Unfolding a degeneracy point of two unbound states: Crossings and anticrossings of energies and widths

0.24 MB·English

by  E. Hernandez

Unfolding a degeneracy point of two unbound states: Crossings and anticrossings of
energies and widths
E. Hern´andez, A. J´auregui , A. Mondrago´n, and L. Nellen
† ‡
Instituto de F´ısica, UNAM, Apdo. Postal 20-364, 01000 M´exico D.F., M´exico
†Departamento de F´ısica, Universidad de Sonora,
Apdo. Postal 1626, Hermosillo, Sonora, M´exico
‡Instituto de Ciencias Nucleares, UNAM, Apdo. Postal 70-543, 04510 M´exico D.F., M´exico
Weshowthatwhenanisolateddoubletofunboundstatesofaphysicalsystembecomesdegenerate
for some values of the control parameters of the system, the energy hypersurfaces representing the
complex resonance energy eigenvalues as functions of the control parameters have an algebraic
branch point of rank one in parameter space. Associated with this singularity in parameter space,
5 the scattering matrix, Sℓ(E), and the Green’s function, G(ℓ+)(k;r,r′), have one double pole in
the unphysical sheet of the complex energy plane. We characterize the universal unfolding or
0
deformation of a typicaldegeneracy point of two unboundstates in parameterspace bymeans of a
0
universal 2-parameter family of functions which is contact equivalent to the pole position function
2
of the isolated doublet of resonances at the exceptional point and includes all small perturbations
n of thedegeneracy condition up to contact equivalence.
a
J
PACSnumbers: 03.65.Nk;33.40.+f;03.65.Ca;03.65.Bz

2
2
I. INTRODUCTION eters (x1,x2) to the other arguments after a semicolon.
1 Theenergyeigenvalues = ¯h2/2m k2 oftheHamil-
v En n
4 Recently, a great deal of attention has been given to tonianHr(ℓ) areobtainedfromth(cid:0)ezeroes(cid:1)oftheJostfunc-
2 the characterization of the singularities of the surfaces tion, f( k;x ,x ) [15], where k is such that
1 2 n
−
1 representing the complex resonance energy eigenvalues
1 at a degeneracy of unbound states. This problem arises f( kn;x1,x2)=0. (1)
0 −
naturallyinconnectionwiththe topologicalphaseofun-
5
0 bound states which was predicted by Hern´andez, Mon- When kn lies in the fourth quadrant of the complex
/ drag´onandJ´auregui[1,3,4]andlaterandindependently k−plane, Rekn > 0 and Imkn < 0, the correspond-
h by W.D. Heiss[5]andwhichwasrecently verifiedina se- ing energy eigenvalue, n, is a complex resonanceenergy
p ries of beautiful experiments by P. von Brentano[6, 7, 8] eigenvalue. E
-
t and the Darmstadt group[9, 10], see also[11]. The condition (1) defines, implicitly, the functions
n
k (x ,x ) as branches of a multivalued function [15]
a n 1 2
whichwillbe calledthe wave-numberpole positionfunc-
u
q tion. Each branch kn(x1,x2) of the pole position func-
II. DEGENERACY OF RESONANCE ENERGY
: tionisacontinuous,single-valuedfunctionofthe control
v EIGENVALUES AS BRANCH POINTS IN
parameters. When the physical system has an isolated
i PARAMETER SPACE
X doubletofresonanceswhichbecomedegenerateforsome
r exceptional values of the external parameters, (x∗1,x∗2),
a In this short communication, we will consider the res- the corresponding two branches of the energy-pole posi-
onanceenergyeigenvaluesofaradialSchr¨odingerHamil- tion function, say (x ,x ) and (x ,x ), are equal
n 1 2 n+1 1 2
tonian,Hr(ℓ),withapotentialV(r;x1,x2)whichisashort (crossorcoincide)Eatthatpoint. AEswillbeshownbelow,
rangedfunctionofthe radialdistance,r,anddepends on at a degeneracy of resonances, the energy hypersurfaces
at least two external control parameters (x1,x2). When representingthecomplexresonanceenergyeigenvaluesas
the potential V(r;x1,x2) has two regions of trapping, functionsoftherealcontrolparametershaveanalgebraic
the physical system may have isolated doublets of res- branchpointofsquareroottype(rankone)inparameter
onances which may become degenerate for some spe- space.
