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S. M. Tashpulatov, Spectra and bound states of the energy operator of two- magnon systems in a non-Heisenberg ferromagnet with spin one and nearest-neighbor coupling, TMF , 2000, Volume 125, Number 2, 282–296

DOI: https://doi.org/10.4213/tmf669

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November 6, 2022, 22:58:09

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^x), ^h¹^(x;^t)⁼^h¹^(x;

−

^x;^t)^¨^f^(x)⁼^f^(x;

−

^x).

§¢¥áâ®,çâ®¥¯à¥àë¢ë©á¯¥ªâà®¯¥à â®à ^H

e

2

¥§ ¢¨á¨â®âäãªæ¨©h

1 (x;t)¨

á®áâ®¨â¨§®âà¥§ª®¢[m

;

M

^]⁼^G^,^£¤¥^m⁼^inf^x^h^(x)^¨

M

⁼^supx h

(x).

®¡áâ¢¥ ïäãªæ¨ï'

∈

^L²^(T⁾^{®¯¥à â®à}^H

e

2

,®â¢¥ç îé ïá®¡áâ¢¥®¬ã§ Ä

ç¥¨îz

∈

^G^,^{§ë¢ ¥âáï}^{®¯¥à â®à}^H

e

2

, ¢¥«¨ç¨ z

{í¥à£¨¥©íâ®£®.

áá¬®âà¨¬®¯¥à â®àK

,

K

(z)f

(x)=

Z

T

h

1 (x;t)

h

(t)

−

^z^f^(t)^dt:

ï¢«ï¥âáï¢¯®«¥¥¯à¥àë¢ë¬®¯¥à â®à®¬¢¯à®áâà áâ¢¥

H e

2

¤«ï§ ç¥¨©z,«¥Ä

¦ é¨å¢¥¬®¦¥áâ¢ G

=Imh

(x)=[m

;

M

]. ¡®§ ç¨¬ç¥à¥§

(z)®¯à¥¤¥«¨Ä

â¥«ìà¥¤£®«ì¬ ®¯¥à â®à E

+K

(z),£¤¥E

{¥¤¨¨çë©®¯¥à â®à¢

H e

2 ,

(z)=1+

X ∞

n=1 1

n!

d

n (z);

d

n (z)=

Z

T

· · · Z

T

det

k

^h¹^(t^k^;^t^j⁾

k

^j=1;n

k =1;n

Q

ⁿ

j=1 (h

(t

j

)

−

^z) ^dt¹^:^:^:^dtⁿ^:

¥¬¬ 2. ¨á«®z=z

0

∈

^G^{ï¢«ï¥âáï}á®¡áâ¢¥ë¬§ ç¥¨¥¬®¯¥à â®à ^H

e

2 â®£¤ ¨â®«ìª®â®£¤ ,ª®£¤ ®®ï¢«ï¥âáïã«¥¬äãªæ¨¨

(z),â.¥.

(z

0 )=0.

(7)

®ª § â¥«ìáâ¢®.Ǳãáâì ç¨á«®z=z

0

{á®¡áâ¢¥®¥§ ç¥¨¥®¯¥à â®à ^H

e

2 ,

'

(x){á®®â¢¥âáâ¢ãîé ïá®¡áâ¢¥ ïäãªæ¨ï,â.¥.

h

(x)'

(x)

−

Z

T

h

1 (x;t)'

(t)dt=z'

(x):

¡®§ ç¨¬

(x)=

h

(x)

−

^z

'

(x).®£¤

(x)

−

Z

T

h

1 (x;t)

'

(t)

−

^z ^(t)^dt⁼^0;

â.¥. ç¨á«®=1¥áâì á®¡áâ¢¥®¥§ ç¥¨¥®¯¥à â®à K

(z). âáî¤ á«¥¤ã¥â, çâ®

(z

0 )=1.

Ǳãáâìâ¥¯¥àìz =z

0

ï¢«ï¥âáïã«¥¬äãªæ¨¨

(z),â.¥.

(z

0

)=0. §â¥®à¥¬ë

à¥¤£®«ì¬ á«¥¤ã¥â,çâ®®¤®à®¤®¥ãà ¢¥¨¥

(x)

−

Z

T

h

1 (x;t)

h

(t)

−

^z ^(t)^dt⁼⁰

¨¬¥¥â¥âà¨¢¨ «ì®¥à¥è¥¨¥. â®®§ ç ¥â,çâ®ç¨á«®z=z

0

ï¢«ï¥âáïá®¡áâ¢¥ë¬

§ ç¥¨¥¬®¯¥à â®à ^H

e

2 .

¥®à¥¬ 1. ǱãáâìJ =2J

1

¨ ¯à®¨§¢®«ì®. ®£¤ ®¯¥à â®à ^H

e

2

¨¬¥¥â à®¢®

¤¢ '

1

¨ '

2

(¡¥§ ãç¥â ªà â®áâ¨ ¢ëà®¦¤¥¨© ¨å í¥à£¨¨) á® § ç¥¨ï¬¨

í¥à£¨¨

z

1

=

−

^2J¹^; ^z²⁼

−

⁽⁴⁺^2)J¹

−

^4J¹

X

i=1 cos

i

;

¯à¨ç¥¬z

1

¨¬¥¥âªà â®áâì¢ëà®¦¤¥¨ï

−

^1,^z²^¥^{¢ëà®¦¤¥®}^¨^zⁱ^<^m^, ⁱ⁼

1;2, ¤«ï¢á¥å

∈

^T^,^â.¥. ^í¥à£¨¨ ^íâ¨å ^«¥¦ â^¨¦¥ ^{®¡« áâ¨}¥¯à¥àë¢®£®

á¯¥ªâà ®¯¥à â®à ^H

e

2 .

