von Neumann-Wigner theorem: level s repulsion and degenerate eigenvalues.

von Numann-Wignr thorm: lvl s rpulsion an gnrat ignvalus. Yu.N.Dmkov an P.Kurasov Abstract. Spctral proprtis of Schröingr oprators with point intractions ar invstigat. Attntion is focus on th intrplay btwn th lvl s rpulsion (von Numann-Wignr thorm) an th symmtry of th points configuration. Explicit solvability of th problm allows to obsrv lvl s rpulsion for two cntrs. For largr numbr of cntrs th familis of point intractions laing to th highst possibl gnracy is invstigat. 1. Introuction Th mtho of oprator xtnsions in mathmatics an its spcial cas call in physics th mtho of zro rang potntials (ZRP), is rapily vloping in th last cas u to its univrsality, applicability to many physical problms laing to ssntial simplifications (usually algbraization). S [5] whr physical applications ar consir an [] whr th sam mol is stui from th mathmatical stanpoint. Rlativ to othr approximativ mthos this on contains from th vry bginning continuous spctrum an prsrvs such proprtis of th xact problms as unitarity (in contrast to th Born approximation). In many problms apparing in physics short rang objcts ar sparat by larg istancs (galaxis, stars, plants, atoms an molculs, nucli an lmntary particls). Apart from vry xotic cass only soli stat osn t satisfy this proprty, but vn thr w can consir xcitations as quasiparticls again rturning to th sam scription. In zro rang potntial mols w hav a uniqu possibility to solv quantum problms xactly. Th mtho is a particular cas of th nonlocal projction oprators mtho whn th projction functions 1

YU.N.DEMKOV AND P.KURASOV ar δ functions an this allows on to prsrv th locality of th oprator. Although th thory is wll vlop th cas whr an infinit numbr or vn a fw zro rang potntials ar prsnt is still not consir in full tails. In th prsnt articl w ar going to iscuss proprtis of ths mols in rlation to invrs problms an Wignrvon Numann thorm. Th clbrat von Numann-Wignr thorm [10] scribs th probability that a finit imnsional matrix has a gnrat ignvalu. This probability is lowr than on may xpct, namly th co-imnsion of th st of matrics having oubl ignvalu is always highr than 1. It follows that for a tim-pnnt Hamiltonian th probability that two nrgy curvs cross ach othr is xtrmly low, an this phnomnon is call lvl s rpulsion. Usually two lvls cross ach othr only if th corrsponing ignfunctions hav iffrnt symmtris. Hnc on xpcts to obsrv gnrat ignvalus for Hamiltonians having crtain symmtris only. In this papr w ar going to stuy this phnomna using Hamiltonians with point intractions in R 1 an R 3. Ths oprators ar wily us in quantum mchanics an atomic physics to mol iffrnt physical procsss (s [5,, 3] an numrous rfrncs thr). Evry Hamiltonian with N point intractions may hav at most N ignvalus. Straightforwar analysis shows that no ignvalu of multiplicity N can appar. Th main goal of this papr is to stuy th possibility for oprators with point intractions to hav ignvalus of th highst possibl multiplicity N 1. Th papr is organiz as follows. Schröingr oprators with point intractions ar fin rigorously in Sction following mainly [, 5]. Sinc two local point intractions cannot prouc any gnrat ignvalu (vn of multiplicity ), w stuy lvl s rpulsion for two cntrs in Sction 3. For th cass of thr, four an fiv cntrs th gnracis of th ignvalus ar stui in Sction 4. It is shown that th maximal gnracy can b obsrv in th cas whr th configuration has a crtain symmtry.. Hamiltonians with lta intractions Schröingr oprators with N local lta intractions at th points{y n } N n=1 ar formally fin by (1) L α = + N α n δ( y n ), n=1

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES.3 whr is th Laplac oprator an δ( y n ) is th lta function with th support at th point y n. Without loss of gnrality w suppos that all points y n ar iffrnt. In what follows w giv prcis finition for th oprator L α. Not that th oprator corrsponing to th formal xprssion (1) is uniquly fin in R 1, but to fin this oprator in R 3 on has to tak into account xtra assumptions. Consir th Laplac oprators () 1 = x in L (R 1 ); an 3 = x 1 x x 3 in L (R 3 ), which ar slf-ajoint whn fin on th Sobolv spacs W (R j ). Hr x an x = (x 1, x, x 3 ) not th coorinats in R 1 an R 3 rspctivly. Thn th oprator corrsponing to th formal xprssion (1) is on of th slf-ajoint xtnsions of th symmtric rstrictions j0 of j, j = 1, 3 to th st of functions vanishing at th intraction points (3) j0 = j {ψ W (R j ):ψ(y n )=0,n=1,,...,N}. Th ficincy lmnts for λ = χ ar just solutions to th quations j0 g + χ g = δ(x y n ) (4) g 1 (x, y n ) = χ x yn an g 3 (x, y n ) = χ x yn, n = 1,,..., N. χ 4π x y n Hnc th ficincy inics of th rstrict oprators ar qual to (N, N). Th omains of th ajoint oprators ar givn by Dom ( 10 ) = W (R 1 \ {y n } N n=1) C(R 1 ); (5) Dom ( 30 ) = W (R 3 \ {y n } N n=1). To scrib slf-ajoint xtnsions of j0 w ar going to us th bounary valus for th functions from th omain of th ajoint oprator ψ(x) x y n 1 x yn ψ n + ψ 0n + o(1), for R 1 ; (6) ψ(x) x y n 1 4π x y n ψ n + ψ 0n + o(1), for R 3. Thn th bounary forms of th ajoint oprators ar givn by N (7) ( j0 )u, v u, ( j0 )v = (u 0n v n u n v 0n ). n=1

