h
1
N
= j A/*
dp(v)T(k ...k -v) u
N
(6.1)
p-Adic Quantum Theories 285
where T(kj,..
.fcjv;v) comes from the Gaussian integral N
6(k)T(k
u
... k ;v) = I[DX*]e*
!
f[ V(k ) j=l
N
}
= s(k) j exp f-|Efe • WU,t/)\U
<Mf&%
This can be expressed in terms of prime form (5.1): r(i,,...**;.)=
I{4tM
/ ' f l r o , * 3 | * * f [ * & ,
(6-2) 1
= B f e , ^ ) ^ . { - i r r „ 4 ( Im 7-)- Im
2 i j
},
In particular for torus (h = 1) one has
Ms
Here i
T (v) =
ef?f[(l-e*' ™)
}
n=l
is the Dedekind eta function,
(1--C)
l/2
x
n
n
~
11 n=l
n
{l-w x)(l-w /x) v
(l_u,«}2 '
The integral runs over the fundamental domain Mi = {v 6 C : Imu > 0, | R e t i | < 1/2} and
N^ISN'^Ki.
r=l,..JV--l.
286
p-Adic Analysis and Mathematical
T h e
i n t e g r a l
i n
T h e
i n t e g r a l
( 6 . 3 )
( 6 . 3 )
l e m
o f
d i v e r g e n c e
p a r t i c l e s
i s
s c a t t e r i n g
a m p l i t u d e
h a s
a
is
i n v a r i a n t
d i v e r g e s
a n d
a v o i d e d
s i m p l e
u n d e r
o n e
i n
a m p l i t u d e
Physics
m o d u l a r
n e e d s
t h e
f o r
t o
u s e
S L ( 2 , 2 ) a
s u p e r s t r i n g
s u p e r s t r i n g
t r a n s f o r m a t i o n s .
r e g u l a r i z a t i o n . t h e o r y .
t h e o r y
T h e
T h e
f o u r
c o n v e r g e s
p r o b -
m a s s l e s s
a n d
1 - l o o p
f o r m
/ TT
,1
T T ,
.1
1.,/J
Mi T h e f o r
r e q u i r e m e n t
l o w
g e n u s u l i
g e n u s =
h
m e a s u r e
c a n
m o d u l a r
u s e
M o r e
2
t h e
o f
e x p r e s s
o n e
M.
s p a c e
m o d u l o
2
t o
t h e
o n
H
S i e g e l
c l a s s
s h o u l d
2
i n v a r i a n c e
m e a s u r e
i s
2
o f
h a v e
i n
o f
t h e
s t r i n g
t e r m s
u p p e r
M
p r e c i s e l y ,
e q u i v a l e n c e
dp
t h e
o f
t h e
h a l f p l a n e
i s o m o r p h i c
t h e
s p a c e
t h e
f o r m
o f
H
t o
2
t o
d i a g o n a l
m e a s u r e
t h e
p e r m i t s
dp
m o d u l a r
f o r m s .
d e s c r i b e
F o r m o d -
H /Sp(4,%.)
s p a c e
p e r i o d
t h e
2
m a t r i c e s .
T)" n^ •
T h e
13
dp
=
| / ( r ) |
2
( d e t l m
( 6 - 4 )
i
h o l o m o r p h i c
w e i g h t
k
i f
/
f u n c t i o n
f{(Ar f o r
/
:
Hh
—*
C
i s
a l l
6
+ B)(CT
+ D)- )
S p ( 2 A , 2 ) .
U n d e r
1
—
Hdn,
=
10.
T h e
m o d u l a r
t h e
t h e
f u n c t i o n
r i n g
o f
f o r m
p r o d u c t
a
g e n u s
t / i
o v e r
I m
/
_
1
g e n u s - t w o
1
0
o f
e v e n
w e i g h t
d e t ( t >
m o d u l a r
d e t ( C r
d e t d e t
t h e r e f o r e
c a l l e d
m o d u l a r
h
f o r m
o f
s a t i s f i e s
+
+
l
D) f(r)
t r a n s f o r m a t i o n s
D ) -
3
o n e
h a s
J T o Y ,
T
I m
| d e t ( 0 + D ) P i n
( 5 . 5 )
m o d u l a r 10
s h o u l d f o r m s
v a n i s h i n g
c h a r a c t e r i s t i c s .
is o n
b e
a
m o d u l a r
k n o w n . t h e
f o r m
T h e r e
d i a g o n a l
e x i s t
p e r i o d
o f
w e i g h t
a
u n i q u e
m a t r i c e s ,
p-Adic Quantum Theoriei
287
Thus we have
i 7. Rigid
Analytic
Geometry
and p-Adic
Strings
We see from the previous discussion that the theory of Rjemann surfaces in particular moduli space and theta functions are used i n string theory. To develop p-adic string theory one needs a p-adic analog of complex analysis including the basic principle of analytic continuation. The p-adic field Q is totally disconnected so one cannot use to this end the "naive" notion of analyticity considered in Sec. 2. Nevertheless an appropriate theory was constructed by Tate and others. It is called rigid analytic geometry. p
The main idea is to perform analytic continuation only with respect to certain admissible open coverings of rigid space. Analytic functions are no longer considered on all open subsets of such a space; one has to restrict oneself to admissible open sets which form the so-called Grothendick topology. There exists a nontrivial notion of connectedness of such a space. This rigid analytic approach was used for the uniformization of algebraic curves and i t is natural to use i t for p-adic strings. We are going to describe shortly some points from rigid analytic geometry. Let T„ = K{Zi,... B
n
z„) be a ring consisting of power series converging on
= {(x ...x ) u
n
£ K
n
: | * i | < l , i = l , , „ ,n}.
Here i f is a complete no n-ar chime dean field for example Q . An affinoid algebra A over K (or Tate-algebra) is a Jf-algebra which is a finite extension of T - That means: there exists a if-algebra homomorphism T —* A which makes A into a finitely generated T -module. The space Sp(A) of maximal ideals of A is called an affinoid space. Every affinoid algebra A has the form T„/I for some ideal I C T and is a Banach space with respect to the quotient norm. One can identify Sp(A) with the points x in B such that f(x) = 0 for all / G J. p
n
n
n
n
n
Let X = Sp(A). Let /o, / i , . -. fn £ A be such that they have no common zeros on X. Then the subspace U C X [/ = { » € X : j / ( z ) | < | / ( * ) | , » = 1,... n } 4
0
288
p-Adic Anatyeit and Mathematical Pkyiics
is called a rational domain. We can define a Grothendick topology on X by taking as open sets all rational domains, A Grothendick topology on a topological space X is defined as follows. Let F be a set of open subsets of X and let Cov(U) be a collection of coverings of U C X. Then (F,Cov) is called a Grothendick topology i f it has the following properties: 1) 2) 3) 4)
, X e F and i f U, V € F then V n V € F, {CO £ Cov(U) for all U G F , i f W G Cov((7) and V C (7 with 17,V G F then W f l V e Cov(V) i f « i € Cov(Ui) and ([/<) G Cov(U) then LJ W, e Cot>(*7).
The elements of F are called admissible open sets and the elements of Cov(U) admissible coverings of U. W i t h a rational domain U one associates the affinoid algebra A
V
= A{zi,...z )/(fi n
- zi/o,.. . / „ - z„f ) 0
.