®ª § â¥«ìáâ¢®.Ǳà¨J =2J

1

¨¬¥¥¬h

(s)

≡

^0,

(z)=

1+ 2J

1

z

⁻

¹

(

1+ 2J

1

z

"

1+ 4J

1

z

X

i=1

(1+cos

i )

#

−

^16J

2

1

z 2

X

i=1 cos

2

i

2

)

:

¥è ïãà ¢¥¨¥

(z)=0,¯®«ãç ¥¬¤®ª § â¥«ìáâ¢®â¥®à¥¬ë.

¬¥ç ¨¥. â¥®à¥¬¥ã«ì-ªà â ï¢ëà®¦¤¥®áâì®§ ç ¥â,çâ®¢¤ ®¬á«ãç ¥

íâ®®âáãâáâ¢ã¥â.

¢¥¤¥¬®¡®§ ç¥¨¥=(;;:::;)

∈

^T^.

(8)

¥®à¥¬ 2. Ǳãáâì=, J

6

⁼^J¹^. ^®£¤ ^{®¯¥à â®à}^H

e

2

¨¬¥¥â¥¤¨áâ¢¥®¥

'á® § ç¥¨¥¬ í¥à£¨¨

z=8(J

−

^2J¹⁾

−

^2(J

−

^J¹^);

¯à¨ç¥¬ãà®¢¥ì íâ®©í¥à£¨¨ -ªà â®¢ëà®¦¤¥. à®¬¥ â®£®, ¥á«¨ J >J

1 ,â®

z<m

, ¥á«¨ J <J

1

, â® z>

M

^. ^Ǳà¨ ^J ⁼^J¹ ^íâ® ^{¨áç¥§ ¥â,} ^{¯®£«®é ïáì}

¥¯à¥àë¢ë¬á¯¥ªâà®¬.

«ï¤®ª § â¥«ìáâ¢ ¨á¯®«ì§ã¥âáïà ¢¥áâ¢®h

(x)=8(J

−

^2J¹⁾^¯à¨⁼^,^â ª¦¥

á®®â¢¥âáâ¢ãîé¨©¢¨¤®¯à¥¤¥«¨â¥«ïà¥¤£®«ì¬

(z).

á«ãç ¥,¥á«¨ =1,¨§¬¥¥¨¥í¥à£¥â¨ç¥áª®£®á¯¥ªâà ®¯¨áë¢ ¥âáï¨¦¥á«¥¤ãîÄ

é¨¬¨â¥®à¥¬ ¬¨.

¥®à¥¬ 3.1. ǱãáâìJ <J

1

¨

∈

^]0;^[ ⁽

∈

^];^2[^):

) ¥á«¨

cos

2

>

−

^J

−

^J1 2J

1

cos

2

<

J

−

^J1 2J

1

;

â® ®¯¥à â®à ^H

e

2

¨¬¥¥â à®¢® ¤¢ '

1

¨ '

2

á® § ç¥¨ï¬¨ í¥à£¨¨, á®®â¢¥âÄ

áâ¢¥®à ¢ë¬¨ z

1

¨ z

2

,¯à¨ç¥¬z

i

<m

, i=1;2;

¡)¥á«¨

cos

2

6 −

^J

−

^J¹

2J

1

cos

2

>

J

−

^J¹

2J

1

;

â®®¯¥à â®à^H

e

2

®¡« ¤ ¥â¥¤¨áâ¢¥ë¬'

1

á®§ ç¥¨¥¬í¥à£¨¨,à ¢ë¬z

1 ,

¯à¨íâ®¬ z

1

<m

.

¥®à¥¬ 3.2. ǱãáâìJ =J

1

¨

∈

^]0;^[ ⁽

∈

^];^2[):

) ¥á«¨0<<

1 (

2

<<2), â®®¯¥à â®à ^H

e

2

¨¬¥¥â¥¤¨áâ¢¥®¥ '

á® § ç¥¨¥¬í¥à£¨¨,à ¢ë¬ z<m

;

¡)¥á«¨¦¥

∈

^[1

;[

∪

^];2

[,â®®¯¥à â®à ^H

e

2

¥ ¨¬¥¥â. ¤¥áì

1

≈

¹⁰⁰

^◦

^,

2

≈

²⁶⁰

^◦

^¨

z=

−

^8J¹

−

8J

1 cos

2

2 1+2

q

3+cos 2

2

3

:

¥®à¥¬ 3.3. ǱãáâìJ

1

<J <2J

1

¨

∈

^]0;^[ ⁽

∈

^];^2[):

) ¥á«¨

cos

2

6

^J

−

^J¹

2J

1

cos

2

> −

^J

−

^J¹

2J

1

;

â® ®¯¥à â®à ^H

e

2

¨¬¥¥â à®¢® ¤¢ '

1

¨ '