4 YU.N.DEMKOV AND P.KURASOV Dfinition 1. Th oprator L j α is th rstriction of th ajoint oprator j0 to th st of functions from ψ Dom ( j0 ) satisfying th bounary conitions (8) ψ0 = α 1 ψ, whr ψ = (ψ 1, ψ,..., ψ N ) T an ψ 0 = (ψ 01, ψ 0,..., ψ 0N ) T ar vctors of bounary valus of th function ψ. In othr wors th oprator L j α is th Laplac oprator fin on th omain of functions satisfying (8). In imnsion on it is possibl to prov that th oprator corrsponing to th formal xprssion (1) is givn by Dfinition 1. In imnsion thr it is ncssary to us crtain aitional assumptions lik th homognity proprtis of th Laplac oprator an th lta istribution in orr to show that th slf-ajoint oprator corrsponing to (1) is givn by Dfinition 1 (s [3], Sction 1.5.1 for tails). W o not want to wll on this point, sinc our furthr stuis ar bas on Dfinition 1 an ar inpnnt of ths assumptions. In what follows only local lta intractions will b consir, i.. intractions corrsponing to th iagonal matrix α. For xhausting scription of local point intractions s [8]. (It is possibl to show that non-iagonal matrics α in (8) la to nonlocal intractions.) Without loss of gnrality w suppos that all cofficints α n ar iffrnt from zro. If it is not th cas thn th st of singular points y n, n = 1,,..., N can simply b ruc. Th rsolvnt of th prturb oprator L α can b calculat using Krins formula [6, 7, 9], sinc ach of th oprators L j α is a finit imnsional prturbation of th Laplac oprator j in th rsolvnt sns. Hnc th ssntial spctrum of L α is purly absolutly continuous an coincis with th intrval [0, ). (It has multiplicity in R 1 an infinit multiplicity in R 3.) Th numbr of ngativ ignvalus cannot xc N (th rank of th prturbation). In aition straightforwar analysis shows that no positiv ignvalus occur. Th iscrt spctrum of th oprator is givn by th zros of th prturbation trminant apparing in Krins formula. Anothr way to obtain th quation for th iscrt spctrum is to consir th following Ansatz for th ignfunction N (9) ψ = a n g j (x, y n ), n=1 whr g j (x, y) ar th Grn functions for th Laplac oprator givn by (4). Th function ψ prsnt by (9) satisfis th ignfunction

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES.5 quation for λ = χ (10) L j0 ψ = χ ψ for any valu of th complx paramtrs a n. It is an ignfunction if an only if it in aition satisfis th bounary conition (8). Consir th bounary valus of th Grns functions (11) g 1 = g 1 (x y n ) g 1 m = g 1 = g 1 (x y n ) g 3 m = { 1, m = n 0, m n, g1 0m = { 1, m = n 0, m n, g3 0m = 1/χ, χ yn y m χ χ/4π, χ yn y m 4π y n y m Th isprsion quations fining th iscrt spctrum in R 1 an R 3 ar (1) t ( 1 + G 1 + χα 1) = 0; m = n m n m = n m n (13) t ( χ + G 3 + 4πα 1) = 0; rspctivly, whr G j ar th following Hrmitian N N matrics (14) G 1 nm = G 3 nm = { χ y n y m, n m { 0, n = m, χ yn y m, n m y n y m 0, n = m. In what follows w ar going to stuy solutions to ths isprsion quations concntrating our attntion to th gnracy of th ignvalus. In aition to xponntially crasing at infinity ignfunctions (corrsponing to ngativ ignvalus) thr xist solutions crasing powr-lik (corrsponing to th ignvalu zro). Dcrasing of ths functions at infinity is rlat to thir angular pnanc. Sphrically symmtric functions ar crasing lik c/r an thrfor ar not normalizabl. All othr valus of th angular momntum ar amissibl (for E = 0).