Then we can define a presheaf Ox on X by associating AJJ to each admissible open set U. A basic result is that Ox is in fact a sheaf for the Grothendick topology. The space X = Sp(A) with its Grothendick topology and the structure sheaf Ox is called an affinoid space. A n analytic space X is defined as follows. X has a Grothendick topology and a sheaf Ox such that there exists a (AT,) G Cov{X) with (Xi,Ox\Xi) is an affinoid space. E x a m p l e . I f X is a complete, non-singular and irreducible curve over K with function field K(X), then for any / G K(X) with / j£ 0 the set Xj — {x G X : \f{x)\ < 1} is an admissible affinoid subset of X. The affinoid algebra Ox, is a completion of algebraic holomorphic functions on X,. A uniformization of X is its representation in the form Y/V where Y is an analytic space and T is an infinite group that acts d is continuously on Y. As i t is known the upper half plane H uniformizes curves of genus g > 2 over C. The problem of uniformization of curves over a non-archimedean field K is more complicated. First consider an elliptic curve X. I f X is defined over C then X has uniformization C / r where T is a lattice. Such an uniformization does not exists over a non-archimedean field K. I f we take T = 2 + S • r with I m T > 0 then we can construct another uniformization of X by using the exponent map. One gets the uniformization C/{q) of X where (g) is the multiplicative group generated by q = e *' . This uniformization does 2
r
p-Adic
Quantum Theories
289
have an analogy over non-archimedean field K. For q € K* with \q\ < 1 an analytic torus A , = K*f{q) is an elliptic curve and is called Tate curve. The j-invariant j ( g ) of X satisfies \j(q)\ > 1. The reduction of X is a rational curve with singularities over K where K denotes the residue field over K. q
9
To uniformize a complete non singular curve X of genus g > 2 one uses the Schottky uniformization. A Schottky group T is a subgroup of PGX(2, K) which is finitely generated, discontinues and has no elements of finite order. I t has a nice fundamental domain: F = IP—(2j open disks). Let L be set of limit points for the action of T in P. Then (P—L)/T is an analytic space and analytically isomorphic to a complete nonsingular and irreducible curve of genus g. Such curve is called a Mumford curve. For Mumford curve there exists a nice theory of automorphic forms, the Jacobian variety, period matrix, non-archimedean Siegel half-space and theta functions. I n particular one can define the prime form
y&x-yfci
V-JM
and t r y to use i t in p-adic string theory simply by replacing the complex variables by p-adic ones. I f we want to develop p-adic string theory in a more systematic way one should start with a classical p-adic action and then quantize i t . Recall that closed strings correspond to compact Riemann surfaces without boundary and open strings correspond to Riemann surfaces with boundary. One can think of the curve X, endowed with a rigid analytic structure, as the non-archimedean analogue of a compact Riemann surface without boundary. Then the definition of Xj suggests that an affinoid space is the non-archimedean analogue of a compact Riemann surface with boundary. One can t r y to define the naive boundary of Xj to be the set B = { £ Xj : \f(x)\ = 1}. However this boundary depends very much on the choice of / . Using rigid geometry we can define the canonical boundary dXj in such a way that B C dXf. X
A n approach to p-adic string theory based on a p-adic analogue of the classical string action is not available at the moment. One attempts to use the 5 i ( 2 , Q )-symmctry of the operator D and consider an action p
=11 F
F
2
{X"{x)~X"{y)) \x — y |
2
-dxdy
290
p-Adic Annlyii) and Mathematical Phyaxcs
where F is a fundamental domain of a Schottky group. One really can obtain p-adic tree amplitudes from this action i f F = Qj>. I t was also suggested to interpret the open p-adic string world sheet as a coset space T/T where T is the Bruhat-Tits tree, T = PGL(2,Q )/PGL(2,Z ) and T is a Schottky group. The tree T is the connected infinite graph with no loops. Each vertex of T is connected with p + 1 neighbor vertices by edges. Note that there is a canonical tree T(X) for any compact subset X of P. P
p
It is an open problem what would be a string measure on p-adic moduli space. Perhaps most principal approach to p-adic string theory would be to start with a theory over the global field of rational numbers in accordance with the discussion in the Introduction. One can expect that the beautiful rigid analytic geometry will also find an application in p-adic gravity. X V . (/-Analysis ( Q u a n t u m G r o u p s ) a n d p - A d i c A n a l y s i s We are going to discuss some remarkable relation between theory of quantum groups (g-analysis) and p-adic analysis. 1. p-Adic
and
q-Integrals
The quantum group S I / , (2) is a Hopf algebra with generators a and c satisfying quadratic relations. I t was shown by Woronovic that there exists a Haar measure m on SU {2). q
P r o p o s i t i o n 1. The Haar measure on 517,(2) coincides with the Haar measure on the field of p-adic numbers Q if q = 1/p. p
3
• I t is known that the cohomology group H (SU {2)} there exist a unique linear functional q
— C, therefore
/ ' ~* °' :r
2
n
such that / d ( = 0 for £ g T , where T is the module of n-differential forms. For coordinate ring of SUq(2) one has j /u wiW2 = 0
m(f)
p-Adic Quantum Theories
where m is the Haar measure, v,- are differential forms on SU {2). polynomial in cc*, we have a
291
I f / is a
oo !
!
m(/) = ( l - ? ! £ / ( 9 " ) 3
2 n
,
0<«<1.
This is in fact the well known Jackson integral in g-analysis
r //(*)<*,* = ( 1 - « ) £ / ( * " ) « " • 0
(1-1)
"=°
Now we recall that the integral over Q with respect to the Haar measure is p
/
f(\x\ )dx
= f l -
p
kl,
^ ) £/(?""}•?-" P
^
J
n
(1-2)
=°
Hence we get that (1.1) is equal to (1.2) if 1 9= P i.e. l
J f(x)d
l/pX
0
=
j
f(\x\ )d*
.
p
m
|*|,<1
There is the following generalization of this observation. Let us consider the operator
\y\,=\A,
I«IP = I
Here / € L^G),
G = ^ U S . , , S-, = [\x\ = pi]. Then p
J f(x)dx
= J
M[f](r)d r, 1/P
292
f-Aiic
Analytit
and Mathematical
Phytic)
vhere
2. Differential
Operators
Let us now consider a relation between differentialoperatorsing-analysis and in p-adic analysis. In ^-analysis one has the following operator of differentiation
One uses also oo
i In p-adic analysis if we consider a real valued function f(x) depending on a p-adic variable x we cannot use the standard definition of differention, because one cannot multiply real and p-adic numbers. I n Sec. 9 it was considered the following operator
s v ^ ^ / M W
cm
Restricting to rational x in (2.1) and (2.2) we see that the expressions (2.1) and (2.2) are very similar, in particular, both are non-local operators. 3. Spectra of the q-Deformed
Oscillator
and the p-Adic
Model
Let us discuss the connections between the spectra of the g-deformed oscillator and of the p-adic model. The spectrum of p-adic equation Dj,( ) x
+ V(\x\ )1>(x) = E# p
for 1+ p+p Pfcls
2
p-Adic
Quantum
Theories
293
has eigenvalues
gn = ["]l/
P
=
P p
." "
P
1
p
".
» € *
(3-1)
For t h e g-deformed oscillator +
N
aa
- qa+a = q~ ,
[JV.a] = - a , [iV,o] = a
+
the s p e c t r u m was found b y B i e d e n h a r n +
H = aa ,
H\n >= [n] \n q
> .
(3.2)
If 5 =
1
P
one gets the p-adic s p e c t r u m (3.1). I n Sec. 10 i t was shown t h a t i n the s p e c t r u m of more general equation D"^)
+ V(\x\ )i>(x) p
= Eif>,
a > 0
there is the following f a m i l y o f eigenvalues £„,!
a
n
= P " + V(p'- ),
1= 1 , 2 , 3 , . . . ;
n = 0, ± 1 , . . .
S i m i l a r s p e c t r u m there appeared for a g-deformed Schrodinger equation. X V I . Stochastic Processes over t h e Field o f p-adic N u m b e r s T h i s section presents elements o f a theory o f stochastic processes over the field of p-adic numbers.
Following a s u m m a r y o f basic definitions of
the p r o b a b i l i t y t h e o r y we consider a B r o w n i a n m o t i o n on the p-adic line as a M a r k o v process.
W e next discuss generalized r a n d o m fields and p-adic
q u a n t u m field theory. 1. Random
Maps
and
Markov
Processes
A measure space is a p a i r {SI, £ } , where SI is a set and E is a cr-algebra o f i t s subsets. A t r i p l e { £ } , £ , F } , where { f I , E } is a measure space and P is a
ff-additive
nonnegative measure o n £ w h i c h satisfies the c o n d i t i o n P(Si)
= 1,
(1.1)
294
p-Adic Analysis and Mathematical Physics
is a probability space. Elements A (E £ are called events and P{A) is the probability of an event A. A random variable £ — £(w) is a E-measurable real-valued function on n, that is C {S) 6 £ for every Borel set S on the real line I . The random variable £ gives rise to a probability measure Pf on BE , defined by 1
1
1
Pi{S)
(1.2)
1
= P(C (S))
1
for any Borel set S o n I . The measure Pc is said to be the distribution of the variable £. The mean o f f , £ ( £ ) is defined by
E(0 = J t(w)dP(u). n I f A, B e £ and P(B) / Owe define
P(A|B) =
P(AnB) P{B)
P(A\B) is called the conditional probability of A given B. Let F be an arbitrary ir-algebra, which is contained in £> £ be a random variable on a probability space (SJ, E, P) with mean equal to zero. Conditional expectation of t\ with respect to the c-algebra F is a random variable E{tl\F}, which is F-measurable and satisfies the relation j E{t:\F}dP
= J
tdp
for an arbitrary B € P . Conditional probability P{v4|F) with respect to a ) is the indicator of the set A. Namely, for fixed A the conditional probability P { . 4 | P } is a F-measurable random variable satisfies the relation
/ for every B € F
P{A\F]dP
= P(A n B)
p-Aiie
Quant-am Tkeoritt
295
Conditional probability P { A | f } with respect to a random variable £ is defined by the formula P{A\F}
=
P{A\F ), (
_1
where F( = {B : B = £ ( S ) , S - belongs to measurable Bore! sets in E] is (T-algebra generated by the map £. Let X be a set and {Y, 23} be a measure space. Random map of a set X to measure space {Y, 23} is a map £ = £(a:,w) : X X CI —• Y which for an arbitrary fixed x is a measurable map from {£2, £ } to {Y, 23} that is for any B e 2}
Below two examples of random maps are considered. The first when X is a subset (semi-axis) of real line, in this case we write ( instead of x. This random map is called random process £((). Independent variable ( is interpreted as time. In the second example X is a space of distributions 7?'(Qp). Here random map reduces to a generalized random field. Let n be an arbitrary positive integer, x^, k — 1,2,... , n be arbitrary points from X. Measures P x ...x (B) on 23" of the form Xl
JW.
I
n
1
n
( 5 ) = P{^t,uU{^h-
-
•*$*».•«) 6 B),
B e 03"
is said to be joint distributions of a random map Under sufficiently wide assumptions a set of joint distributions defines a random map. We consider Markov process £ = £(t) on the semi-axis ( > 0. Let Y be a complete metric space and 23 be the ff-algebra of Borel subsets of Y. A random process £ — f > 0 with vaiues in Y is called a (homogeneous) Markov process, i f the following conditions are valid: 1) for any 0 < t i < ( ; < . . . < t„ < t and B € 23 the relations for conditional probabilities are fulfilled
P{t(t)
e B\atiU(t ),..