cial values of the control parameters. For example, a Isolated doublet of resonances: Let us suppose
double square barrier potential has isolated doublets of that there is a finite bounded and connected region
resonances which may become degenerate for some spe- in parameter space and a finite domain in the
M D
cial values of the heights and widths of the barriers fourthquadrantofthecomplexk plane,suchthat,when
−
[12, 13, 14]. (x ,x ) ǫ , the Jostfunction has two andonly two ze-
1 2
M
In the case under consideration,the regularand phys- roes, k and k , in the finite domain ǫ C, all other
n n+1
D
ical solutions of the Hamiltonian are functions of the ra- zeroes of f( k;x ,x ) lying outside . Then, we say
1 2
− D
dial distance, r, the wave number, k, and the control that the physical system has an isolated doublet of res-
parameters (x ,x ). When necessary, we will stress this onances. To make this situation explicit, the two zeroes
1 2
lastfunctionaldependence by adding the controlparam- off( k;x ,x ),correspondingto the isolateddoubletof
1 2
−
2
resonances are explicitly factorized as
f(−41k(cid:0);kxn1,−x2k)n+=1(cid:1)h2(cid:0)ikg−n,n21+(1k(nk+,xk1n,x+21)).(cid:1)2− (2) ∂2f(−k21;xh1(cid:16),x−2∂1)(cid:0)kn(x1,x2)∂+xk1nn+h1(cid:16)(x∂12,xf2(∂)−(cid:1)xk1(cid:17);∂xxk12,ix2k)d(cid:17)=x2ik=kd−
When the physical system moves in parameter space h(cid:16) ∂k2 (cid:17)x∗1,x∗2ik=dd
pfkrnoo+imn1t(xt(1hx,e∗1x,2xo)r∗2,d)c,ionatalhreyescetpwoinointotsimo(nxpe1le,dxoz2ue)brolteeosz,etrhkoenk(xde1(xx,cx∗1e,p2x)ti∗2oa)nniandl h(cid:16)∂2f(−∂kk;2x1,x12)(cid:17)x∗1,x∗2ik=kd 13h(cid:16)∂3f(−∂kk;3x1,x2)(cid:17)x∗1,x∗2ik=kd
the fourth quadrant of the complex k plane. ∂f(−k;x1,x2) .
If the external parameters take valu−es in a neighbour- ×h(cid:16) ∂x1 (cid:17)x2ik=kdo
(8)
hood of the exceptional point (x ,x ) ǫ and k ǫ ,
∗1 ∗2 M D Fromtheseresults,thefirsttermsinaTaylorseriesex-
we may write
pansion of the functions 1/2 k (x ,x )+k (x ,x )
n 1 2 n+1 1 2
gn,n+1(k;x1,x2)≈gn,n+1(kd,x∗1,x∗2)6=0. (3) and 1/4 kn(x1,x2) − kn+1(x(cid:0)1,x2) 2 about the excep(cid:1)-
Then, tional po(cid:0)int (x∗1,x∗2), when substitu(cid:1)ted in eq.(5), give
k 1 k (x ,x )+k (x ,x ) 2 kˆn,n+1(x1,x2)=kd(x∗1,x∗2)+∆kd(x1,x2)+
n 1 2 n+1 1 2
1 k (x ,x ) hk− 2((cid:0)x ,x ) 2 f(−k;x1,x2(cid:1))i −, (4) r14 c(11)(x1−x∗1)+c(21)(x2−x∗2) (9)
4(cid:16) n 1 2 − n+1 1 2 (cid:17) ≈ gn,n+1(kd;x∗1,x∗2) (cid:2) (cid:3)
for (x ,x ) in a neighbourhood of the exceptional point
1 2
1
the coefficient gn,n+1(kd;x∗1,x∗2) − multiplying (x∗1,x∗2). Thisresultmayreadilybetranslatedintoasim-
f( k;x1,x2) may (cid:2)be understood a(cid:3)s a finite, non- ilarassertionfortheresonanceenergy-polepositionfunc-
−
vanishing, constant scaling factor. tion (x ,x )andtheenergyeigenvalues, (x ,x )
n,n+1 1 2 n 1 2
E E
The vanishing of the Jost function defines, implicitly, and (x ,x ), of the isolated doublet of resonances.