6 YU.N.DEMKOV AND P.KURASOV 3. Lvl s rpulsion for two cntrs in R 1 an R 3. Two local lta intractions cannot prouc a gnrat ignvalu. Thrfor w ar stuying hr th lvl rpulsion. Th oprator with two lta potntials can b paramtriz by thr ral paramtrs: > 0 - th istanc btwn th cntrs, α j, j = 1, - th strngths of th lta intractions. Consir first th cas of two point cntrs in imnsion on. Th isprsion quation is givn by th following formula in this cas (15) (1 + χα 1 1 )(1 + χα 1 ) χ = 0. It is convnint to us th following two paramtrs (16) γ i = 1 α i, i = 1,. Thn th isprsion quation is (17) (χ γ 1 )(χ γ ) γ 1 γ χ = 0. Lt us not th corrsponing oprator by L 1 (γ 1, γ ). Th isprsion quation for two cntrs in R 3 is givn by (18) ( χ + 4πα 1 1 )( χ + 4πα 1 ) χ = 0. Lt us introuc two nw paramtrs (19) γ j = 4π/α j, j = 1,. W gt th isprsion quation (0) (χ γ 1 )(χ γ ) χ = 0. Th corrsponing oprator will b not by L 3 (γ 1, γ ). Th paramtrs γ j just introuc for on- an thr-imnsional problms can b intrprt as th nrgis of th boun stats associat with ach of th two point cntrs. Consir th Schröingr oprator with on lta intraction + αδ(x). Thn th corrsponing oprator has xactly on boun stat with th nrgy (1) E = γ = α, provi α < 0 in imnsion on 4

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES.7 an () E = γ = (4π), provi α > 0 in imnsion thr. α Th nrgis corrsponing to singl intractions can b obtain from th isprsion quations (17) an (0) consiring th limit. Th xponntial function tns to zro an th two isprsion quations transform into th following quation (χ γ 1 )(χ γ ) = 0, having two solutions χ = γ 1, an fining th nrgis of th two boun stats E 1 = γ1 an E = γ provi that γ 1, > 0. Anothr way to obtain ths boun stats is to consir th limit whr th intraction at on of th cntrs vanishs. Not that vanishing intraction α j = 0 corrspons formally to γ j = 0 for R 1 an to γ j = for R 3. Th limits of th quations (17) an (0) whn γ 0, rspctivly γ ar givn by χ γ 1 = 0. Th last quation trmins th uniqu boun stat with th nrgy E 1 = γ 1 (provi γ 1 < 0). W fin it mor convnint to us paramtrs γ j insta of α j to paramtriz th oprators with lta intractions. Lt us stuy th numbr of ignvalus pning on th valus of th two paramtrs. Without loss of gnrality w can assum that γ 1 γ. W introuc th following notation P 1 (χ) = (χ γ 1)(χ γ ) γ 1 γ ; P 3 (χ) = (χ γ 1 )(χ γ ). Th ignvalus of th oprator L(γ 1, γ ) corrspon to positiv (ral) solutions of th isprsion quation. Not that quation (17) has on parasit solution χ = 0 which is not physical, sinc no ignfunction corrspons to E = 0 in this cas. Th corrsponing function os not blong to th Hilbrt spac. Lt us stuy th thr cass covring all possibilitis (th cass whn γ 1 or γ ar qual to zro can b xclu from th consiration), sparatly for on an thr imnsional point intractions γ γ 1 < 0 For positiv χ th functions P 1 (χ) an χ satisfy th following inqualitis P 1 (χ) 1 χ.

8 YU.N.DEMKOV AND P.KURASOV Th lattr inquality is strong for χ 0. Thrfor quation (17) has no solution on th intrval (0, ) in this cas. Th function P 3 is growing to infinity for positiv χ an th function χ / is crasing. Comparing th valus of th functions at th origin on can uc that q. (0) has on solution iff (3) γ 1 γ < 1. γ < 0 < γ 1 Th functions P 1 (χ) an χ ar qual to 1 at χ = 0. Thir scon rivativs ar ngativ an positiv, rspctivly. Thrfor quation (17) has at most on positiv solution an this solution xists if an only if χ ( χ ) χ=0 < χ P 1 (χ) χ=0 1 γ 1 + 1 γ <. Th solution blongs to th intrval (0, γ 1 ). Lt us not this solution by χ 1. Equation (0) always has on solution in this cas, sinc th function P 3 is growing to infinity an has positiv zro an th function χ / is positiv an crass to zro. Th solution blongs to th intrval (γ 1, ). 0 < γ γ 1 Th function P 1 is qual to zro at th points χ = γ 1 an χ = γ. It incrass to + on th intrval (γ 1, ). Thrfor thr xists a solution to th isprsion quation (17) on th intrval (γ 1, ). This solution will b not by χ 1 in what follows. Th scon solution is situat on th intrval (0, γ ) if an only if th following conition is satisfi χ ( χ ) χ=0 < χ P 1 (χ) χ=0 1 γ 1 + 1 γ <. Th scon solution will b not by χ. Similarly quation (0) has two solutions χ 1 an χ situat on th intrvals (0, γ ) an (γ 1, ) if an only if th following conition is satisfi (4) γ 1 γ > 1.