.
2
- PtfW 6 B]t(t )} n
{mod
P);
2) there is a function P(f, i , B) which satisfies the following conditions: i) it is 23-measurable on x for fixed t and B; ii) i t is a probability measure on the space (Y, B, P) for fixed i and x\ hi) i t satisfies the Chap man-Kolmogorov equation P(s + t,x,B)
= j Y
P(s x dy)P(t y,B), l
l
y
s,t>0;
296
-p-Adic Analysis and Mathematical Physics
iv) i t coincides with probability 1 with conditional probability
P(t,x,B) = P{S(s + t) G B K ( « ) = x),
8,( > 0.
The function P(t, x, B) is called the transition function of Markov process £((). I f a probability measure u on {Y, 0, x,y G Y, is called the transition density i f the following conditions are valid: 1) P ( * . « . » ) > 0 , t > 0 , x,y£Y; 2) p(t,x:y) for fixed t > 0 is a 0J x OS-measurable function of i , y ; 3) j > ( M , 3 / M r f ! / ) < i> « > o , z e y ; y 4) p(s + i , i , z ) = / p ( s , r , y ) p ( r , , , r ) f { d ! / ) , s , r > 0 ; 3
i
1
i,zeY.
y
If a transition density p(t,ar,!/) is given then the function
f Sp(t^,y)pi{dy), t > 0, ^ 7 , P(t,z,5) = <
flGS
B
defines a transition function which corresponds to some Markov process, A transition function defines a set of linear operators on the Banach space E of all bounded measurable complex valued functions f(x), i t V with the norm | | / | | = sup|/(x)| by the formula
T f(x)=
I P(t,x,dy)f(y),
t
t > 0.
J
Y The set T ,t > 0 is a contraction semigroup of operators, that is bounded operators T which satisfy the following condition t
t
T,
+i
= T,T
HUH < i ,
u
t,t>0, t > o.
p-Adic Quantum Theories
297
An infinitesimal generator A of this semigroup is also called an infinitesimal operator of a function P(t, x,B). The domain DA consists of all functions / for which the limit in the relation
J
P(t,x,dy)f(y)-f{x)
does exist uniformly with respect to x G Y. Let C(Y) be a space of continuous bounded complex valued functions on Y. A Markov process is stochastically continuous if (
Hm Ttf = f,
v/eqy),
and is called Feller's process, i f T C(Y)cC(Y),
VOO.
t
There is the following criterion. I f for a transition function P{t, x, B) for any compact B the following conditions are valid: Vs > 0 l i m &uoP(t,x,B) = 0, (A) and Ve > 0 lim
supP{t,x,U {x))
= 0,
c
(BJ
where U (x) is a disk of radius E with the center at x, and U (x) = Y\U {x), then the corresponding Markov process is bounded, continuous from the right and has no discontinuitiej of the second kind with probability 1. t
2. Brownian
c
Motion
on the p-Adic
c
Line
It is known that a Brownian motion on the real line can be described by means of the diffusion equation 2
du du = a dt dx • 2
(2.1)
A solution of the equation (2.1) defines a transition probability of Brownian motion considered as a Markov process. Here we consider the equation
296
p-Adic Analysis and Mathematical Physics
and associate with it a Markov process, which we shall call a Brownian motion on p-adic line. Here real valued function U = U(t,x) in (2.2) depends on variables t € ' K and x e Q . Let us consider a fundamental solution of the Cauchy problem p
x,0) =
(2.3)
for the equation (2.2). I t can be given by using the following functions: p(t,x
:
y) = K(t,z-
y) = / (t(x X
2
- y)) exp(-f K| )d£
(2.4)
where t > 0, x,y e Qp. T h e o r e m 1. The function p(t,x,y) (2.4) satisfies the conditions l ) - 4 ) for a transition density ifY = Q , p(dz) — dx is a Haar measure on Q , and hence defines a Markov process which we call the Brownian motion on the p-adic line. p
p
R e m a r k . In contrast to the Winer process, trajectories of which are continuous, there are no nonconstant continuous functions from a real segment to Q . p
• Let us consider the function (2.5) We prove the following properties: (i) K (z)>0, (ii) / K {x)dx = 1, t
t
(iii) K,(z) -> S(z), (iv) K *Kf = K. t
t+t 7
t -> +0 in V, t,t' > 0.
Using the formula (3.3) of Sec. 4.3 for / ( a ) = e " (x)=(i-1-)
Kt
ki;
1
i > - e - " - ^ ' -
, a l
we have
i « r ^
w
'
a
1=0
> 0. y=0
p-Adic
Quantum Theories
299
This gives the proof of (i). Then / K (x)dx
= (A",,A ) =
t
T
l f l i
U ) = (e-'IffJ, t
m
= ( e - ^ i : , ^ ) = J 6tM)*~ '%
A*)
—'1,
k-^eo
and thus we have the property ( i i ) . One has m
K (x)
= F[e~ '}
t
— , I = 6(x),
t -* + 0 in V
and (iii) is proved. Taking into account the Theorem of Sec. 7.5 we get the property (iv): m
i% * A V - F{r *>
• e-'
,|E|
J] =
•rp*w#m\ = Kt+v
-
•
The function Kt(x) is the transition function of a Markov process which satisfies the conditions (A) and (B). 3. Generalized
Stochastic
Processes
Let L be a locally convex space, V dual space and { f i , £ , P} a probability space. A linear map
,
tffa),...
,
W]
is called a cylinder set. Here tpi 6 £ , 1 = 1 , . . . , n and A is a Borel set in ffi". The family of cylinder sets Eg is an algebra. Any subset N from the factor space L ' / $ where $ is the annulator is called a base. A cylinder measure p is a real valued function fi(c) on EQ with the following properties: l) 0 < u(c) < 1 for every c £ Eo ii) " ( E ) = 1 iii) i f a set c is a disjoint union of cylinder sets c\,. .. ,c ,... 0
n
based on
CO
N then pic) = £ / « 0 h ) n=l
iv) u(c) = inf/i((7) for every cylinder set C where U runs over all cylinder sets containing C. I f L is a nuclear space then the cylinder measure p. can be extended to a
300
v-Adic Analysis and Mathematical Physics
countably additive measure on E where E is the Borel ir-algebra generated by E . According to the Minlos theorem i f J : L —• C is a continuous function of positive type and J(Q) = 1 then there is a probability measure (i on {L', £ } such that 0
J{) = j e x p ( i * ( ? ) ) < f c { » .
J is called the characteristic function of u. Let B{) be a continuous inner product on a nuclear space L . Then
is a continuous characteristic function and thus there is a probability measure fi on { L ' , E } so that exp (-^B(
=
Jexp(i>(v))d/i(^). L'
The measure u is called a Gaussian measure on V and 4>(
Field
Theory
Let us take L = V(Q ) as the space of real test functions which was considered in Sec. 6. I t is a locally convex nuclear space. Consider the following inner product p
Q;
where a(k) > c > 0 is some function of k. This inner product gives rise to a Gaussian generalized process. I f we take n 2
+ m ,
m > 0
i=l
then by analogy with the Euclidean formulation of quantum field theory one can call the corresponding Gaussian process the free scalar p-adic quantum
p-Adic Quantum Theories 301 field. This field is invariant under group S0(n,
2
a(k) = | | t | | + m ,
Another natural choice
||*|| = m a x | j f c , V
p-adic white noise is obtained i f a(k) = 1. The measure dji can be formally written in the form
where the free action S o O ) = j 4>(x)a{D)
(3.1)
Here a(D) is a pseu do differential operator. The expression (3.1) is not defined for an arbitrary generalized functions
S(
with some function V{tj>).
V(
B ib lio gr aphy In the following we present a brief chapter by chapter bibliographic discussion of main ideas and results in p-adic mathematical physics. This branch of mathematical physics being rather new is related with different branches of mathematics. Namely, it is closely connected with very traditional mathematics having a long history, such as number theory as well as with modern abstract algebraic geometry and representation theory. The list of bibliography contains references to books on p-adic analysis. This list is not complete and the selected books have proved as more suitable for current mathematical physics applications. There are references to pure p-adic topics as well as to applications. Physical papers motivated p-adic considerations are also collected in the list. We apologize in advance for any errors or oversights and possible omissions.
Introduction We are not going to review enormous literature on the space-time geometry, we point out only a few references on this subject. A general mathematical discussion of the notion of the physical space has been performed by many authors, notably by Poincare [172] and H.Weyl [228]. Riemann [177] considered a continuous (renowned Riemannian geometry) as well as discrete models of the space. Our intuitive understanding of properties of space is expressed in the axioms of elementary geometry. A complete list of geometrical axioms was presented in the Hilbert famous "Grundlagen der Geometry'' [104]. The role of the Archimedean axiom and a possibility of construction of a non-Archimedean geometry was pointed out by Veroneze and Hilbert. An absolute limitation on length measurements in quantum gravity and string theory is discussed in [176, 171, 147, 229, 200, 109, 220] and [7, 94]. As Witten summarized, 'There are many reasons to believe that in the 302
Bibliography 303
string theory there is no such thing as distances less than the fundamental length -Jo7 A t distances below -JcP, not just physics as we know it but local physics altogether has disappeared. There will be no distances, no times, no energies, no particles, no local signals — only differential topology, or its string theoretic successor." [232]* A hypothesis on a possible non-Archimedean p-adic structure of spacetime at the Planck scale was suggested and considered by Volovich [221]. The basic role of rational numbers was stressed and the idea of fluctuating number fields was suggested [219-221]. p-Adic numbers were introduced in 1899 by K.Hensel. The books by Koblitz [121], Mahler [138] and Schikhof [186] can serve as an introduction to p-adic numbers and p-adic analysis. As a background reading on algebra and number theory one can use Borevich and Shafarevich [37], Kostrikin [126], Leng [128], Serre [189], Vinogradov [204] and Weil [225], A possible role of p-adic analysis in mathematical physics in the context of super analysis and supersymmetry was noted by Vladimirov and Volovich [211]. An application of ultrametricity in solid state physics was discussed by Mezard, Parisi, Rammal, Sourlas, Toulouse and Virasoro [157, 178]. The Parisi matrix [169] describing a hierarchical structure leads naturally to ultrametricity. A discussion of a possible role of number theory in physics one can find in Manin [142, 145], Atiyah [25] and Bott [39]. Comments on references related with p-adic analysis , p-adic quantum mechanics and strings will be done under discussion of corresponding chapters.
