/ICJi j JJL /

/

/ICJi j JJL /

.1

P.Exner

_gj;/t _ яy

сообщения обьеаиненного

ИНСТИТУТа ядерных иссJiедований дУб на

Е2-84-49

IMPROVED ROSEN' S CONDITIONS ON BOUND STATES

OF SCHRODINGER OPERATORS

1984

1. Introduction

V8rious quslit8tive methods for Qn8lyzing the discrete вpectrum

of Schr~dinger oper8torв h8ve been el8bOr8ted during the ^l8вt three dec8des- for 8 review вее Ref.1, Section XIII.3 • They ыrе of 8 gre8t physic8l intereвt, eвpec18lly bec8uee they c8n provide uв with 8n information 8bout ~disc(H) for the few-body 8nd m8ny-body H8gil- ton18ne, for which the qu8ntitвtive methods 8re uвually difficultto Ье

applied.It is not strange,therefore, that new results of this typc continue to 8ppe8r.

Recently, Rosen/ 2/ employed 8 Sobolev inequality to deduce 8 necess8ry condition for exiвtence of bound et8teв of 8 Schr~dinger

oper8tor Н=-^{6 +} V on L2(1R3 ) , 8nd 8 lower bound to the ground- state energy of Н • Тhеве reвultв 8re formul8ted in termв of 1nteg- r8le of cert81n powerв of 1 V 1 , 8nd no вyПIDietry of the potent18l is required. On the other h8nd, convergence of ^th~ mentioned 1ntegr8ls

demandв IVI to decreaвe r8pidly enough 8t infinity, excluding thuв

m8ny phyвic8lly intereвting с8вев. In ^thiв note, we are going to give 8n improvement which m8kes it ровв1Ьlе to rel8X thiв reвtriction. At the S8me time, we preвent exteneior. of the reвultв to 8DY dimenвion

d <: 3 of the configur8tion вр8се.

on

2. Тhе ;,;дin Resul t

We Bh8ll ^Ье concerned with the Schr~dinger oper8torв

~

=

L2(1id) , d ~ 3 • For 8 given Е. ^, we denote V_(t,x) m8x

1

О , Е. -V ( х)

J

H=-6~V

( 1 )

The functionв V_(t,.) 8re eвsent18l in formul8tion of our result

~ ^{' ,,~,.at} ~-!C'i'-l'yt

1

I 11' "1' ;..;ot•,

w~ 1

БИ&Л!-·

--

Theorem: Suppo8e that Н 18 8elf-adjoint on D(H) =D(-A)(')D(V) (а) ^{Let V} (~,.)е Ld/2(Rd) , then the condition

1

^{d/2 _d} Jt'1/2(~d(d-2) )d/2

v_<t,x) dx > кd е d- 1 ^(~) (2)

нd 2

r

₂

18 nece88ary for the operator Н to have an eigenvalue below ~ ,

(Ь)

^Let

V_(L,.)eL~/

²

(Rd)

^{for 8ome}

Е.

^and

1~1,

then any eigen- value Е of Н fulfil8 the inequвlity

( 1 1 ^{1 )}

^2/d(f-1)

Е ~ Е. - ( 1- ,{ 1 )

J'

-d 2 Kdd d V- ( t 'х) fd 2 dx R

(3)

In partic~lar, thiв 18 true for the ground-вtate energy _Во Proof : Since the improvementв of Ro8en'в argument8 follow the 8ame line, we limit ourselves with 8ketching the proof of the aввertion (Ь) ; the detail8 will Ье given in а forthcoming paper. The 8tarting point 18 the Sobolev inequality

11 Vf'U

₂ ~ К

4 111f11q

(4)

where q = 2d/(d-2) and Kd 18 given Ьу (2). It iв the Ьевt con8tant for the inequalij{ (4) whoвe V8lue wa8 found Ьу Talenti/3/, and earlier

Ьу

^{Ro8en 4/ for d = 3 • If}

'f

¹⁸

а

normalized eigenvector,

R'f/1

_{2 = 1 ,} correвponding to Е , then

Е

⁼

1/Vf/1~

⁺

J

V(x) ly,<x)l2 dx

~

Rd

~

^1/

Vfll~

⁺

Е.

- jd V _ (

~, х) l1f< х) 1

² dx R

во the inequality (4) giveв

Е ~E.+K~П"fU~-jd

V_(E,x)/'f'<x)/2

dx (5а)

IR

Next one ha8 to eвtimate the la8t term on the r.h.в. using twioe the Helder 1nequal1ty

J:

V_(t,x)/'f(x)/2 dx ~

R

~

⁽

~

^{V_(t,x)fd/2dx Y/fd}

ll"fll~(f- ¹ )/cl'll"f'll~/cf

R

2

(5Ь)

!

]

l1nally, one hав to aax1DI1ze the expre881on follow1ng from the 1nequa- 11t1eв (5) over 111J!Uq to get the .deвired re8ult.

3. Examples and Remarks

( 1) The Ro8en '8 re8ul t8 are recovered 1f d = 3 , t. = О and the potent1al V 18 non-po81t1ve. In part1cular, he treated the 8pher1- cally 8ymaetr1o exponential well, V(r) =-V

0 exp(-r/a) • Here the nece88ary cond1 t1on g1 ven Ьу ( 2) d1ffer8 Ьу О. б~ from the or1 t1cal value of v0a2 , lurthermore, the 1nequal1ty (3) for v0a2 >>1 and

f

~ 1. 7262 (

v

₀ а²) ^{1/ 3} g1veв

[ 2 -1/3 2 -2/3

J

в0 ^~-v0 1-1.73791(V0a) +O<<v0a) ) , wh1le the exact value 18 ea81ly found to Ье

в

0

"' -V

0 [ 1 - 2. 33810 < v

0a 2 )- 1/ 3 + 1. 45779 (V

0a 2 )-2/3 + O(v

0

^{1 а -2)}

J

(11) Another в1mple three-d1men81onal example concern8 the 8phe- r1cally 8yaaetr1c вquare well, V(r) =-V

0

^{Э<a-r) • In} th1в саве, the neoe88ary cond1t1on follow1ng from (2) 18 about 15~ from the cr1t1- cal value of

v

_0a2 , The Ьевt lower bound from (3),

Е

0

^{~ -v0} [1- 2.10809 (V0a2 )- 1 ] ,

1в now ach1eved w1th а non-zero ~ ' and 1t 18 1ndependent of

r .