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES.9 Othrwis th quation has uniqu solution situat on th intrval (γ 1, ). Thus w prov onc mor that th iscrt spctra of th two problms consist of at most two istinct ignvalus situat on th ngativ half-axis. Lt us stuy th invrs spctral problm for ths oprator: Rconstruct th coupling constants γ 1 an γ so that th oprators L 1 (γ 1, γ ) an L 3 (γ, γ 3 ) hav givn χ 1 = E 1 < E = χ < 0 as ignvalus provi that th istanc btwn th intraction points is fix. Without loss of gnrality w can suppos that = 1. This problm can b solv in th following way. Consir first th isprsion quation corrsponing to th on imnsional problm. Suppos that χ 1 an χ ar solutions of (17). Thn th paramtrs γ 1, satisfy th following systm of quations (χ 1 γ 1 )(χ 1 γ ) = γ 1 γ χ 1 (5), (χ γ 1 )(χ γ ) = γ 1 γ χ which can b writtn as follows χ γ 1 γ = 1 χ (χ 1 χ ) χ 1 (1 χ ) χ (1 χ 1) A(χ 1, χ ) (6) γ 1 + γ = χ 1 (1 χ ) χ (1 χ 1) χ 1 (1 χ ) χ (1 χ 1) B(χ 1, χ ) Each of th two quations is linar with rspct to γ i. Thrfor this systm of quations can b solv for xampl by rsolving th first of th two quations with rspct to γ 1 an substituting it into th scon quation. On obtains in this way th quaratic quation, which can always b solv: (7) γ Bγ + A = 0. But on cannot guarant that th solutions, i.. paramtrs γ 1, ar ral. Only ral paramtrs γ j fin a slf-ajoint oprator. Hnc th nrgis of th boun stats ar not arbitrary, but satisfy th inquality (8) D(χ 1, χ ) B (χ 1, χ ) 4A(χ 1, χ ) 0. Similarly for th thr imnsional problm w hav th systm of quations { (χ1 γ (9) 1 )(χ 1 γ ) = χ 1, (χ γ 1 )(χ γ ) = χ ;.

10 YU.N.DEMKOV AND P.KURASOV (30) { γ 1 γ = χ 1 χ + χ 1 χ χ χ 1 χ 1 χ A(χ 1, χ ), γ 1 + γ = χ 1 + χ χ 1 χ χ 1 χ B(χ 1, χ ). Again thr xists a slf-ajoint oprator if an only if th iscriminant givn by (8) is non-ngativ. Thorm 1. Lt χ 1 < χ b two ignvalus of th oprator L j (γ 1, γ ), j = 1, 3. Assum that th nrgy E 1 = χ 1 is fix. Thn all possibl valus of χ fills in th intrval [0, χ max ] whr χ max = χ max (χ 1 ) is th valu of χ corrsponing to th symmtric intraction γ 1 = γ. This χ max is th uniqu solution to th following quations (31) 1 χmax χ max = 1 + χ 1 χ 1 for R 1 an R 3 rspctivly. an χ max + χmax = χ 1 χ 1, Proof. W ar going to prov th thorm for th on an thr imnsional cass sparatly. Dimnsion on. Th nrgis χ 1 an χ max corrsponing to th symmtric cas γ 1 = γ γ can b calculat from th following quation (χ γ) = γ χ χ γ = ±γ χ 1 + χ 1 = 1 χmax. χ 1 χ max To prov th thorm it is nough to show that th iscriminant of th systm (7) is positiv for all χ < χ max an ngativ for χ max < χ. Taking into account that th iscriminant is qual to zro for χ = χ max ( γ 1 = γ in this cas) it is sufficint to show that th rivativ χ D(χ 1, χ ) is ngativ. Consir th function f(x) = (1 x )/x. Dirct calculations show that 0 f(x), f (x) 0, 0 f (x), provi x > 0. It is asy to s that A = χ 1 χ f(χ ) f(χ 1 ). Thn th main valu thorm implis 0 A(χ 1, χ ) 1/. Using th fact that B = χ 1 + A(χ 1, χ )f(χ 1 ) th rivativ of th iscriminant can b valuat D(χ 1, χ ) χ = ((χ 1 + A(χ 1, χ )f(χ 1 ))f(χ 1 ) ) A(χ 1, χ ) χ.