2

á® § ç¥¨ï¬¨ í¥à£¨¨, á®®â¢¥âÄ

áâ¢¥®à ¢ë¬¨ z

1

¨ z

2

,¯à¨ç¥¬z

1

<m

, z

2

>

M

;

¡)¥á«¨¦¥

cos

2

>

J

−

^J¹

2J

1

cos

2

<

−

^J

−

^J¹

2J

1

;

â® ®¯¥à â®à ^H

e

2

¨¬¥¥â à®¢® âà¨ '

1 , '

2 , '

3

á® § ç¥¨ï¬¨ í¥à£¨¨, á®®âÄ

¢¥âáâ¢¥®à ¢ë¬¨ z

1 ,z

2 ,z

3

,¯à¨íâ®¬ z

1

<m

, z

i

>

M

^, ⁱ⁼^2;^3.

(9)

¥®à¥¬ 3.4. Ǳãáâì2J

1

<J <3J

1

¨

∈

^]0;^[ ⁽

∈

^];^2[):

) ¥á«¨

cos

2

>

J

−

^J¹

2J

1

cos

2

<

−

^J

−

^J¹

2J

1

;

â®®¯¥à â®à ^H

e

2

®¡« ¤ ¥âà®¢® ¤¢ã¬ï'

1

¨'

2

á® § ç¥¨ï¬¨í¥à£¨¨,á®®âÄ

¢¥âáâ¢¥®à ¢ë¬¨ z

1

¨ z

2 ,£¤¥z

i

<m

, i=1;2;

¡)¥á«¨¦¥

cos

2

6

J

−

^J¹

2J

1

cos

2

> −

^J

−

^J¹

2J

1

;

â® ®¯¥à â®à ^H

e

2

®¡« ¤ ¥â ¥¤¨áâ¢¥ë¬ '

1

á® § ç¥¨¥¬ í¥à£¨¨, à ¢ë¬

z

1

<m

. íâ®¬ á«ãç ¥ ¢â®à®¥ ¨áç¥§ ¥â, ¯®£«®é ïáì ¥¯à¥àë¢ë¬ á¯¥ªâÄ

à®¬.

¥®à¥¬ 3.5. ǱãáâìJ =3J

1

¨

6

⁼^0. ^®£¤^{®¯¥à â®à} ^H

e

2

¨¬¥¥â¥¤¨áâ¢¥®¥

'á® § ç¥¨¥¬í¥à£¨¨

z=4J

1

−

^cos²

2

<m

:

¥®à¥¬ 3.6. Ǳãáâì J >3J

1

¨

6

⁼^0. ^®£¤ ^{®¯¥à â®à} ^H

e

2

¨¬¥¥â à®¢® ¤¢

'

1

¨ '

2

á® § ç¥¨ï¬¨ í¥à£¨¨, á®®â¢¥âáâ¢¥® à ¢ë¬¨ z

1

¨ z

2

, ¯à¨ç¥¬

z

1

<m

, z

2

>

M

.

á«ãç ¥=1¨=0®¯¨á ¨¥¨§¬¥¥¨ïí¥à£¥â¨ç¥áª®£®á¯¥ªâà ¤ ¥âá«¥¤ãîé ï

â¥®à¥¬ .

¥®à¥¬ 4. 1) ¥á«¨ J <J

1

¨ =0, â® ®¯¥à â®à ^H

e

2

®¡« ¤ ¥â à®¢® ¤¢ã¬ï

'

1

¨ '

2

á® § ç¥¨ï¬¨ í¥à£¨¨, á®®â¢¥âáâ¢¥® à ¢ë¬¨ z

1

¨ z

2

, ¯à¨ç¥¬

z

i

<m

, i=1;2;

2)¥á«¨J =J

1

¨=0,â®®¯¥à â®à^H

e

2

¨¬¥¥â¥¤¨áâ¢¥®¥'á®§ ç¥¨¥¬

í¥à£¨¨

z=

−

⁶⁴

3 J

1

<m

;

3) ¯ãáâì J

1

<J <2J

1

¨ =0. ®£¤ ®¯¥à â®à ^H

e

2

¨¬¥¥âà®¢® ¤¢ '

1

¨

'

2

á® § ç¥¨ï¬¨í¥à£¨¨, á®®â¢¥âáâ¢¥®à ¢ë¬¨ z

1

¨ z

2

,¯à¨íâ®¬ z

1

<m

,

z

2

>

M

^;

4) ¯ãáâì2J

1

<J <3J

1

¨ =0. ®£¤ ®¯¥à â®à ^H

e

2

®¡« ¤ ¥â à®¢® ¤¢ã¬ï

'

1

¨ '

2

á® § ç¥¨ï¬¨ í¥à£¨¨, á®®â¢¥âáâ¢¥® à ¢ë¬¨ z

1

¨ z

2 , £¤¥ z

i

<m

,

i=1;2;

5) ¯ãáâìJ =3J

1

¨=0. ®£¤ ®¯¥à â®à ^H

e

2

¥ ¨¬¥¥â;

6) ¥á«¨ J >3J

1

¨ =0, â® ®¯¥à â®à ^H

e

2

¨¬¥¥â¥¤¨áâ¢¥®¥ 'á® § ç¥Ä

¨¥¬í¥à£¨¨ z>

M

^.