n+1 1 2
E
the pole position function kn,n+1(x1,x2) of the isolated Energy-pole position function: Let us take the
doublet of resonances. Solving eq.(2) for kn,n+1, we get square of both sides of eq.(5), multiplying them by
¯h2/2m and recalling = ¯h2/2m k2, in the approxi-
1 En n
kn,n+1(x1,x2)= kn(x1,x2)+kn+1(x1,x2) + (cid:0)mation (cid:1)of (9), we get (cid:0) (cid:1)
2
(cid:16) (cid:17)
1 2 ˆ (x ,x ) = (x ,x )+∆ (x ,x )
r4(cid:16)kn(x1−x2(cid:1)−kn+1(cid:0)x1,x2(cid:1)(cid:17) (5) En,n+1 1 2 + Eǫˆnd,n+∗11(x∗21,x2),Ed 1 2 (10)
with (x ,x ) ǫ . Since the argument of the square-
1 2 M where
root function is complex, it is necessary to specify the
branch. Hereandthereafter,thesquarerootofanycom- 1
plex quantity F will be defined by ǫˆn,n+1(x1,x2)=r4 (R~ ·ξ~)+i(I~·ξ~)
(cid:2) (cid:3)
1 (11)
√F = √F exp i argF , 0 argF 2π (6)
| | (cid:0) 2 (cid:1) ≤ ≤ The components of the real fixed vectors R~ and I~ are
so that √F = F and the F plane is cut along the the real and imaginary parts of the coefficients C(1)
| | | | − i
real axis. p of (x x ) in the Taylor expansion of the function
Equation (5) relates the wave number-pole position 1/4 i −(x ,∗ix ) (x ,x ) 2 and the real vector ξ~ is
n 1 2 n+1 1 2
functionofthedoubletofresonancestothewavenumber- E −E
the(cid:0)position vector of the po(cid:1)int (x1,x2) relative to the
polepositionfunctions oftheindividualresonancestates
exceptional point (x ,x ) in parameter space.
in the doublet. ∗1 ∗2
The analytical behaviour of the pole-position ξ x x
funTchteiodneraitvatthiveesexocfepthtieonfaulncptoioinnst:1/2 k (x ,x ) + ξ~=(cid:18)ξ12(cid:19)=(cid:18)x21−−x∗2∗1(cid:19), (12)
n 1 2
k (x ,x ) and 1/4 k (x ,x ) k (x(cid:0),x ) 2 are fi-
n+1 1 2 n 1 2 − n+1 1 2 Re C(1) Im C(1)
nite at the e(cid:1)xceptiona(cid:0)l point. They may be co(cid:1)mputed R~ = 1 , I~= 1 . (13)
fromtheJostfunctionwiththehelpoftheimplicitfunc- (cid:18)Re C2(1)(cid:19) (cid:18)Im C2(1)(cid:19)
tion theorem [17],
The real and imaginary parts of the function
2 ˆǫ (x ,x ) are
∂ kn(x1,x2)−kn+1(x1,x2) = n,n+1 1 2
∂2hf(cid:16)(−k(cid:0);x1,x−28) ∂x1 h(cid:16)∂(cid:1)f((cid:17)−∂xkx2;x1i1k,=x2k)d(cid:17)x2ikd, (7) Reǫˆn,n+1(x1,x2)=±2√12h+q(cid:0)R~ ·ξ~(cid:1)2+(cid:0)I~·ξ~(cid:1)2+R~·ξ~i1/2
h(cid:16) ∂k2 (cid:17)x∗1,x∗2ik=kd (14)
3
Imǫˆn,n+1(x1,x2)= 1 + R~ ξ~ 2+ I~ ξ~ 2 R~ ξ~ 1/2 Along the line LR, excluding the exceptional point
±2√2h q(cid:0) · (cid:1) (cid:0) · (cid:1) − · i (x∗1,x∗2),
(15)
Re (x ,x )=Re (x ,x ) (21)
and n 1 2 n+1 1 2
E E
but
sign Reǫ sign Imǫ =sign I~ ξ~ (16)
n,n+1 n,n+1
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) · (cid:17) Im (x ,x )=Im (x ,x ). (22)
n 1 2 n+1 1 2
E 6 E
It follows from (14), that Reǫˆn,n+1(x1,x2) is a two Similarly, along the line I, excluding the exceptional