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES. 11 To prov that th rivativ A(χ 1,χ ) χ thorm again is positiv w us th main valu A(χ 1, χ ) = f(χ 1) (f(χ ) + (χ 1 χ )f (χ )) χ (f(χ 1 ) f(χ )) an tak into account that th scon rivativ of f is positiv. Th xprssion in th brackts is ngativ χ 1 f(χ 1 ) + A(χ 1, χ )f (χ 1 ) χ 1 f(χ 1 ) + f (χ 1 )/ 0. Th last inquality follows irctly from th proprtis of th function f(x). It follows that D(χ 1, χ ) is positiv for χ < χ max an ngativ for χ max < χ. Dimnsion thr W introuc nw paramtrs { ξ = 1 (3) (γ { 1 + γ ) γ1 = ξ + η η = 1(γ 1 γ ) γ = ξ η Disprsion quation (0) is thn givn by (33) (χ ξ) η χ = 0. Sinc χ 1 is a solution of th last quation w gt (34) Q 3 (ξ, χ) (χ ξ) (χ 1 ξ) χ + χ 1 = 0. Thn th partial rivativ χ η sinc χ η χ=χ = can b stimat η(χ 1 χ ) (χ 1 ξ)(χ ξ + χ ) < 0, χ ξ + χ = χ + η + χ χ ( χ 1) < 0. It follows that χ attains its maximal valu whn η = 0 i.. in th symmtric cas γ 1 = γ γ. This cas corrspons to χ 1 an χ = χ max satisfying th quation (χ γ) = χ χ 1 χ max = χ 1 + χmax. Hnc for all χ : 0 χ χ max th following stimat hols (35) χ 1 χ χ 1 + χ. Lt us show that th iscriminant is positiv for all χ χ max. Th iscriminant of th quaratic quation (30) on γ 1 an γ is D(χ 1, χ ) = (χ 1 χ ) ( χ 1 + χ ) + ( χ 1 χ ) (χ 1 χ ).

1 YU.N.DEMKOV AND P.KURASOV Th sum of th first an th thir trms (χ 1 χ ) + ( χ 1 χ ) (χ 1 χ ) can b stimat by ( χ 1 + χ ) + ( χ 1 χ ) taking into account ( χ 1+ χ ) that (χ 1 χ ) χ 1 χ in accoranc with (35). Thn th iscriminant can b stimat from blow D(χ 1, χ ) ( χ 1 + χ ) ( χ 1 + ) χ + ( χ 1 χ ) ( χ 1 + χ ) = 0. It follows that th systm (30) has ral solutions for any 0 χ χ max (χ 1 ). Th thorm stats that th systm of two local point intractions nvr has a multipl ignvalu. Th istanc btwn th ignnrgis is minimal in th symmtric cas γ 1 = γ. This can b illustrat by th following figur. 5 5 4 4 3 3 x x 1 1 0 0 1 x1 3 4 5 0 0 1 3 4 x1 5 R 1 R 3 Figur 1. Lvl s rpulsion for two cntrs. Th ara lying btwn th two curv lins is forbin, i.. it is impossibl to fin two point intractions at th istanc = 1 trmining two ignvalus in this rgion. In th limit χ 1, th curvs approach th lin χ 1 = χ. It follows that two p ignvalus can b situat rathr clos to ach othr. Th curvs crosss th corrsponing axs at th sam point which is th uniqu solution of th quation x = 1 + x. W hav provn that th Schröingr oprator with two local point intractions cannot hav a gnrat ignvalu. Morovr th two ignvalus cannot b situat arbitrarily clos to ach othr. This is a crtain gnralization of von Numann-Wignr thorm [10]. Not

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES. 13 that th last statmnt hols u to th spcial form of th bounary conitions scrib by iagonal matrics. Consiring nonlocal point intractions on can obtain oprators with two ngativ ignvalus situat arbitrarily. 4. Dgnrat ignvalus for svral cntrs Th on-imnsional Schröingr oprator cannot hav gnrat ignvalus (xcpt th cas whr th oprator can b rprsnt by an orthogonal sum of oprators on two intrvals). Thrfor w rstrict our consiration to th cas of th thr-imnsional Schröingr oprator with point intraction scrib in Sction. Morovr w ar going to stuy th maximal possibl gnracy for th sak of simplicity. Low-orr gnracis can b scrib by stuying clustrs consisting of a smallr numbr of cntrs. Lt us consir th possibility for th maximal gnracy N 1 for th systm of N point potntials. Th quation trmining th ignvalus in this systm ras as follows χ + γ χ 1 χ 13 1 1 13... χ 1N 1N χ 1 1 χ + γ χ 3 3... χ N N (36) t χ 31 χ 3 31 3 χ + γ 3... χ 3N = 0, 3N............... χ N1 N1 χ N N χ N3 N3... χ + γ N whr γ j = 4πα 1 j. This quation trmins at most N ngativ ignvalus. W ar intrst in th cas whr on of ths ignvalus has maximal possibl multiplicity N 1. This happns if an only if all rows in th matrix ar paralll, i.. th trminants of all minors ar zro. Lt us rckon for which numbr of cntrs this is possibl. For larg nough N (N 3) configuration of th cntrs is trmin by N gom = 3(N ) istancs. In aition thr ar N xt = N paramtrs trmining th xtnsions. Hnc th matrix contains N par = N gom + N xt + 1 = 4N 5 paramtrs incluing in aition th spctral paramtr χ. Th total numbr of minors is qual to ( n(n 1) ), but th numbr of inpnnt quations rucs to n(n 1) u to th symmtry an spcial form of th matrix. Hnc in gnral thr ar N con = n(n 1) constraints on th paramtrs, which guarant th maximal gnracy of th ignvalu. Sinc th numbr of constraints is growing quaratically, but th numbr of paramtrs - just linarly, it is impossibl to fin a configuration laing to th maximal gnracy for sufficintly larg N. In th tabl blow w