Chapter I p-Adic analysis is treated in [8, 38, 61, 82, 85, 121, 122, 179, 186]. A homeomorphism (see subsect.1.6) of p-adic numbers Q to some Cantor subset of the field of real numbers K was constructed by Zelenov [240]. In the presentation of the theory of additive characters we follow the approach of the Pontryagin book [172]. Gaussian integrals on an arbitrary locally compact abelian group were considered by Weil [225]. The explicit calculations for a special case of the field Q were performed by authors p
p
* Here \ZcV is the string theory notation for the Planck length.
304
p-Adic Analysis and Mathematical Physics
[212, 215, 218] and independently by Alacoque, Ruelle, Thiran, Verstegen and Weyers [3, 182] and by Meurice [151]. The number theory function Ap(a) was introduced and investigated by Vladimirov and Volovich [212], and by Alacoque at al. [3], The proof of the adelic formula for A (n) is presented here for the first time. p
A theory of distributions on an arbitrary locally compact group has been developed by Bruhat [43] and on a locally compact disconnected field by Gel'fand, Graev and Piyatjeckii-Shapiro [82]. We present elements of this theory for the field Q) along the line of the paper [206]. A distinguished feature of the theory of p-adic distributions as compare with Sobolev-Schwartz distributions over real numbers [205] is the fact that in the p-adic case from the linearity of a functional follows its continuity. A theory of convolution and multiplication of p-adic distributions using the Fourier transformation was firstly developed by Vladimirov [206]. Dealing with the theory of one-dimensional homogeneous distributions we follow the book by Gelfand, Graev and P y ate tsskii-Shapiro [82] and for multidimensional case the paper by Smirnov [193]. Properties of the two-dimensional Green function were considered by Bikulov [35]. p
Chapter I I A notion of a pseudo-differential operator on the space Q) has been introduced by Vladimirov [208]. A reader can find the notions of the spectral theory, used in this chapter, in Reed and Simon [175], Dunford and Schwartz [60] and Yosida [237]. p
The non-local operator of fraction differentiation and integration D° was introduced and investigated by Vladimirov [206] (see also [15, 168, 196, 245]). The spectral theory of the operator D a > 0, acting in Q has been performed by Vladimirov [207]. This treatment includes a calculation of an explicit form of eigenfunctions (see also [218]). The orthonormal basis of eigenfunctions of operator D", (including p = 2) in Q is firstly presented in subsection 9.5. p
p
Spectral theory of a pscudodiffcrcntial operator of the form a" + V ( i ) on an open-closed set G ( bounded or unbounded) with functions a(() and V{x) being bounded from below and going to +00 at the infinity has been constructed in [208], There one can also find an explicit form of the
Bibliography 305
eigenfunctions and eigenvalues of the operator D",a > 0 in the disk 5 and on the circle S-, ( i f p ^ 2). Here all these results are reproduced, the case of p = 2 is also considered, and the inversion of the main theorem is done in the stronger form. 7
The method of finding of invariant eigenfunctions ^ ( | x | ) of the p-adic Schrodinger operator D" + V(\x\ ) has been developed by Vladimirov and is given here for the first time. Further developments of spectral theory of the operator D -f- V ( | i r | ) , a > 0 in Q without a condition at the infinity have been performed by Kochubej [123]. p
p
a
p
p
A decomposition of the field Q Vladimirov [207].
p
on sectors has been suggested by
Stationary and non-stationary p-adic Schrodinger equations were suggested by Vladimirov and Volovich [214].
Chapter I I I About foundations of quantum mechanics see for example Dirac [52], Holevo [107], Mackey [137], Maslov [149]. A formalism of p-adic quantum mechanics based on a triplet ( X ( Q ) , W(z), U(t)) was suggested by Vladimirov and Volovich [212, 213). This formalism there appeared as a quantization of p-adic classical mechanics. Slightly different approaches were considered by Freund and Olson [74], Alacoque et al. [3] and Meurice [154]. Instead of the dynamical operator U(t) these approaches use a unitary representation of a non-abelian group and therefore are restricted by quadratic Hamiltonians. Several ways of construction of p-adic quantum mechanics were discussed by Parisi [168]. A Lagrangian formalism and Feynman path integral were briefly discussed in [213] and have been developed by Zelenov [240]. Another approach to p-adic quantum mechanics based on the probability measures on the space of distributions was also pointed out in [213]. Quantum mechanics with padic valued functions is also sensible [212]. A general approach to quantum mechanics with p-adic valued functions based on the theory of Gaussian distributions was elaborated by Khrennikov [114, 116, 120]. p-Adic Hilbert spaces are discussed by Bayod [29]. A l l the above-mentioned approaches are equivalent over real numbers but in the case of p-adic numbers their equivalence is questionable and only fragmentary information concerning relations between different approaches is available at the moment. Note p
p
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that in the construction of p-adic quantum mechanics ideas and results by H.Weyl [227], Mackey [136], Segal [187] and A.Weil [225] are essentially used. The theory of Jacquet and Langlends [81] should be helpful in further development of p-adic quantum mechanics. Much recent effort have been concentrated on studying spectral theory in p-adic quantum mechanics. An appealing property of this theory is the appearance of a rich spectrum for a rather simple system (p-adic harmonic oscillator). Spectral theory in p-adic quantum mechanics was carried out by authors [218], similar results were obtained by Ruelle et al. [182] and Meurice [151]. An investigation of p-adic Weyl systems in finite and infinite dimensional cases and study of coherent states and eigenfunctions for p = 3 (mod 4) were performed by Zelenov [241-243]. Formulations of p-adic string theory with p-adic valued and complex valued amplitudes as convolutions of characters were suggested by Volovich [219-221], Freund and Olson [73] made an important proposal to consider the p-adic 5-functions as string amplitudes. Freund has stressed an idea of using p-adic calculus in order to simplify calculations for real systems. A connection of p-adic string theory and the Weil conjecture in the number theory was discussed by Grossman [95]. An idea of adelic approach was suggested by Manin [145]. A n important adelic formula was pointed out by Freund and Witten [75]. Problems of regularization of this formula and an alternative adelic formula were considered by Aref'eva, Dragovic and Volovich [16]. Five-point p-adic amplitudes were considered by Marinary and Parisi [146] and by Frampton and Okada [65]. Appoint p-adic amplitudes were elaborated by Frampton and Okada [65,68] and by Brekke et al. [40, 42]. They derived an effective field theory with the remarkable soliton solution. Interesting papers by Frampton, Nishino, Okada and Ubriaco [69, 164, 165, 167] are devoted to p-adic o--models. The action from Sec. 16,4 was considered in quantum field theory by Parisi [168], Aref'eva and Volovich [15], Lerner and Missarov [131] and in string theory by Spokoiny [196] and Zhang [245]. This action uses the operator D considered by Vladimirov. p-Adic multiloop amplitudes on the Bruhat-Tits tree were considered by Chekhov, Mironov and Zabrndin [238, 44, 45]. Subsections 14.4-14.6, which deal with multiloop amplitudes, is a short extraction of enormous current literature on string theory. We shall indicate
Bibliography 307