branched function of (ξ1,ξ2) which may be repre- point, L
sented as a two-sheeted surface S , in a three di-
R
mensional Euclidean space with cartesian coordinates Im n(x1,x2)=Im n+1(x1,x2), (23)
E E
(Reǫˆ ,ξ ,ξ ). The two branches of Reǫˆ (ξ ,ξ )
n,n+1 1 2 n,n+1 1 2 but
are represented by two sheets which are copies of the
plane (ξ1,ξ2) cut along a line where the two branches of Re En(x1,x2)6=Re En+1(x1,x2). (24)
the function are joined smoothly. The cut is defined as
Equality of the complex resonance energy eigenvalues
thelocusofthepointswheretheargumentofthesquare-
(degeneracy of resonances), (x ,x ) = (x ,x ) =
root function in the right hand side of (14) vanishes. En ∗1 ∗2 En+1 ∗1 ∗2
(x ,x ), occurs only at the exceptional point with co-
Therefore, the real part of the energy-pole position Ed ∗1 ∗2
ordinates (x ,x ) in parameter space and only at that
function, (x ,x ),asafunctionoftherealparame- ∗1 ∗2
En,n+1 1 2 point.
ters (x ,x ), has an algebraic branch point of square root
1 2 Inconsequence,inthecomplexenergyplane,thecross-
type (rank one) at the exceptional point with coordinates
ing point of two simple resonance poles of the scattering
(x ,x ) in parameter space, and a branch cut along a
∗1 ∗2 matrix is an isolated point where the scattering matrix
line, , that starts at the exceptional point and extends
in theLpRositive direction defined by the unit vector ξˆ sat- has one double resonance pole.
c Remark: In the general case, a variation of the vec-
isfying.
tor of parameters causes a perturbation of the energy
eigenvalues. In the particular case of a double complex
I~ ξˆ =0 and R~ ξˆ = R~ ξˆ (17)
· c · c −| · c| resonanceenergyeigenvalueEd(x∗1,x∗2),associatedwitha
chain of length two of generalized Jordan-Gamow eigen-
A similar analysis shows that, the imaginary part of
functions [19], we are considering here, the perturbation
the energy-pole position function, Im (x ,x ), as
a function of the real parameters (x1,x2E)n,,na+ls1o h1as a2n al- steerrmiesseoxfptahnesisomnaollf pthaeraemigeetnevralξue,seEqns.,(E1n8+-210)a,btoaukteEsdthine
gebraic branch point of square root type (rank one) at the | |
form of a Puiseux series
exceptional point with coordinates (x ,x ) in parameter
∗1 ∗2
sstpaarctes,aatntdheaelsxocehpatisonaablrpaoninchtacnudteaxlotenngdsainlinteh,eLneIg,atthivaet En,n+1(x1,x2)=Ed(x∗1,x∗2)+|ξ|1/2q14 (R~ ·ξˆ)+i(I~·ξˆ)
+∆ (x ,x )+O ξ 3/(cid:2)2 (cid:3)
direction defined by the unit vector ξˆc satisfying eqs.(17). Ed 1 2 (cid:16)| | (cid:17)
The branch cut lines, and , are in orthogonal (25)
R I
subspaces of a four dimeLnsional LEuclidean space with with fractional powers ξ j/2, j = 0,1,2,... of the small
| |
coordinates (Reǫ ,Imǫ ,ξ ,ξ ), but have one parameter ξ [17, 18].