14 YU.N.DEMKOV AND P.KURASOV calculat th maximal xpct imnsion D of th st of paramtrs which guarant th maximal gnracy. N N gom N xt N par N con D 1 0 1 1 1 1 4 1 3 3 3 3 7 3 4 4 6 4 11 6 5 5 9 5 15 10 5 6 1 6 19 15 4 7 15 7 3 1 8 18 8 7 8 W s that for low N our naiv calculations giv th corrct rsult. Thus in th cas of on point intraction thr is a on-paramtr family of xtnsions having an ignvalu. For two point intractions th family of oprators having an ignvalu is scrib by thr paramtrs: two xtnsion paramtrs an on istanc. For larg N (N 8) this tabl inicats that no ignvalu of maximal multiplicity is possibl. In th currnt sction w ar going to stuy th cas of intrmiat valus of N. Thr cntrs. Thr ar 9 minors. Sinc th matrix is symmtric, only 6 minors ar inpnnt. Du to th spcial structur of th matrix, th numbr of quations rucs to 3 (37) χ + γ 1 = χ + γ = χ + γ 3 = 3 1 13 χ( 3 13 1) 13 χ( 13 1 3) 1 3 1 χ( 1 13 3) 13 3 Thorm. For any configuration {y 1, y, y 3 } of thr points in R 3 an any ngativ numbr E = χ thr xists a uniqu st of paramtrs α 1, α, α 3 such that th Schröingr oprator with thr lta intractions of strngths α 1, α, α 3 concntrat at y 1, y, y 3 rspctivly has a gnrat ignvalu with th nrgy E = χ. Proof. Consir th systm of quations (37) for arbitrary st of positiv paramtrs 1, 13, 3, χ. Ths quations allow on to calculat irctly 3 positiv ral numbrs γ j = 4π/α j an hnc rconstruct th paramtrs trmining th lta intractions at th points y j. Th thorm shows that for any fix configuration of points supporting lta intractions th st of paramtrs trmining such intractions laing to oubl ignvalus can b paramtriz by on

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES. 15 ral paramtr - th nrgy of th gnrat ignvalu. Thn th st of paramtrs laing to a gnrat ignvalu can b scrib by 4 paramtrs as it was prict by th tabl. Four cntrs. Thr ar 36 minors, but u to th symmtry of th matrix only 1 minors ar inpnnt. Spcial form of th matrix rucs th numbr of inpnnt quations to 6. It is natural to ivi ths quations into two systms of four an two quations in ach systm rspctivly (38) an χ + γ 1 = χ + γ = χ + γ 3 = χ + γ 4 = 3 13 1 χ( 3 13 1) 34 4 3 χ( 34 4 3) 14 χ( 14 34 13) 34 13 1 χ( 1 14 4) 14 4 (39) χ( 1+ 34 ) 1 34 = χ( 13+ 4 ) 13 4 = χ( 14+ 3 ) 14 3. Th scon systm in th gnral cas scribs a crtain rlation btwn th istancs an th nrgy paramtr. It follows that not all configurations of four cntrs la to a tripl ignvalu. In gnral if this configuration is amissibl thn it trmins uniquly th possibl nrgy of th tripl boun stat (xcpt spcial cass scrib by Thorm 3). Thn th istancs an th nrgy of th tripl ignvalu can b us to trmin th strngths of th point intractions from th first four quations (38). For th systm of four cntrs lt us introuc som primtric coorinats - th sums of lngths of opposit gs in th ttrahron trmin by y 1, y, y 3, y 4 (40) D 1 = 1 + 34, D 13 = 13 + 4, D 14 = 14 + 3. Ths coorinats can b us to calculat th primtrs of all thr possibl four-angls in an asy way: th primtrs ar qual to th sums of th corrsponing two primtric coorinats. Thorm 3. Consir th Schröingr oprator in L (R 3 ) with four point intractions of strngths α 1, α, α 3, α 4 situat at th points y 1, y, y 3, y 4. This oprator has a tripl ignvalu if an only if on of th following conitions hol:

16 YU.N.DEMKOV AND P.KURASOV a) If all thr primtric coorinats ar iffrnt, thn th istancs btwn th cntrs shoul satisfy on of th following thr quivalnt conitions (41) ln 1 + ln 34 ln 13 ln 4 = ln 1 + ln 34 ln 14 ln 3 < 0, 1 + 34 13 4 1 + 34 14 3 (4) ln 1 + ln 34 ln 13 ln 4 1 + 34 13 4 = ln 13 + ln 4 ln 14 ln 3 13 + 4 14 3 < 0, (43) ln 1 + ln 34 ln 14 ln 3 1 + 34 14 3 = ln 13 + ln 4 ln 14 ln 3 13 + 4 14 3 < 0. Th nrgy of th tripl ignvalu is trmin uniquly by th gomtry of th cntrs ( ) ln 1 + ln 34 ln 14 ln 3 (44) E =. 1 + 34 14 3 Th uniqu valus of th constants α j ar trmin by (38) (taking into account (19)). b) If two of th primtric coorinats coinci, say D 1 D 13 = 1 + 34 13 4 = 0, thn th lngths in ths pairs shoul b qual, i.. (45) { 1 = 13 34 = 4 or { 1 = 4 34 = 13 Thn th tripl ignvalu xists only if conition (41) hols, its valu is uniquly trmin by th gomtry an is givn by (44). Th uniqu valus of th constants α j ar trmin by (38) (using (19)). c) If all thr paramtric coorinats ar qual thn th tripl ignvalu xists if an only if th four cntrs ar situat at th vrtics of a ttrahron with at last on si givn by a rgular triangl an th thr othr sis qual. Th nrgy of th tripl ignvalu is arbitrary ngativ an th valus of th paramtrs α j (all qual) ar uniquly trmin by this nrgy. Proof. Lt us consir all thr cass sparatly. a) If all thr paramtric coorinats ar iffrnt thn th paramtr χ which trmins th nrgy of th tripl boun stat can b

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES. 17 calculat from (39) using thr iffrnt qualitis as follows χ = ln 1 + ln 34 ln 13 ln 4 1 + 34 13 4, χ = ln 1 + ln 34 ln 14 ln 3 1 + 34 14 3, χ = ln 13 + ln 4 ln 14 ln 3 13 + 4 14 3. Excluing χ from th last quations in thr iffrnt ways w gt (41,4,43) taking into account that χ must b positiv. Thn th paramtrs α j (or γ j ) can b calculat from (38). b) If any two of thr primtric coorinats coinsi, say D 1 = D 13, thn th systm of quations (39) is quivalnt to th following two quations ln 1 + ln 34 ln 13 ln 4 = ln 1 + ln 34 ln 14 ln 3 > 0; 1 + 34 13 4 1 + 34 14 3 1 34 = 13 4. Taking into account that th two primtric coorinats coinci w conclu that th latr quation implis (45) an that th nrgy of th tripl ignvalu is givn by (44). Th xtnsion paramtrs ar again trmin by (38). c) If all thr primtric coorinats coinci thn (39) is quivalnt to 1 34 = 13 4 = 14 3, an it follows that th four cntrs form a ttrahron with at last on si givn by a rgular triangl. Th thr rmaining sis ar qual. In this cas quation (39) is satisfi for arbitrary valu of th nrgy paramtr E = χ. Aftr choosing this paramtr qual to arbitrary ngativ numbr on can calculat uniqu valus of th strngth paramtrs from (38). Th first family of point intractions is scrib by 5 inpnnt paramtrs as it was xpct (s th tabl). Th othr two familis ar scrib by 4 an 3 paramtrs rspctivly. Th last family is th most intrsting, sinc it inclus th rgular ttrahron - th most symmtric configuration of four cntrs. Fiv cntrs. W stuy th possibility that this systm has an ignvalu of multiplicity 4. It is not obvious that th systm of quations is solvabl. Th tabl pricts that th family of solutions is scrib by 5 paramtrs. W ar abl to show that thr xists a two-paramtr family. Consir th most symmtric configuration of 5 points:

18 YU.N.DEMKOV AND P.KURASOV points y, y 3, y 4, y 5 ar situat at th cornrs of a rgular ttrahron, point y 1 is situat in th cntr of th ttrahron. W not by th lngth of th ttrahron s gs an by r th raius of th scrib sphr containing th four vrtics of th ttrahron. Thn th systm (36) ras as follows χ + γ χr χr χr χr 1 r r r r χr χ + γ χ χ χ r (46) t χr χ χ + γ χ χ r 3 = 0. χr χ χ χ + γ χ r 4 χr χ χ χ χ + γ r 5 All rows of th matrix ar linarly pnnt (th ignvalu has multiplicity 4) if an only if an χ + γ = χ + γ 3 = χ + γ 4 = χ + γ 5 = χ γ = γ 3 = γ 4 = γ 5 = χ + χ / ( ) χr χ + γ 1 = / χ γ 1 = χ + r r χ( r). Hnc for any χ on can fin paramtrs α j, j = 1,..., 5 such that th Schröingr oprator with 5 point intractions has an ignvalu of multiplicity 4. Th last family appars to b th most intrsting, sinc it conyains th rgular ttrahron - th most symmtric configuration of four cntrs. Six an mor cntrs. Consir th systm of six cntrs. Th tabl pricts that th st of paramtrs laing to th maximal gnracy in this cas is scrib by 4 paramtrs. Lt us xamin th most symmtric configuration of six cntrs - th points situat at th summits of th octahron. Th matrix taks th following form χ + γ χ χ 1 χ χ χ χ + γ χ χ χ χ χ χ + γ χ χ 3 χ χ χ χ + γ χ 4 χ χ χ χ χ + γ χ 5 χ χ χ χ χ χ χ χ χ χ + γ 6,