here the sources of most of the material presented in these subsections, and attempt to direct the reader to additional papers. A nice short introduction to string theory is the old Scherk review [184]. Excellent reference on string theory is Green, Schwartz and Witten [92]. Multiloop calculations based on the path-integral method suggested by Polyakov [173] have been recently developed into a field of its own, and references on more detailed aspects of multiloop calculations are [6, 31, 87, 108, 143, 203], Multiloop calculations are closely related with geometry of complex manifolds [93, 162]. Properties of moduli space described in subsect. 14.5 are due to Mumford [160, 161] and Deligne and Mumford [50]. About p-adic Schottky groups and p-adic uniformization see Manin [141] and Gerritzen and van der Put [85]. Rigid analytic geometry was introduced by Tate [198]. I t is considered in Gerritzen and van der Put [85], Bosch, Guntzer and Remmert [28] and Fresnel and van der Put [72]. The theory of p-adic differential equations is presented in Dwork [61]. Gervais [86] suggested an idea of using p-adic variables to extend conformal symmetry to higher-dimensional case. An approach to p-adic string theory with p-adic valued amplitudes using the Morita gamma function was suggested by Volovich [221] and Grossman [95]. By means of Gross-Koblitz formula it leads to string theory on the Galois fields. The Galois-Veneziano-Jacobi string amplitudes [221] can be expressed in terms of Frobenius action on the space of etale cohomology. This approach is connected with the theory of motives and L-functions [158, 100, 48, 49, 28]. Witten proposed that at short distances there is a phase in which general covariance is unbroken. This phase is described by topological quantum field theories which are systems without local, propagating degrees of freedom. These theories are associated with the cohomology of various moduli spaces [233, 51]. I t seems it would be a natural next step to consider analogous theories over p-adic and other number fields. Note that recently methods of quantum field theory proved useful for investigation of properties of moduli spaces of Riemann surfaces [99, 170, 234, 124]. Section 15, which deals with relations between (-analysis and quantum groups is mainly taken from [22]. Other connections between them were pointed out by Macdonald [134] and Freund [78,79]. In the last years there has been considerable work done on the theory of quantum groups. Some references are Drinfeld [58], Jimbo [80], Faddeev, Reshetichin and Tachtadjan [62], Woronowicz [235] and Manin [144]. The g-deformed oscillator
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was considered by Biedenharn [34] and Macfarlane [135]. A general approach to non-com mutative geometry has been developed by Connes [47]. Non-commutative differential calculus was considered by Wess and Zumino [224]. About possible applications of quantum groups in field theory see Aref'eva and Volovich [23]. g-Analysis is discussed in Andrews [10] and Askey [24]. Koornwinder [125] discussed a relation between g-analysis and quantum groups. Stochastic processes and generalized random fields are considered by many authors. Note in particular Gelfand and Vilenkin [83], Gihman and Skorokhod [88], Accardi, Frigeiro and Lewis [2], Hida [103] and Heyer [102]. The probability theory approach to quantum field theory is considered by Glimm and Jaffe [89] and Simon [91]. Diffusion on p-adic numbers is considered by Albeverio and Karwowski [4]. Recently new interesting approaches to the description of the space-time structure at small distances have been suggested by Bennet, Nielsen and Picek [33], Alvarez, Cespedes and Verdaguer [4] and Isham, Kubyshin and Renteln [111]. Basic quantum variable is the two-point distance on a metric space. Such consideration naturally includes also the ultrametric. The first steps in investigation of p-adic Einstein equation and quantum p-adic gravity were performed by Arefeva, Dragovic, Frampton and Volovich in [20, 71, 21, 54, 57]
References 1. S.Abyankar: Local analytic geometry, N.Y.: Academic Press, 1964. 2. L.C.Accaidi, A.Frigerio and J.T.Lewis: Quantum stochastic processes. Publications RIMS 18, 97 (1982). 3. L.C.Alacoque, P.Ruelle, E.Thiran, D.Verstegen and J.Weyers: Quantum amplitudes on p-adic fields. Phya. Lett. 211B, 59-62 (1988). 4. S.Albeverio, and W.Karwowski: Diffusion on p-adic numbers, in "Gaussian Random Fields", The third Nagoya Levy Seminar, 15-20 Aug. 1990, eds. K.Ito, T.Hida, World Scientific Publ. 1991. 5. E.Alvarez: Phya. Lett. 210B, 73-78 (1988); E.Alvarez, J.Cespedesm and E.Verdaguer: Dynamical generation of space time dimensions. Preprint C E R N - T H . 5764/90 (1990). 6. L.Alvarez-Gaume, J.-B.Bost, G.Moore, P.Nelson and C.Vafa: Bosonization on Higher Genus Riemann Surfaces. Comm. Math. Phys. 112, 503-569 (1987). 7. D.Amati, M.Ciafaloni and G.Veneziano: Phys. Lett- 197, 81 (1987); Int. Mod. Phya. 3 A , 1615 (1988); Phya. Lett. 216, 41 (1989). 8. Y.Amis: Les nombres p-adiques. Presses Universitaires de France, 1975. 9. Y.Andre: G-functions and Geometry, Aspects of Mathematics E l 3 , Vieweg, 1988. 10. G.E.Andrews: ^-Series: Their develpment and application in analysis, number theory, combinatorics, physics, and computer algebra, in "Regional Conference Series in Math." v.66, Amer. Math. Soc., Providence, R I , 1986. 11. S.J.Arakelov: An intersection theory for divisors on an arithmeic surface. Izv, Akad Nauk SSSR, Ser. Math. 38, 1179-1192 (1974). 12. V . I , Arnold: Mathematical Methods of Classical Mechanics. Moscow: Nauka, 1974. 13. I.Ya.Aref'eva: String Field Theory, in "Conformal Invariace and String Theory", eds, P.Dita and V.Gergescu, Boston: Acadic Press, 1989. 14. I.Ya.Aref'eva: Physics at the Planck Length and p-adic Field Theory, in "Differential Geometry Methods in Theoretical Physics", eds. L . L . Chau and W.Nahm, N.Y.:Plenum Press, 1990. 15. I.Ya.Aref'eva and I.V.Volovich: "String, gravity and p-adic space-time", in: Quantum Gravity, eds. M.A.Markov, V.A.Berezin and V.P.Frolov, World Scientific Publishing, 1988. 16. I.Ya.Aref'eva, B.Dragovic and LV.Volovich: On the adelic string amplitudes. Phys. Lett. B 2 0 9 , 445-450 (1988). 17. I.Ya.Aref'eva, B.Dragovic and I.V.Volovich: On the p-adic summability of the anharmonk oscillator. Phys. Lett. B200, 512-514 (1988). 18. I.Ya.Aref'eva, B.Dragovic and I.V.Volovich: Open and closed p-adic strings and quadratic extensions of number fields. Phys. Lett. B 2 1 2 , 283-289 (1988). 19. I.Ya.Aref'eva, B.Dragovic and I,V,Volovich: P-adic superstrings. Phys. Lett. B 2 1 4 , 339-346 (1988). 20. I.Ya.Aref'eva, B.Dragovic, P.Frampton and I.V.Volovich: Wave function of the universe and p-adic gravity. Mod. Phys. Lett. A 6 , 4341-4358 (1991). 309
310
p-Adic Analysis and Mathematical Physics
21. I.Ya.Aref'eva and P.H.Frampton: Beyond Planck energy to non-Archimedean geometry. Mod. Phys. Lett. A 6 , 313-316 (1991). 22. I.Ya.Aref'eva and I.V.Volovich: Quantum groups particles and non-Archimedean geometry. Phys. Lett. B 2 6 8 , 179-193 (1991). 23. I.Ya.Aref'eva and I.V.Volovich: Quantum group gauge fields. Mod. Phys. Lett. A 6 , 893-906 (1991). 24. R.Askey: The 3-gamma and j-beta functions, Appl. Anal. 8, 125-146 (1978). 25. M.F.Atiyah: Commentary on the article of Manin. Led. Notes in Math. 1111, pp.103-109, Springer, 1985 26. M.F.Atiyah and R.Bott: The Yang-Mills equation over Riemann surfaces, Phil. Trans. R. Soc. London A 3 0 8 , 523-615 (1982). 27. I. At irk and E . Witten: The Hagedorn transition and the number of degrees of freedom of string theory, Preprint IASSNS-HEP-88/4 (1988). 28. F.Baldassarri: p-Adic interpolation of Evans sums and deformation of the Selberg integrals: an application of Dwork's theory. In: Special Differencial Equations. Proc. of the Taniguchi Workshop, p.7-35, 1991. 29. J.M.B. Bayod: Productos Internos en Espacios Normados no Arquimedianos. Memoria presentada para optar al grado de Doctor en Ciencias (Seccion de Matematicas). Univ. de Bilbao, 1976. 30. J.M. Bayod, N. De Grande-DeKimpe and J.Martinez-Maurica: p-Adic functional analysis. N.Y.: Marcel Dekker, Inc., 1992. 31. A.A.Betavin and V.G.Knizhnik: Algebraic geometry and the geometry of quantum strings. Phys. Lett. B 1 6 8 , 201 (1986). 32. S. Ben-Menahem; p-adic iterations. Preprint TAUP-1627-88, 1-17 (1988). 33. D.L.Bennett, H.B.Nielsen and I.Picek: Phys. Lett. B 2 0 8 , 275-284 (1988). 34. L.C.Biedenharn: J. Phys. A. Math. Gen. 22, L873 (1989). 35. A.H.Bikulov: Investigation of the p-adic Green function. Theor. Math. Phys. 87, 376-390 (1991). 36. J.-M. Bismut, H.Gillet and C.Soule: Analytic torsion and holomorphic determinant bundles I, II, III. Comm. Math. Phys. 115, 301-350 (1988). 37. Z.I.Borevich and I.R.Shafarevich: Number theory. N.Y.: Academic Press, 1966. 38. S.Bosch, U.Guntzer and R.Remmert: Non-Archimedean Analysis,Berlin: Springer-Verlag, 1984. 39. R.Bott: On the shape of the curve. Advances in Mathematics, 16, 144-159 (1975). 40. L.Brekke, P.G.O.Freund, E.Melzer and M.Olson: Adelic A -point amplitudes. Phys. Lett. B 2 1 6 , (1989). 41. L.Brekke and M.Olson: p-Adic diffusion and relaxation in glasses. Preprint UTTG-16-89, EFI-89-23 (1989). 42. L.Brekke, P.G.O.Freund, M.Olson and E.Witten: Non-archmedean string dynamics. Nucl. Phys. B 3 0 2 , 365-402 (1988). 43. F.Bruhat: Distributions sur tin groupe localemont compact et applications a I'etude des representations des groupes p-adiques. Bulletin Soc. Mathem. de France 89, 43-75 (1961). r