n,n+1 n,n+1 1 2 | |
pointincommon,theexceptionalpointwithcoordinates
(x ,x ).
∗1 ∗2 III. UNFOLDING OF THE DEGENERACY
The individual resonance energy eigenvalues are con-
POINT
ventionally asociated with the branches of the pole posi-
tion function according to
Let us introduce a function fˆ ( k;ξ ,ξ ) such that
doub 1 2
−
σR(mEˆm)2(√1ξ21,hξ+2q)=(R~E·dξ(~0)2,0+)+(I~∆·ξ~E)n2,n++1(R(~ξ1·,ξ~ξ)2i)1+/12/2+ (18) fˆdoub(−k;ξ1,ξ2) = h1k−(~(cid:16)kξ~d)(0+,0i)(~+∆ξ~)(1),kd(ξ1,ξ2)(cid:17)(i226)
iσI(m)2√12h+q(R~ ·ξ~)2+(I~·ξ~)2−(R~ ·ξ~)i , and − 4(cid:16) R· I· (cid:17)
with m=n,n+1, and
∆(1)k (x ,x )= 2 d(1)ξ (27)
d 1 2 i=1 i i
σ(n) = σn+1 = ReEn−ReEn+1 , (19) Close to the exceptional poPint, the Jost function
R − R Re n Re n+1 f( k;ξ ,ξ )andthe family offunctionsfˆ ( k;ξ ,ξ )
| E − E | 1 2 doub 1 2
− −
are related by
Im Im 1
σ(n) = σn+1 = En− En+1 (20) f( k;ξ ,ξ ) fˆ ( k;ξ ,ξ ) (28)
I − I Im n Im n+1 − 1 2 ≈ gn,n+1(kd;0,0) doub − 1 2
| E − E |
4
1
the term gn,n+1(kd,0,0) − may be understood as a
non-vanish(cid:2)ing scale factor.(cid:3)
Hence, the two-parameters family of functions
2
fˆ ( k;ξ ,ξ )= k k +∆(1)k (ξ ,ξ ) 0.08
doub 1 2 d d 1 2
− h −(cid:16) (cid:17)i −
1 ~ ξ~+i~ ξ~
4(cid:16)R· I· (cid:17) (29) 0.04
is contactequivalenttothe Jostfunctionf( k;ξ ,ξ ) at
− 1 2 d*
the exceptional point. It is also an unfolding [16, 20] of d-
0.00
f( k;ξ ,ξ ) with the following features:
1 2
−
1. It includes all possible small perturbations of the
-0.04
degeneracy conditions C^n+1( )
n+1
∂f( k;ξ ,ξ ) -0.08 ^
f( k;ξ ,ξ )=0, − 1 2 =0 (30) Cn( ) 0.08
− 1 2 (cid:16) ∂k (cid:17)kd 0.08 0.04 n 0.04
0.00
0.00
∂2f(−k;ξ1,ξ2) =0 (31) Im -0.04 -0.08 -0.08-0.04 Re
(cid:16) ∂k2 (cid:17)kd 6
up to contact equivalence.