Ǳà¨¢¥¤¥¬íáª¨§¤®ª § â¥«ìáâ¢ â¥®à¥¬3¨4. è¥¬á«ãç ¥ãà ¢¥¨¥¤«ïá®¡áâÄ

¢¥ëå§ ç¥¨©ï¢«ï¥âáï¨â¥£à «ìë¬ãà ¢¥¨¥¬á¢ëà®¦¤¥ë¬ï¤à®¬. Ǳ®íâ®¬ã

(10)

®®íª¢¨¢ «¥â®á¨áâ¥¬¥«¨¥©ëå®¤®à®¤ëå «£¥¡à ¨ç¥áª¨åãà ¢¥¨©. §¢¥áâÄ

®,çâ®á¨áâ¥¬ «¨¥©ëå®¤®à®¤ëå «£¥¡à ¨ç¥áª¨åãà ¢¥¨©¨¬¥¥â¥âà¨¢¨ «ì®¥

à¥è¥¨¥â®£¤ ¨â®«ìª®â®£¤ ,ª®£¤ ¤¥â¥à¬¨ âá¨áâ¥¬ëà ¢¥ã«î.Ǳ®íâ®¬ã¢à áÄ

á¬ âà¨¢ ¥¬®¬á«ãç ¥ãà ¢¥¨¥

(z)=0íª¢¨¢ «¥â®à ¢¥áâ¢ãã«î¤¥â¥à¬¨ Ä

â íâ®©á¨áâ¥¬ë. ëà ¦ ï¢á¥¨â¥£à «ë,¢å®¤ïé¨¥¢ãà ¢¥¨¥

(z)=0,ç¥à¥§¨Ä

â¥£à «

J

∗

(z)=

Z

T dt

h

(t)

−

^z^;

¯®«ãç ¥¬,çâ®ãà ¢¥¨¥

(z)=0íª¢¨¢ «¥â®ãà ¢¥¨î

J

∗

(z)=

8(J

−

^2J¹^)(J

−

^5J¹⁾^cos²

2

−

^(J

−

^J¹^)[z

−

^8(J

−

^2J¹^)]

×

128J

1

(J

−

^2J1 )

2

cos 4

2

+8(J

−

^2J1 )(J+J

1 )cos

2

[z

−

^8(J

−

^2J1 )]+

+(J

−

^J¹^)[z

−

^8(J

−

^2J¹^)]²

⁻

¹

: (7)

Ǳ®áª®«ìªã1=(h

(t)

−

^z)^{¥¯à¥àë¢ ïäãªæ¨ï¯à¨z

∈

^[m^;

M

^]^¨

[J

∗

(z)]

0

=

Z

T

dt

[h

(t)

−

^z]² ^>^0;

äãªæ¨ï J

∗

(z) ï¢«ï¥âáï ¢®§à áâ îé¥© äãªæ¨¥© z ¯à¨ z

∈

^[m

;

M

]. à®¬¥â®£®,

J

∗

(z)

→

⁰^¯à¨^z

→ −∞

^,^J

^∗

^(z)

→

⁺

∞

^¯à¨^z

→

^m

−

^0,^J

^∗

^(z)

→ −∞

^¯à¨^z

→ M

+0¨

J

∗

(z)

→

⁰^¯à¨^z

→

⁺

∞

^.^áá«¥^¤ãï^â¥¯¥àì^{ãà ¢¥¨¥}⁽⁷⁾^¢¥^{¬®¦¥áâ¢}^G

=[m

;

M

],

¯®«ãç¨¬¤®ª § â¥«ìáâ¢®â¥®à¥¬3¨4.

áá¬®âà¨¬á«ãç ©=2¨®¯¨è¥¬¨§¬¥¥¨¥í¥à£¥â¨ç¥áª®£®á¯¥ªâà á¨áâ¥¬ë¤«ï

§ ç¥¨©¯®«®£®ª¢ §¨¨¬¯ã«ìá ,¨¬¥îé¨å¢¨¤=(

1

;

2 )=(

0

;

0

).á«¨¯ à ¬¥âÄ

àëJ,J

1

¨

0

ã¤®¢«¥â¢®àïîâãá«®¢¨ï¬â¥®à¥¬3¨4,â®¥âàã¤®¢¨¤¥âì,çâ®¨¬¥îâ¬¥áÄ

â®á®®â¢¥âáâ¢ãîé¨¥ãâ¢¥à¦¤¥¨ïíâ¨åâ¥®à¥¬.Ǳ®ï¢«ï¥âáï«¨èì®¤®¤®¯®«¨â¥«ì®¥

'~á®§ ç¥¨¥¬í¥à£¨¨,à ¢ë¬z,~¯à¨ç¥¬z~<m

(~z>

M

^),^¥á«¨^J^>^J¹ ^(J ^<^J¹^).

á«¨¦¥J=J

1

,â®®¯¥à â®à^H

e

2

¥¨¬¥¥â¤®¯®«¨â¥«ì®£®.