VON NEUMANN-WIGNER THEOREM:LEVEL S REPULSION AND DEGENERATE EIGENVALUES. 19 whr is th lngth of th octahrons g. Th rows ar paralll only if χ = χ ln, χ = ( 1) < 0. But th paramtr χ has to b positiv. It follows that this systm cannot hav an ignvalu of multiplicity 5. Th last quation trmins a rsonanc insta of th ignvalu. W conjctur that th systm of 6 point intractions cannot hav an ignvalu of multiplicity 5. Similarly w o not xpct ignvalus of th multiplicity N 1 for any systm N point intraction, provi N > 6. W ar going to rturn back to this problm in on of our forthcoming publications. Conclusions Th authors ar gratfull to Th Swish Royal Acamy of Scincs an Founation for Basic Rsarch (Vtnskapsråt) for financial support. Rfrncs [1] M.N. Aamov, Yu.N. Dmkov, V.D. Ob kov, an T.K. Rban, Mol of a small-raius potntial for molcular systms, Tor. Eksp. Khimiya, 4 (1968), 147 153. [] S. Albvrio, F. Gsztsy, R. Høgh-Krohn, an H. Holn, Solvabl mols in quantum mchanics. Scon ition. With an appnix by Pavl Exnr. AMS Chlsa Publishing, Provinc, RI, 005. [3] S. Albvrio an P. Kurasov, Singular prturbations of iffrntial oprators. Solvabl Schröingr typ oprators, Lonon Mathmatical Socity Lctur Not Sris, 71. Cambrig Univrsity Prss, Cambrig, 000. [4] Yu.N. Dmkov, G.F. Drukarv, an V.V. Kuchinskii, Dissociation of ngativ ions in short rang potntial approximation, J. Exp. Thor. Phys., 58 (1970), 944 951. [5] Yu. Dmkov an V. Ostrovsky, Zro-rang potntials an thir applications in atomic physics, Plnum, Nw York, 1988 (Translation from th 1975 Russian original: Dmkov,.N., Ostrovski, V.N., Mto potncialov nulvogo raiusa v atomno fizik, Iz vo Lningraskogo Univrsitta, Lningra, 1975 ) [6] M. Krin, On Hrmitian oprators whos ficincy inics ar 1, Compts Rnu (Doklay) Aca. Sci. URSS (N.S.), 43, 131-134, 1944. [7] M. Krin, On Hrmitian oprators whos ficincy inics ar qual to on. II, Compts Rnu (Doklay) Aca. Sci. URSS (N.S.), 44, 131-134, 1944. [8] P. Kurasov an A. Posilicano, Finit sp of propagation an local bounary conitions for wav quations with point intractions. Proc. Amr. Math. Soc. 133 (005), 3071 3078. [9] M. Naimark, On spctral functions of a symmtric oprator, Bull. Aca. Scincs URSS, 7, 85-96, 1943. [10] J. von Numann an E. Wignr, Übr as Vrhaltn von Eignwrtn bi aiabatischn Prozssn, Phys. Zit., 30 (199), 467-470.

0 YU.N.DEMKOV AND P.KURASOV Particulars about th authors Pavl Kurasov (corrsponing author) Physics Rsarch Institut, St.Ptrsburg Univrsity Russia, 198904 St.Ptrsburg, St. Ptrhof Ul yanovskaya 1, St.Ptrsburg Univrsity, Physics Rsarch Institut, Laboratory of Quantum Ntworks an Dpt. of Mathmatics, Lun Univrsity, 1 00 Lun, Swn -mail: pak@math.su.s Contact phon numbr: (81) 340791 Yurii N. Dmkov Physics Rsarch Institut, St.Ptrsburg Univrsity Russia, 198904 St.Ptrsburg, St. Ptrhof Ul yanovskaya 1, St.Ptrsburg Univrsity, Physics Rsarch Institut, Dpt. of Quantum Mchanics Contact phon numbr: (81) 3067 Titl: von Numann-Wignr thorm: lvl s rpulsion an gnrat ignvalus. Authors: Yu.N.Dmkov an P.Kurasov Abstract Spctral proprtis of Schröingr oprators with point intractions ar invstigat. Attntion is focus on th intrplay btwn th lvl s rpulsion (von Numann-Wignr thorm) an th symmtry of th points configuration. Explicit solvability of th problm allows to obsrv lvl s rpulsion for two cntrs. For largr numbr of cntrs th familis of point intractions laing to th highst possibl gnracy is invstigat. UDC 517.984, 530.. Kywors: Wignr-von Numann thorm, zro-rang potntials, xtnsion thory, invrs spctral problm.