References
311
44. L.Chekhov: A note on multiloop calculus in p-adic string theory. Mod. Phys. Lett. A 4 , 1151-1158 (1989). 45. L.O.Chekhov, A.D.Mironov and A.Zabrodin: Multiloop calculations in p-adic string theory and Bruhat-Tits trees 1,11. Preprints ITP-89-41E, 42E (1989); Mod. Phys. Lett. A 4 , 1127-1235 (1989). 46. G.Christol: Solutious algebriques des equatious differenielles p-adique. Prog. Math. 38, 51-58 (1983). 47. A.Connes: Non Commutative geometry. Publ. Math. I H E S , 62 (1986). 48. P.Deligne: Hodge Cycles on Abelian Varieties. In: P.Deligne, J.S,Milne, A.Ogus and Shih Kuang-yen, Hodge Cycles, Motives and Shimura Varieties. £ect. Notes in Math., No.900, pp.9-100, Springer-Verlag, 1982. 49. P.Deligne: Valeurs de functions L et periodes g'integrales. Proc. Symp. Pure Math. v . X X X I I I , 313-336, Amer. Math. Soc. Providence, 1979. 50. P.Delinge and D.Mumford: The irreducibility of the space of curves of a given genus, Publ.I .H.E.S. 36, 1969. 51. R.Dijkgraaf, H.Verlinde and E.Verhnde: Notes on Topological String Theory and 2D Quantum Gravity. Preprint PUPT-1217, IASSNS-HEP-90/80 (1990). 52. P.A.M.Dirac: The principles of Quantum Mechanics. Oxford: At the Clarendon Press, 1958. 53. B . G . Dragovic:: Adelic summability of perturbation series. Preprint IF-19/89, 1-15 (1989). 54. B . G . Dragovic, P.H.Frampton and B.V.Urosevic: Classical p-adic space-time. Mod. Phys. Lett. A 5 , 1521-1528 (1990). 55. B . G . Dragovic: p-Adic perturbation series and adelic summability. Phys. Lett. 2 5 6 B , 392-396 (1991). 56. B.G.Dragovic: On factorial perturbation series. Preprint IF-91-11, 1-8 (1991). 57. B.G.Dragovic: On signature change in p-adic space-times. Mod. Phys. Lett, A6, 2301-2307 (1991). 58. V.G.Drinfeld: Quantum groups. In: Proc. Intern. Congr. Math., Berkeley, 1986, pp.798-820, 59. B.A.Dubrovin, S.P.Novikov and A.T.Fomenko, Modern Geometry.Moscow: Nauka, 1979. 60. N.Dunford and J.T.Schwartz: Linear Operators, part 2. Spectral Theory. Selfadjoint Operators in Hilbert Space. N.Y.-London: Interscience Publishers, 1963. 61. B.M.Dwork: Lectures on p-adic differential equations, N.Y.: Springer-Verlag, 1977. 62. L.D.Faddeev, N.Reshetikhin and L.A.Tachtadjan: Alg. Anal. 1, 129 (1988). 63. G.Faltings: Calculus on arithmetic surfaces. Ann. of Math. 119, 387-424 (1984). 64. G.M.Fichtengolz: Differential and integral Calculus. Vol.3, Gostechizdat, 1949. 65. P.H.Frampton and Y-Okada: The p-adic string N-point function. Phys. Rev. Lett. 60, 484-488 (1988).
312
p-Adic Analysis and Mathematical Physics
66. P.H.Frampton, Y.Okada and M.R.Ubriaco: On adelic formulas for the p-adic string. Phys. Lett. B 2 1 3 , 260-264 (1988). 67. P.H.Frampton, Y.Okada and M.R.Ubriaco: New p-adic strings from old dual models. Phys. Rev. D39, 1152-1156 (1989). 68. P.H.Frampton and Y.Okada: Effective scalar field theory of p-adic string. Phys. Rev. D37, 3077-3079 (1989). 69. P.H.Frampton and H.Nishino: Theory of p-adic closed strings. Phys. Rev. Lett. 62, 1960-1964 (1989). 70. P.H.Frampton: p-Adic number fields and string tree amplitudes. In: Problems in high energy physics and field theory. Proc. of the X I workshop, Protvino 1988, Nauka, 1989. 71. P.H.Frampton and I,V.Volovich: Cosmogenesis and primary quantization. Mod. Phys. Lett. A 4 , 1825-1832 (1990). 72. J.Fresnel and M.van der Put: Geometrie Analytique Rigide et Applicatiions, Boston: Birkhauser, 1981. 73. P.G.O.Freund and M.Olson: Non-archimedean strings. Phys. Lett. B 1 9 9 , 186¬ 190 (1987). 74. P.G.O.Freund and M.Olson: Nucl. Phys. B 2 9 7 , 86-102 (1988). 75. P.G.O.Freund and E.Witten: Adelic string amplitudes. Phys. Lett. B 1 9 9 , 191-194 (1987). 76. P.G.O.Freund: Real, p-adic and adelic strings. Proceedings of IXth Int. Congress on Mathematical physics, Swansea, 1988, Adam Hilder, 282-285 (1989). 77. P.G.O.Freund: Scattering on p-adic and on adelic symmetric spaces. Preprint EFI-90-87, 13-17 (1990). 78. P.G.O.Freund: in Superstrings and Particle Theory, L.Clavelle and B.Harms eds., World Scientific Publishing, Singapore, 251 (1990). 79. P.G.O.Freund: On the quantum group - p-adic connection. Chicago preprint (1991). 80. M. Jimbo: A g-difference analogue of U(g) and the Yang-Baxter equation, ieff. Math. Phys. 10, 63-69 (1985). 81. H.Jacquet and R.P.Langlands: Automorphic Forms on G i ( 2 ) , Lecture Notes in Mathematics, v.114, Berlin: Springer-Verlag, 1970. 82. I.M.Gelfand, M.I.Graev and I.I.Pjatetskii-Shapiro: Representation Theory and Automorphic Functions. London: Saunders, 1966. 83. I.M.Gelfand and N.Ya.Vilenkin: Generalized functions, vol.IV, Moscow, Nauka, 1961 84. L.Gerritzen: p-Adic Siegel halfspace. Groupe Etude Anal. Ultrametrique 3, J9, 1981/82. 85. L.Gerritzen and M. van der Put: Schottky Groups and Mumford Curves. Lecture Notes in Mathematics, v.817. Berlin: Springer-Verlag, 1980. 86. J.L.Gervais: p-Adic analyticity and Virasoro algebras for conformal theories in more than two dimensions. Phys. Lett. B 2 0 1 , 306-310 (1988). 87. S.W. Giddings: Fundamental strings Preprint HUTP-88/A061 (1988).
References 313
88. I.I.Gihman ajid A.V.Skorokhod: Introduction to the theory of random processes. Moscow: Nauka, 1965. 89. J.Glimm and A.Jaffe: Quantum Physics, A Functional Intyegral Point of View, N.Y.: Springer-Verlag, 1981. 90. J.A.Gracey and D.Verstegen: The O(N) Gross-Neveu and supersymmetric ir-models on p-adic fields. Mod. Phys. Lett. A 5 , 243-254 (1990). 91. N.De Grande-De Kimpe and L.Van Hamme (eds.) Proceedings of the Conference on p-adic analysis, Houthalen, BrussehVrije Universiteit Brussel, 1986. 92. M.B. Green, J.H.Scwarz and E.Witten: Superstring Theory. London: Cambridge University Press, 1987. 93. Ph.Griffiths and J.Harris: Principles of Algebraic Geometry, N.Y.: J.Wiley and Sons, 1978 94. D.Gross and P.F.Mende: String theory beyond the Planck scale. Nucl. Phys. 303, 407 (1988). 95. B.Grossman: p-Adic strings, the Weyl conjectures and anomalies. Phys. Lett. B197 101-105 (1987). 96. B.Grossman: The adelic components of the Dirac operator. / . Phys. A 2 1 , L1051-L1060 (1988). 97. B.Grossman: Adelic conformal field theory. Phys. Lett. B215, 14-19 (1988). 98. B.Grossman: Arithmetic Directions in Topologycal Quantum Field Theory and Strings. Preprint Rockefeller Univ. DOE/ER40 325-52-Task.B. (1988). 99. J.Rarer and D.Zagier: The Euler characteristic of the moduli space of curves. Invent. Math. 185, 457-486 (1986). 100. R.Hartshorne: Algebraic Geometry, N.Y.: Springer-Verlag, 1977. 101. F.Herrlich: Moduli and Teichmuller space for degenerating curves. Proc. Conf. p-adic anal. Hendelhoef, 1986, pp.83-95. 102. H.Heyer: Probability measures on locally compact groups, Berlin: Springer, 1977. 103. T.Hida: Brownian motion. Springer-Verlag, 1980. 104. D.Hilbert: Grundlagen der Geometrie, Leipzig, 1930. 105. Z.Hlousek and D.Spector; Scattering amplitudes in p-adic string theory. Phys. Lett. B214, 19-25 (1988). 106. Z.Hlousek and D.Spector: p-Adic string theory. Preprint CLNS 88/832, 1988. 107. A.S.Holevo: Probabilistic and statistical aspects of quantum theory. Moscow: Nauka, 1980. 108. E.D'Hoker and D.Phong: The geometry of string perturbation theory, Preprint PUPT-103,1988. 109. G.'t Hooft: Nucl.Phys, B256, 1985: "Gravitational Collapse and Quantum Mechanics'':, Lectures given at the 5th Adriatic Meeting on Particle Physisc, Dubrovnik, 1986. 110. C.J.Isham, Topological and global aspects of quantum theory, in:Relativity, groups and topology I I , eds. B.S.DeWitt and R.Stora , 1983. 111. C.J.Isham, Y.Kubyshin and P.Renteln: Quantum norm theory and the quantization of metric topology. Class. Quant. Grav. 7, 1053-1074 (1990).