«ï¤®ª § â¥«ìáâ¢ íâ®£®ãâ¢¥à¦¤¥¨ï § ¬¥â¨¬, çâ®¢ á«ãç ¥, ¥á«¨ =2¨ =

(

0

;

0

),äãªæ¨ï

(z)¨¬¥¥â¢¨¤

(z)=

"

1

−

^2(J

−

^J¹⁾

Z

2

cos

0

2

−

^t¹

−

^cos ⁰

2

−

^t²

²

dt

1 dt

2

h

(t

1

;t

2 )

−

^z

#

(z); (8)

(11)

£¤¥

(z)=

(

1

−

^8J¹

Z

T 2

1+cos

0

−

^cos⁰

2

cos

0

2

−

^t¹

+cos

0

2

−

^t²

h

(t)

−

^z ^dt

)

×

"

1

−

^4(J

−

^J¹⁾

Z

T 2

cos

0

2

−

^t1

cos

0

2

−

^t1

+cos

0

2

−

^t2

−

²^cos⁰

2

h

(t)

−

^z ^dt

#

−

^64(J

−

^J¹^)J¹

×

× Z

T 2

1+cos

0

−

^cos⁰

2

cos

0

2

−

^t1

+cos

0

2

−

^t2 cos

0

2

−

^t1

h

(t)

−

^z ^dt¹^dt²

×

× Z

T 2

cos

0

2

−

^t1

−

^cos⁰

2

h

(t)

−

^z ^dt¹^dt²^:

Ǳ®íâ®¬ããà ¢¥¨¥

(z)=0íª¢¨¢ «¥â®ãà ¢¥¨î

1

−

^2(J

−

^J¹⁾

Z

T 2

cos

0

2

−

^t¹

−

^cos ⁰

2

−

^t²

²

dt

1 dt

2

h

(t

1

;t

2

)

−

^z ⁼⁰ ⁽⁹⁾

¨

(z)=0: (10)

¥âàã¤®¢¨¤¥âì,çâ®ãà ¢¥¨¥(9)¨¬¥¥â¥¤¨áâ¢¥®¥à¥è¥¨¥z~<m

¯à¨ãá«®¢¨¨

J >J

1

; ¥á«¨¦¥J <J

1

, â®íâ®à¥è¥¨¥ã¤®¢«¥â¢®àï¥â ãá«®¢¨îz~>

M

. Ǳà¨J =

J

1

ãà ¢¥¨¥(9)à¥è¥¨ï¥¨¬¥¥â. ëà ¦ ï¢á¥¨â¥£à «ë,¢å®¤ïé¨¥¢ (10), ç¥à¥§

¨â¥£à «

J

∗

(z)=

Z

T 2

dt

1 dt

2

h

(t

1

;t

2 )

−

^z^;

¯®«ãç ¥¬ãà ¢¥¨¥¢¨¤

(z)J

∗

(z)=

(z);

£¤¥

(z)=(J

−

^J¹^)~^z²⁺^16(J

−

^2J¹^)(J⁺^J¹⁾^cos²⁰

2

~ z+512J

1

(J

−

^2J¹⁾^cos⁴ ⁰

2

;

(z)=16(J

−

^2J1

)(J

−

^5J1 )cos

2

0

2

−

^(J

−

^J1 )~z:

¤¥áìz~=z

−

^16(J

−

^2J1

).á¢®î®ç¥à¥¤ì,íâ®ãà ¢¥¨¥¯à¨

(z)

6

⁼⁰íª¢¨¢ «¥â®

ãà ¢¥¨î¢¨¤

J

∗

(z)=

(z)

: (11)

á¯®«ì§ãï¬®®â®®áâìäãªæ¨¨J

∗

(z)¯à¨z

∈

^[m^;

M

^]^¨^{¨áá«¥¤ãï}^{ãà ¢¥¨¥}⁽¹¹⁾^¢¥

¬®¦¥áâ¢ G

,¯®«ãç¨¬ãâ¢¥à¦¤¥¨ï, «®£¨çë¥ãâ¢¥à¦¤¥¨ï¬â¥®à¥¬3¨4.

Ǳà¨®áâ «ìëå§ ç¥¨ïå¯®«®£®ª¢ §¨¨¬¯ã«ìá =(

1

;

2 ),

1

6

⁼²^,^{áãé¥áâ¢ãÄ}

îââ ª¨¥¬®¦¥áâ¢ ¯ à ¬¥âà®¢J,J ¨,®¡®§ ç ¥¬ë¥G , j=0;5,çâ®¢ª ¦¤®¬

(12)

¬®¦¥áâ¢¥G

j

®¯¥à â®à^H

e

2

¨¬¥¥âà®¢®j(áãç¥â®¬ªà â®áâ¨¢ëà®¦¤¥¨©¨åí¥àÄ

£¨¨)á®§ ç¥¨ï¬¨í¥à£¨¨,á®®â¢¥âáâ¢¥®à ¢ë¬¨z

k

, k=1;5,¨z

k

∈

^G^.