(111) The aввert1on (а) can Ье regarded al8o ав а conвequence of

Cw1kel-L1eb-Ro8enЬljum theorea (вее Ref.1, Theorem XIII.12, and Ref.5) except for the value of the conвtant. Moreover, the latter can Ъе ob- ta1ned from another Sobolev-1nequal1ty argument (cf. Ref.б, or Ref.7, Seotion III.9), The Roвen-type proof 1в, however, much more в1mple

and 8tra1ghtforward. Notioe al8o that for d = 3 and t. "'О , (2) be- oome8 а part1cular са8е of the GMGT-cond1tion (вее Ref.б, or Ref.1, Theorem XIII. 9) w1 th р = 3/2 •

(1v) In order to 11lu8trate how powerfull .the obtained cond1t1onв

Dlight Ье, we pre8ent two d-d1menв1onal example8 concern1ng thefollow- 1ng exactly вolveЬle proЬlea8 :

harmon1c oвc1llator V(r)

=

^{AJ2r2 (ба)}

hydrogen-l1ke atom V(r) = 201'/r, (бЪ)

3

where r ,. ²

Z:

^d xj • In both ² св в ев, the 1neqll8l1 ty ( ' ' yieldв the lower bound j=l for 8 non-zero Е ; in the 11m1t

f_.l+

we get

Ьевt

Ео ~ 21/d(d(d-2) > 1/2 с.>

2

(78)

Е0 ^{~ -4} «. /d(d-2) ^(7Ь)

respectively (for det81lв вее the forthcoming p8per мentioned 8bove).

Since the ground-вt8te energieв 8re E0^=~d for the oscillator, and Е

0

^{х -4~2/(d-1)2} for the hydrogen-like 8toшf⁸/, we see thвt both the lower bounds (7) 8re aвymptot1c8lly ex8ct with d~oo

References

1. Reed м. 8nd Simon в., Methods of Modern M8them8t1cal Рhувiсв,

Vol.4, Acsdemic Presв, New York, 1978.

2. Rosen G., Phyв.Rev.Lett., 1982, v.49, рр.1885-1887.

,. Tslenti G., Annali M8t.Pilr8 et Appl., 1976, v.110, рр.'5'-'72.

4. Roвen G., SIAИ J.of Appl.И8th., 1971, v.21, рр.,О-,2.

5. Lieb Е.Н., The N11mber of Bo11nd St8teв of One-Body Schrбdinger

Oper8tor вnd the Weyl ProЬlem, in "Geometry of the Lapl8ce Oper8tor" (R.Oвserm8n, A.Weinвtein, eds.), Proceedingв of Sympos18 in P11re M8them8t1cs, Vol.,б, рр.241-252, Amer.

Кath.Soc., Providence, Rhode Isl8nd, 1980.

б. Gl8ser V., M8rt1n А., Grбsвe Н. 8nd Th1rr1ng W., А F8m1ly of Opt1m8l Conditionв for the АЪвеnсе of Bound St8teв in 8 Potent18l, in "St11dies in M8them8tic8l Рhувiсв : Ess8ye in Honor of V.B8rgm8nn" (E.Lieb et вl., еdв. ), рр.169-194,

Princeton University Рrевв, 1976.

7. Simon Б., F11nction8l Integrвtion snd Q118nt11m Phyвics, Acsdemic

Рrевв, New York, 1979.

8. Bargmsnn V., Helv.Phyв.Act8,1972, v.45, рр.249-257.

Received Ьу PllЬ11sh1ng Department on January 30,1984

4

Экснер П.

Улучшенные условия Розена на связанные состояния операторов Шредингера

Е2-84-49

Выводится необходимое условие для того, чтобы оператор Шредингера

Н • -!!. + V на I} (Rd ), d > 3, имел связанное состояние ниже заданной энергии

(, а также 011енка снизу -на энергию основного состояния оператора Н. Эти ус­

ловил выражаются только в терминах потенциала V и обобщают недавние результаты Розенана случаи размерности d>3 и для потенциалов, которые не должны быстро убывать на бесконечность.

Работа выполнена в Лаборатории теоретической физики ОИЯИ.

Сообщение Объединенного института ядерных исследований. Дубна \984

Exner

Р.

Improved

Rosen' s Condi ti ons on Bound States

of Schrodinger Operators

Е2-84-49

We derive а

necessary condition on

а

Schrodinger operator

н

=

^{-!!. + v}

on

I} (Rd ), d ~ 3

to

have а Ьound state below а

qiven energy ( ,and

а lower Ьound

to the ground-state energy of

н.

These conditions are expressed in terms of the potential

v

alone, and generalize the recent results of Rosen to the dimensions

d > 3

and to the potential s that are not necessarily rapidly decreasinq. Some examples are given.

The investigation ·

has

Ьееn

performed at the Laboratory of Theoretical Physics, JINR.

Coaaunication of the Joint Institute for Nuclear Research. Dubna 1984