314
p-Adic Analysis and Mathematical Physics
112. E.Kani: Potential theory on curves, N S E R C preprint,1987. 113. G.Kato: Frobenius map of Fermat curves for p-adic strings. Princeton preprint, 1988. 114. A.Yu.Khrennikov: Quantum mechanics over non Archimedean number field. TAeor. Math. Phys. 83, 406-418 (1990). 115. A.Yu.Khrennikov: Quantum mechanics over Galois extantions of number fteld. Dokl. Acad. Nauk USSR 315, 840-864 (1990). 116. A.Yu.Khrennikov; Mathematical methods of non-Archimedean physics. Usp. Math. Nauk. 45, 79-110 (1990). 117. A.Yu.Khrennikov: Pseudodifferential operators over non Archimedean spaces. Diff. Equal. 26, 1044-1053 (1990). 118. A.Yu.Khrennikov: Real and non-Archimedean structure of the space-time. Theor. Math. Phys. 86, 177-190 (1991). 119. A.Yu.Khrennikov: Generalized functions and Gauss path integrals over nonArchimedean functional spaces, hv. Acad. Nauk USSR 55, 780-814 (1991). 120. A.Yu.Khrennikov: p-Adic quantum mechanics with p-adic valued functions. J. Math. Phys. 32, 932-936 (1991). 121. N.KobHtz: p-Adic numbers, p-adic analysis, and zeta-functions. Berlin, Heidelberg, New York: Springer, 1984. 122. N.Koblitz: p-Adic analysis: a short course on recent work. London: Cambridge University Press, Mathematical Society Lecture Notes, series 46, 1980. 123. A.N.Kochubej: Schrodinger-type operator over the field of p-adic numbers. Tfieor. Math. Phys. 86, 323-333 (1991). 124. M.Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy functions. Max-Planck-Institute preprint MP1/91-77 (1991). 125. T.H.Koornwinder: Orthogor.al polynomials in connection with quantum groups. In "Orthogonal PolynomialsiTheory and Practice'', ed. P.Nevai, NATO ASI Series C , v.294, Kluwer Academic Publ.,1990. 126. A.I.Kostrilcin: Introduction to algebra, M.:Nauka, 1977. 127. S.Lang: Fundamentals of Diophantine Geometry. N.Y.: Springer-Verlag, 1983. 128. S. Lang: Algebra. Addison-Wesley, 1965. 129. D.R.Lebedev and A.Yu.Morosov; An attempt of p-adic one-loop compulations. Preprint 1 T E P 163-88 (1988) 130. E.Y.Lerner and M.D.Missarov: p-Adic Feynman string amplitudes. Commun. Math. Phy$. 121, 35-48 (1989). 131. E.Yu.Lerner and M.D.Missarov: Scalar models in p-adic quantum field theory and hierarchical models. 77ieor. Math. Phys. 78, 248-257 (1989). 132. L.Leroy: Particle in a p-adic box. Mod. Phys. Lett. A 5 , 1359-1364 (1990). 133. J.L.Lucio and Y , Meurice: Asymptotic properties of vandom walks on p-adic spaces. Preprint U. of Iowa 90-33, 1-5 (1990). 134. I.G.Macdonald: Orthogonal polynomias associated with root systems, preprint. 135. A.J.Macfarlane: J. Phys. A. Math. Gen. 22, 4581 (1989).
References 315 136. G.W.Mackey: Unitary Group Representations in Physics, Probability, and Number Theory, N.Y.: Addison-Wesley Publ. Comp.,, Inc, 1989. 137. G.W.Mackey: Mathemetical foundation of quantum mechanics, W.A. Benjamin, N . Y . , 1963. 138. K . Mahler: p-Adic numbers and their functions, London: Cambridge University Press, Cambridge tracts in mathematics 76,1980. 139. B.Mandelbrot: The Fractal Geometry of Nature. Freeman, 1982. 140. Yu.I.Manin and V.G.Drinfeld: Periods of p-adic Schotktky groups, J.retne angew.Math., 2 6 2 / 2 6 3 , 239 (1973). 141. Yu.I.Manin. p-Adic Automorphic Functions, Itogi Naukii Tekhnilci, Sovremennie Problemy Mathematiki, 3, 5-92, 1974. 142. Yu.I.Manin: New dimensions in geometry. Lect. Notes in Math. 1111, 59¬ 101, Springer, 1985. 143. Yu.I.Manin: Quantum strings and algebraic curves, Proc. Int. Cong. Math., Berkeley, 1986, p.1286-1296. 144. Yu.I.Manin: Multiparametric deformations of the general linear supergroup. Comm. Math. Phys. 123, 163 (1989). 145. Yu.I.Manin: Reflections on Arithmetical Physics. In "Conformal Invariace and String Theory", eds. P.Dita and V.Gergescu, p.293-303. Boston: Acadic Press, 1989. 146. E.Marinari and G.Parisi: On the p-adic five point function. Phys. Lett. 203 B . 52-56 (1988). 147. M.A.Markov: Progr.Theor. Phys. Suppl. 85, (1965). 148. A.Marshakov and A.Zabrodin: New p-adic string amplitudes. Phys. Lett. (1991). 149. V.P.Maslov: Perturbation theory and asymptotic methods. Moscow: Moscow University Press, 1965. 150. V.P.Maslov: Operator Methods. Moscow: Nauka, 1973. 151. Y.Meurice: Quantum mechanics with p-adic numbers. Int. J. Mod. Phys. A 4 , 5133-5137 (1989). 152. Y.Meurice: A path integral formulation of p-adic quantum mechanics. Phys. Lett. 2 4 5 B , 99-104 (1990). 153. Y.Meurice: A discretization of p-adic quantum mechanics. Comm. Math. Phys. 135, 303-312 (1991). 154. Y.Meurice: The classical harmonic oscillator on Galois and p-adic fields. Intern. Journ. of Mod. Phys. A 4 , 2211-2233 (1989). 155. Y.Meurice: Quantum mechanics with p-adic time. Preprint C I N V E S T A V FIS-12-89, 1-14 (1989). 156. Y.Meurice: Symanzik's Field Theory on p-adic Spaces, Iowa Preprint, (1991). 157. M.Mezard, G.Parisi, N.Sourlas, G.Toulouse and M.A.Virasoro: J. Phys. (Paris) 45, 843 (1984). 158. J.S.Milne: Etale Cohomology. Princeton: Princeton University Press, 1980. 159. M.D.Missarov: Random fields on the adele ring and Wilson's renormalization group. Ann. Inst. H. Poincare 49, 357-367 (1989).
316
p-Adic Analysis and Mathematical Physics
160. D.Mumford: An analytic construction of degenerating curves over complete local fields. Composite* Math. 24, 129 (1972). 161. D.Mumford: Stability of projective varieties. Enseign. Math. 23, 39-100 (1977). 162. D.Mumford: Tata lectures on Tetha 1,11. Boston: Birkhauaer, 1983,1984. 163. M.A.Naimark: Normed rings, moscow: Nauka, 1968. 164. H.Nishino, Y.Okada and M.R.Ubriaco: Effective field theory from a p-adic string. Phys. Rev. D40, 1153-1157 (1989). 165. H.Nishino and Y.Okada: Beta function approach to p-adic string. Phys Lett. B219, 258-262 (1989). 166. Y.Okada: p-Adic string amplitude. Nucl.Phys. B6, 177-182 (1989). 167. Y.Okada and M.R.Ubriaco; Renormalization of O(N) Nonlinear ff-Model on a p-adic Field. Phys. Rev. Lett. 61, 1910-1913 (1988); A 3 , 639-643 (1988). 168. G.Parisi: On p-adic functional integrals. Mod. Pkys. Lett. A4, 369-374 (1988). 169. G.Parisi: /. Phys. A13, 1887-1892 (1980). 170. R.C.Penner: Perturbation series and the moduli space of Riemann surface. J. Diff. Geom. 27, 35-59 (1988). 171. A.Peres and N.Rosen: Quantum limitation to the measurement of gravitational field. Phys. Rev. 118, 335-376 (1960). 172. H.Poincare: La Science et l'Hypothese, Paris: Flammarion, 1923. 173. A.M.Polyakov: Quantum geometry of bosonic strings. Phys. Lett. B103, 207-212 (1981). 174. L.S.Pontrjagin: Continous groups. Moscow:Nauka, 1972. 175. M.Reed and B.Simon: Methods of Modern Mathematical Physics. 1: Functional Analysis, 1972; 4: Analysis of Operators, 1978,-Academic Press. 176. T.Regge: Nuovo Cimento 7, 215 (1958). 177. B.Riemann: Uber die Hypothesen die der Geometrie Zugrunde leigen, Nachr. Ges. Gottingen, Bd.13, S.133 (1868). 178. R.Rammal, G.Toulouse and M.A.Virasoro: Ultrametricity for physicist. Rev. Mod. Phys. 58, 765-821 (1986). 179. A. van Rooij: Non-archimedean functional analysis. N.Y.: Marcel Dekker Inc., 1978. 180. B.O.B.Roth: A general approach to quantum fields and strings on adeles. Phys. Lett. B213, 263-268 (1988). 181. P.Ruelle, E.Thiran, D.Verstegen and J.Weyers: Adelic string and superstring amplitudes. Mod. Phys. Lett. A4, 1745-1753 (1989). 182. P.Ruelle, E.Thiran, D.Verstegen and J.Weyers: Quantum mechanics on padic fields. J. Math. Phys. 30, 2854-2859 (1989). 183. Z.Ryzak: Scattering amplitudes from higner dimensional p-adic world sheets. Phys. Lett. B208, 411-416 (1988). 184. J.Scherk; An introduction to the theory of dual models and strings. Rev. Mod. Phys. 47, 123-165 (1975).