¥©áâ¢¨â¥«ì®,¢íâ®¬á«ãç ¥¯à¨ =2äãªæ¨ï

(z)¨¬¥¥â¢¨¤

(z)=

a

1 a

2 a

3

b

1 b

2 b

3

c

1 c

2 c

3

;

£¤¥

a

1

=1

−

^4J1

Z

T 2

g

(t)dt

h

(t)

−

^z^;

a

k +1

=

−

^4(J

−

^J¹⁾

Z

T 2

f

k (t

k )dt

h

(t)

−

^z^; ^k⁼^1;^2;

b

1

=

−

^4J1

Z

T 2

g

(t)'

1 (t

1 )

h

(t)

−

^z ^dt;

b

2

=1

−

^4(J

−

^J¹⁾

Z

T 2

'

1 (t

1 )f

1 (t

1 )

h

(t)

−

^z ^dt;

b

3

=

−

^4(J

−

^J¹⁾

Z

T 2

'

1 (t

1 )f

2 (t

2 )

h

(t)

−

^z ^dt;

c

1

=

−

^4J¹

Z

T 2

g

(t)'

2 (t

2 )

h

(t)

−

^z ^dt;

c

2

=

−

^4(J

−

^J1 )

Z

T 2

'

2 (t

2 )f

1 (t

1 )

h

(t)

−

^z ^dt;

c

3

=1

−

^4(J

−

^J¹⁾

Z

T 2

'

2 (t

2 )f

2 (t

2 )

h

(t)

−

^z ^dt:

¤¥áì

g

(t)=

X

2

i=1

1+cos

i

−

²^cosⁱ

2 cos

i

2

−

^ti

;

f

k (t

k )=cos

k

2

−

^tk

−

^cos^k

2

;

'

k (t

k )=cos

k

2

−

^tk

; k=1;2;

∈

^T²^; ^t

∈

^T²^:

ëà ¦ ï¢á¥¨â¥£à «ë,¢å®¤ïé¨¥¢ãà ¢¥¨¥

(z)=0,ç¥à¥§J

∗

(z),¯®á«¥¥ª®Ä

â®àëå «£¥¡à ¨ç¥áª¨å¯à¥®¡à §®¢ ¨©¥£®¬®¦®á¢¥áâ¨ªãà ¢¥¨î¢¨¤

(z)J

∗

(z)=

(z); (12)

£¤¥

(z) ï¢«ï¥âáï ¬®£®ç«¥®¬¯ïâ®©áâ¥¯¥¨®âz,

(z){¬®£®ç«¥¡®«¥¥ ¨§Ä

ª®©áâ¥¯¥¨®âz. á¯®«ì§ãï¬®®â®®áâìJ

∗

(z)¯à¨z

∈

^[m^;

M

^]^¨^{¨áá«¥¤ãï}^ãà ¢¥Ä

¨¥(12)¢¥¬®¦¥áâ¢ G

â¢ G .

(13)

¨¬¥¥â¢¨¤=(

1

;

2

;

3 )=(

0

;

0

;

0

). á«¨¯ à ¬¥âàë

0 ,J ¨J

1

ã¤®¢«¥â¢®àïîâ

ãá«®¢¨ï¬â¥®à¥¬3¨4,â®¨¬¥îâ¬¥áâ® «®£¨çë¥ãâ¢¥à¦¤¥¨ïíâ¨åâ¥®à¥¬.Ǳ®ï¢«ïÄ

¥âáï«¨èì®¤®¤®¯®«¨â¥«ì®¥

~

'á®§ ç¥¨¥¬í¥à£¨¨,à ¢ë¬

~

z.Ǳà¨ç¥¬ãà®¢¥ì

~

~ z<m

(

~

z>

M

^),^¥á«¨^J ^>^J¹ ^(J ^<^J¹^). ^Ǳà¨

J =J

1

íâ®¤®¯®«¨â¥«ì®¥¨áç¥§ ¥â,¯®£«®é ïáì¥¯à¥àë¢ë¬á¯¥ªâà®¬.

«ï¤®ª § â¥«ìáâ¢ § ¬¥â¨¬«¨èì,çâ®¢íâ®¬á«ãç ¥äãªæ¨ï

(z)¨¬¥¥â¢¨¤

(z)=

"

1

−

^2(J

−

^J¹⁾

Z

T 3

cos

0

2

−

^t¹

−

^cos ⁰

2

−

^t²

²

dt

1 dt

2 dt

3

h

(t)

−

^z

#

²

e

(z);

t

∈

^T³^;

£¤¥

e

(z)=

"

1

−

^4J¹

Z

T 3

3+3cos

0

−

²^cos⁰

2

P

³

i=1 cos

0

2

−

^tⁱ

h

(t)

−

^z ^dt

#

×

× (

1

−

^4(J

−

^J¹⁾

Z

T 3

cos

0

2

−

^t¹

P

³

i=1 cos

0

2

−

^tⁱ

−

³^cos⁰

2

h

(t)

−

^z ^dt

)

−

^48J¹^(J

−

^J¹⁾

Z

T 3

cos

0

2

−

^t¹

−

^cos⁰

2

dt

h

(t)

−

^z

×

× Z

T 3

3+3cos

0

−

²^cos⁰

2

P

³

i=1 cos

0

2

−

^tⁱ ^cos ⁰

2

−

^t¹

h

(t)

−

^z ^dt:

Ǳ®íâ®¬ããà ¢¥¨¥

(z)=0íª¢¨¢ «¥â®ãà ¢¥¨ï¬

"

1

−

^2(J

−

^J¹⁾

Z

T 3

cos

0

2

−

^t¹

−

^cos ⁰

2

−

^t²

²

dt

1 dt

2 dt

3

h

(t)

−

^z

#

²

=0 (13)

¨

e

(z)=0: (14)

¥âàã¤®¢¨¤¥âì,çâ®ãà ¢¥¨¥(13)¨¬¥¥â¥¤¨áâ¢¥®¥¤¢ãªà â®¥à¥è¥¨¥z

0

,¥á«¨

J

6

⁼^J¹^,^¯à¨ç¥¬^z

⁰

^<^m ^(z

⁰

^>

M

^),^¥á«¨^J ^>^J¹ ^(J^<^J¹^). ^ëà ¦ ï^¢á¥^{¨â¥£à «ë,}

¢å®¤ïé¨¥¢ãà ¢¥¨¥(14),ç¥à¥§

J

∗

(z)=

Z

T 3

dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3 )

−

^z^;

¯®«ãç ¥¬ãà ¢¥¨¥

e

(z)J

∗

(z)=

e

(z); (15)

£¤¥

e

(z)=(J

−

^J1 )~z

2

+24(J+J

1

)(J

−

^2J1 )cos

2

0

~

z+1152J

1

(J

−

^2J1 )

2

cos 4

0

;

(14)

e

(z)=24(J

−

^2J¹^)(J

−

^5J¹⁾^cos²⁰

2

−

^(J

−

^J¹^)~^z:

¤¥áìz~=z

−

^24(J

−

^2J1

).á¢®î®ç¥à¥¤ì,ãà ¢¥¨¥(15)¯à¨ãá«®¢¨¨,çâ®

e

^(z)

6

⁼^0,

íª¢¨¢ «¥â®ãà ¢¥¨î

J

∗

(z)=

e

(z)

e

(z)

: (16)

áá«¥¤ãïãà ¢¥¨¥ (16)¢¥ ¬®¦¥áâ¢ G

¨ ¨á¯®«ì§ãï ¬®®â®®áâìJ

∗

(z) ¯à¨

z

∈

á«ãç ¥

6

⁼⁽⁰^;⁰^;⁰⁾^á¨áâ¥¬^{®¡« ¤ ¥â}^¥^¡®«¥¥^ç¥¬^á¥¬ìî^(á^ãç¥â®¬^ªà âÄ

k

, k=0;7,

¯ à ¬¥âà®¢,J¨J

1

,çâ®¢ª ¦¤®¬¬®¦¥áâ¢¥G

k

, k=0;7,á¨áâ¥¬ ¨¬¥¥âà®¢®k.

¥à£¨¨íâ¨å«¥¦ â¢¥¬®¦¥áâ¢ G

.Ǳà¨¯¥à¥å®¤¥®â®¤®£®¨§íâ¨å¬®¦¥áâ¢ª

¤àã£®¬ãã®¯¥à â®à ^H

e

2

¢®§¨ª îâ«¨¡®¤®¯®«¨â¥«ìë¥,«¨¡®¥ª®â®àë¥áãé¥áÄ

â¢ãîé¨¥¨áç¥§ îâ.íâ®¬á«ãç ¥äãªæ¨ï

(z)¨¬¥¥â¢¨¤

(z)=

a

1 a

2 a

3 a

4

b

1 b

2 b

3 b

4

c

1 c

2 c

3 c

4

d

1 d

2 d

3 d

4

;

£¤¥

a

1

=1

−

^4J1

Z

T 3

g

(t)dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3 )

−

^z^;

a

k +1

=

−

^4(J

−

^J¹⁾

Z

T 3

f

k (t

k )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^; ^k⁼^1;^2;^3;

b

1

=

−

^4J1

Z

T 3

g

(t)'

1 (t

1 )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^;

b

2

=1

−

^4(J

−

^J¹⁾

Z

T 3

'

1 (t

1 )f

1 (t

1 )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^;

b

k +1

=

−

^4(J

−

^J1 )

Z

T 3

'

1 (t

1 )f

k (t

k )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^; ^k⁼^2;^3;

c

1

=

−

^4J¹

Z

T 3

g

(t)'

2 (t

2 )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^;

c

k +1

=

−

^4(J

−

^J¹⁾

Z

T 3

'

2 (t

2 )f

k (t

k )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^; ^k⁼^1;^3;

c

3

=1

−

^4(J

−

^J¹⁾

Z

T 3

'

2 (t

2 )f

2 (t

2 )

h

(t

1

;t

2

;t

3

)

−

^z^dt¹^dt²^dt³^;

d

1

=

−

^4J¹

Z

T 3

g

(t)'

3 (t

3 )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^;

d

k +1

=

−

^4(J

−

^J¹⁾

Z

3 '

3 (t

3 )f

k (t

k )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^; ^k⁼^1;^2;

(15)

d

4

=1

−

^4(J

−

^J¹⁾

Z

T 3

'

3 (t

3 )f

3 (t

3 )dt

1 dt

2 dt

3

h

(t

1

;t

2

;t

3

)

−

^z ^:

¤¥áì

g

(t)=

X

3

i=1

1+cos

i

−

²^cosⁱ

2 cos

i

2

−

^tⁱ

;

f

k (t

k )=cos

k

2

−

^t^k

−

^cos^k

2

;

'

k (t

k )=cos

k

2

−

^t^k

; k=1;2;3:

ëà ¦ ï¢á¥¨â¥£à «ë,¢å®¤ïé¨¥¢ãà ¢¥¨¥

(z)=0,ç¥à¥§J

∗

(z),¯®á«¥¥ª®â®Ä

J

∗

(z)=

A

(z)

B

^(z)^;

£¤¥B

.