References 317 185. W.H.Schikhof: Non-Archimedean Harmonic Analysis, Ph.D. Thesis, Rotterdam University, 1967. 186. W.H.Schikhof: Ultrametric calculus. An introduction to p-adic analysis. Cambridge University Press, 1984. 187. I.E.Segal: Mathematical Problems of Relativistic Physics. Amer. Math. Soc.
Providence, R.I., 1963. 188. I.R.Shafarevich: Foundations of Algebraic geometry. Moscow: Nauka, 1980. 189. J.P.Serre: Cours d'Arithmetique. Presses Universitaires de France, Paris, 1970. 190. J.P.Serre: Lie Algebras and Lie groups. - N.Y.: Benjamin, 1985. 191. B.Simon: The P(<&) Euclidean (Quantum) Field Theory. Princeton, N . J . 1974. 192. V.A.Smirnov: Renormalization in p-adic quantum field theory. Mod. Phys. Lett. A 6 , 1421-1427 (1991). 193. V . A . Smirnov: p-Adic Feynman amplitudes. Preprint MPI-Ph/91-48, 1-13 (1991). 194. O.G.Smolianov: Analysis on linear topological spaces and its applications. Moscow. Moscow University Press, 1979. 195. C.Soule: Geometric d'Arakelov des surfaces arithmetiques. Seminare Bourbaki, 1988-89, No.713, Asterisque 177-178 (1989), p.327-343. 196. B.L.Spokoiny: Quantum geometry of non-Archimedean particles and strings. Phys. Lett. 2 0 7 B , 4 G H 0 6 (1988). 197. B.L.Spokoiny: Non-Archmedean geometry and quantum mechanics. Phys. Lett. B 2 1 1 , 120-125 (1989). 198. J.Tate: Rigid Analytic spaces. Invent. Math. 12, 257-293 (1971). 199. E.Thiran, D.Verstegen and J.Weyers: p-Adic Dynamics. I. Stat. Phys. 54, 893-913 (1989). 200. H.-J.Treder: in "Relativity, Quants and Cosmology", ed. F.de Finis, N . Y . , 1979. 201. M.R.Ubriaco: Fermions on the field of p-adic numbers. Phys.Rev. D41, 2631¬ 2636 (1990). 202. M.R.Ubriaco: Field quantization in non arch imedean field theory. Preprint LTP-012-UPR. 203. E.Verlinde and H.Verlinde: Chiral bosonization, de terminals and string partition functions. Preprint Utrecht, 1986. 204. I.M.Vinogradov: Elements of Number theory. Moscow: Nauka, 1972. 205. V.S.Vladimirov: Generalized functions in mathematical physics. Moscow: Nauka, 1979. 206. V . S .Vladimirov: Generalized functions over p-adic number field. Usp. Mat. Nauk 43, 17-53 (1988). 207. V.S.Vladimirov: p-Adic analysis and p-adic quantum mechanics. Ann. of the NY Ac. of Sci: Symposium in Frontiers of Math. (1988). 208. V.S.Vladimirov: On the spectrum of some pseudo-differential operators over p-adic number field. Algebra and analysis 2, 107-124 (1990).
318
p-Adic Analysis and Mathematical Physics
209. V.S. Vladimirov: An application of p-adic numbers in quantum mechanics. In: Selected topics in Statistical mechanics, 22-24 August 1989, Dubna, USSR, World Scientific Publishing, 1990, p.282-297. 210. V.S.Vladimirov: Applications of p-adic numbers in mathematical physics, Delph: Culturel Center, 1989. 211. V.S. Vladimirov and L V . Volovich: Super analysis. Differential calculus. Theor. Math. Phys. 59, 3-27 (1984). 212. V.S.Vladimirov and I.V.Volovich; P-Adic Quantum Mechanics. Soviet. Math,. Dokl. 303, No.2, 320-323 (1988). 213. V.S.Vladimirov and I.V.Volovich: P-Adic Quantum Mechanics, Commun. Math. Phys. 123, 659-676 (1989). 214. V.S.Vladimirov and I.V.Volcvich: P-adic Schrodinger-type equation. Lett, in Math. Phys. 18, 43-53 (1989). 215. V.S.Vladimirov and I.V.Volovich: A vacuum state in p-adic quantum mechanics. Phys. Lett. B 2 1 7 , 411-414 (1989). 216. V.S.Vladimirov and I.V.Volovich: Applications of p-adic numbers in Mathematical physics. Trudi MIAN 200, 88-99 (1991). 217. V.S.Vladimirov, I.V..Volovich and E.I.Zelenov: Spectral theory in p-adic quantum mechanics and representation theory. Soviet. Math. Dokl. 41, 40-44 (1990). 218. V.S.Vladimirov, I.V.Volovich and E.I.Zelenov; Spectral theory in p-adic quantum mechanics and representation theory. Izv. Acad. Nauk USSR 54, 275-302 (1990); Math. USSR Izvestya 36, 281-309 (1991). 219. I.V.Volovich: p-Adic space-time and string theory. Theor. Math. Phys. 71, 337-340 (1987). 220. I.V.Volovich: Number theory as the ultimate physical theory. Preprint C E R N - T H . 87, 4781-4786 (1987). 221. I.V.Volovich: p-Adic string. Class. Quantum Grav. 4, L83-L87 (1987). 222. I.V.Volovich: Harmonic analysis and p-adic strings. Lett. Math. Phys. 16, 61-66 (1988), 223. I.V.Volovich: p-Adic quantum theory and srtings. In: Proceedings of IXth Int. Congress on Mathematical physics, Sweansea, 1988, Adam Hilger, 286¬ 293 (1989). 224. J.Wess and B.Zumino: Covariant differential calculus on the quantum hyperplane. Preprint CERN-TH-5697/90 (1990). 225. A.Weil: Sur certains groupes d'operateurs unitaries. -4c(a Math. I l l , 143¬ 211 (1964). 226. A.Weil: Basic Number Theory. Berlin: Springer, 1985 227. H.Weyl: The theory of groups and quantum mechanics. New York: Dover, 1931. 228. H.Weyl: Philosophy of Mathematics and Natural Science. Princeton University Press, 1949. 229. J.A.Wheeler: "Superspace and the Nature of Quantum Geometrodynamics". In; C.M.DeWitt and J.A.Wheeler, eds., Battell Rencontres: 1967 Lectures in
References
319
Mathematical Physics, pp.242-307, N.Y.: Benjamin, 1968. 230. B.De Witt: The Quantization of Geometry. In: "Gravitation: an Introduction to Current Research", ed. L.Witten, N.Y.:J.WUey and Sons, 1962. 231. E.Witten: Free fermions on an algebraic curve. In: Proc. of the Symposium on the Mathematical Heritage of Herman Weyl, Durham, North Carolina, 1987. 232. E.Witten: The search for higher Symmetry in String Theory, Preprint IASSNS-HEP-88/55 (1988). 233. E.Witten: Topological Quantum Field Theories. Comm. Math. Phys. 117, 353-391 (1988). 234. E . Witten: Two dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom. 1, 243-289 (1991). 235. S.Woronovicz: Comm. Math. Phys. I l l , 613 (1987); 122, 125 (1989); Publ. RIMS Kyoto Univ. 23, 117 (1987). 236. H.Yamakoshi: Arithmetic of strings. Phys. Lett. B207, 426-428 (1988). 237. K.Yosida: Functional analysis. Springer-Verlag, 1965. 238. A.V.Zabrodin: Non-archimedean strings and Bruhat-Tits trees. Mod. Phys. Lett. A 4 , 367-376 (1989). 239. E.I.Zelenov: p-Adic quantum mechanics for p = 2. Theor. Math. Phys. 80, 253-263 (1989). 240. E.I.Zelenov: p-Adic path integrals. J. Math. Phys. 32, 147-152 (1991) 12. 243. E.I.Zelenov: p-Adic quantum mechanics and coherent states. 1. Weyl systems. Theor. Math. Phys. 86, 210-220 (1991). 242. E.I.Zelenov: p-Adic quantum mechanics and coherernt states 2. Oscillator eigenfunctions. Theor. Math. phys. 86, 375-384 (1991). 243. E.I.Zelenov: Representations of commutation relations of p-adic systems of infinitely many degrees of freedom. J. Math. Phys. 33, 178-188 (1992). 244. E.I.Zelenov: p-Adic Heisenberg group, Maslov index and metaplectic representation. Preprint Print-91-0439, 1-57 (1991). 245. R.B.Zhang: Lagrangian formulation of open and closed p-adic strings. Phys. Lett. B 2 0 9 , 229-232 (1988).