«ï¯à®¨§¢®«ì®£® >3¨=(

1

;

2

;:::;

)=(

0

;

0

;:::;

0

)¢á«ãç ¥,¥á«¨¯ Ä

à ¬¥âàëJ,J

1

¨

0

ã¤®¢«¥â¢®àïîâãá«®¢¨ï¬â¥®à¥¬3¨4,¨¬¥îâ¬¥áâ® «®£¨çë¥

e

2

¥áâì«¨èì®¤®¤®¯®«¨Ä

â¥«ì®¥¯à¨J

6

⁼^J¹^. ^Ǳà¨ç¥¬^í¥à£¨ï^íâ®£®¤®¯®«¨â¥«ì®£®z (

−

^{1)-ªà â®}

¢ëà®¦¤¥ .à®¬¥â®£®,z<m

(z>

M

^),^¥á«¨^J ^>^J¹ ^(J ^<^J¹^).^Ǳà¨^{®áâ «ìëå}^¦¥

§ ç¥¨ïå¯®«®£®ª¢ §¨¨¬¯ã«ìá á¨áâ¥¬ë®¯¥à â®à^H

e

2

¨¬¥¥â¥¡®«¥¥ç¥¬2+1

¬®¦¥áâ¢ G

.

à¥¤£®«ì¬ à áá¬ âà¨¢ ¥¬ëå ®¯¥à â®à®¢. ëà ¦ ï ¢á¥ ¨â¥£à «ë ¢

(z) ç¥à¥§

J

∗

(z),¬®¦® ¯à¨¢¥áâ¨ãà ¢¥¨¥

(z)=0ª ¢¨¤ã

J

∗

(z)=

C

^(z)

D

^(z)^; ⁽¹⁷⁾

£¤¥

D

(z)ï¢«ï¥âáï¬®£®ç«¥®¬(2+1)-áâ¥¯¥¨®âz,

C

(z)â ª¦¥ï¢«ï¥âáï¬®£®Ä

D

(z). áá«¥¤®¢ ¨¥ãà ¢¥¨ï(17)¢¥

¬®¦¥áâ¢ G

¥®à¥¬ 5. ǱãáâìJ =J

1

¨ç¨á«® ¯à®¨§¢®«ì®. ®£¤ ®¯¥à â®à^H

e

2

¨¬¥¥â¥

¡®«¥¥®¤®£® ,¯à¨ç¥¬ãà®¢¥ìí¥à£¨¨ z ¥¢ëà®¦¤¥¨z<m

.

®ª § â¥«ìáâ¢®. ¬¥â¨¬,çâ®¯à¨J =J

1

¨¬¥îâ¬¥áâ®á®®â®è¥¨ï

h

1

(x;t)=

−

^4J¹

X

i=1

1

−

^cosⁱ

2 cos

i

2

−

^xⁱ

+cos

i

;

h

(x)=

−

^8J¹

X

i=1

1

−

^cosⁱ

2 cos

i

2

−

^xⁱ

:

á¯®«ì§ãï¢¨¤®¯à¥¤¥«¨â¥«ïà¥¤£®«ì¬

(z)¨à¥è ïá®®â¢¥âáâ¢ãîé¥¥ãà ¢¥¨¥,

¯®«ãç ¥¬ãâ¢¥à¦¤¥¨¥â¥®à¥¬ë5.

(16)

¯¨á®ª«¨â¥à âãàë

[1] H.A.Bethe. Z.Phys.1931.V.71.P.205.

[2] N.Fukuda,M.Wortis. J.Phys.Chem.Solids.1963.V.24.P.1675.

[3] M.Wortis. Phys.Rev.1963.V.132.ò1.P.85.

[4] I.Majumdar. J.Math.Phys.1969. V.132.ò10.P.85.

[5] I.Ono,S.Mikado, T.Oguchi. J.Phys.So c.Japan.1971.V.30.ò2.P.358.

[6] ..®ç¥¢. .1973..15.ò1..120.

[7] ..®ç¥¢. .1971..61.ò10..1674.

[8] .. è¯ã« â®¢. .1996..107.ò1..155.

[9] .. è¯ã« â®¢. §.1994.ò1..7.

[10].. è¯ã« â®¢. §.1994.ò4..7.

[11].. è¯ã« â®¢. §.1996.ò12..4.

[12]E.Schrodinger. Pro c.RoyIrish.Acad.A.1941.V.48.P.39.

[13]R.Micnas. Phys.Stat.Sol.(b).1974.V.66.ò2.P.75.

[14]A.A.Brown. Phys.Rev.B.1971.V.4.ò1.P.115.

[15]H.H.Chen,P.Levy. Phys.Rev.Lett.1971.V.27.ò20.P.1383.

[16]D.A.Pink,R.Ballard. CanJ.Phys.1974.V.52.ò1.P.33.

[17]D.A.Pink,P.Tremblay. Can.J.Phys.1972.V.50.ò15.P.1728{1735.

[18].. è¯ã« â®¢. .1996..107.ò2..251.

[19].. è¯ã« â®¢. .1996..107.ò2..262.

«¨§..:¨à,1977.

Ǳ®áâã¯¨« ¢à¥¤ ªæ¨î13.IV.2000£.