FIG. 1: The curves Cˆn(π1) and Cˆn+1(π1) are the trajecto-
2. Itusestheminimumnumberofparameters,namely riestracedbythepointsEˆn(ξ1,ξ¯2(1))andEˆn+1(ξ1,ξ¯2(1))onthe
two,whichisthecodimensionofthedegeneracy[2]. hypersurface Eˆn,n+1(ξ1,ξ¯2(1)) when the point (ξ1,ξ¯2(1)) moves
The parameters are (ξ1,ξ2). along the straight line path π1 in parameter space. In the
figure, the path π runs parallel to the vertical axis and
Therefore, fˆdoub(−k;ξ1,ξ2) is a universal unfolding crosses the line LI1at a point (ξ1,c,ξ¯2(1)) with ξ1,c < ξ1∗ and
[p1o6in]towf htheereJtohsetdfuegnecntieornacfy(o−fku;nξ1bo,ξu2n)dasttathteeseoxccceuprtsi.onal ξp¯2(l1a)ne<(Iξm2∗.E,Tξh1)eaprreosjeeccttiioonnssoofftChˆen(sπu1r)faacendSICˆ;nt+he1(pπr1o)jeocntiothnes
The vanishing of fˆdoub(−k;ξ1,ξ2) defines the approxi- of Cˆn(π1) and Cˆn+1(π1) on the plane (ReE,ξ1) are sections
mate wave number-pole position function of the surface SR. The projections of Cˆn(π1) and Cˆn+1(π1)
ontheplane(ReE,ImE)arethetrajectoriesoftheS−matrix
kˆn,n+1(ξ1,ξ2)=kd+∆(n1,)n+1kd(ξ1,ξ2)±h41(cid:0)R~·ξ~+iI~·ξ~(cid:1)(i312/)2 poles in thecomplex energy pl.ane. In the figure,d−d∗ =ξ1
and the corresponding energy-pole position function
ˆ (ξ ,ξ ) given in eq.(10).
n,n+1 1 2
E Since the functions ˆn(ξ1,ξ2) and ˆn+1(ξ1,ξ2) are ob- parameter, ξ1, keeping the other constant, ξ2 = ξ¯2(i). A
tained from the vaniEshing of the uEniversal unfolding crossing of energies occurs if the difference of real ener-
fˆdoub( k;ξ1,ξ2) of the Jost function f( k;ξ1,ξ2) at the gies vanishes, ∆E =0, for some value ξ1,c of the varying
− − parameter. An anticrossing of energies means that, for
exceptional point, we are justified in saying that, the
family of functions ˆn(ξ1,ξ2) and ˆn+1(ξ1,ξ2), given in all values of the varying parameter, ξ1, the energies dif-
E E fer, ∆E = 0. Crossings and anticrossings of widths are
eqs.(18) and (19-20), is a universal unfolding or defor-
6
mation of a generic degeneracy or crossing point of two similarly described.
The experimentally determined dependence ofthe dif-
unboundstateenergy eigenvalues, which is contact equiv-
alenttotheexactenergy-poleposition functionoftheiso- ference of complex resonance energy eigenvalues on one
control parameter, ξ , while the other is kept constant,
lated doublet of resonances at the exceptional point, and 1
includes all small perturbations of the degeneracy condi-
tions up to contact equivalence . Eˆn(ξ1,ξ¯2(i))−Eˆn+1(ξ1,ξ¯2(i))=ǫˆn,n+1(ξ1,ξ¯2(i)) (33)
has a simple and straightforward geometrical inter-
pretation, it is the intersection of the hypersurface
IV. CROSSINGS AND ANTICROSSINGS OF
RESONANCE ENERGIES AND WIDTHS ˆǫn,n+1(ξ1,ξ2) with the hyperplane defined by the con-
dition (ξ ,ξ¯(i)).
1 2
Crossings or anticrossings of energies and widths are Torelatethegeometricalpropertiesofthisintersection
experimentally observed when the difference of complex with the experimentally determined properties of cross-
energy eigenvalues (ξ ,ξ¯) (ξ ,ξ¯) = ∆E ings andanticrossingsof energiesandwidths, letus con-
n 1 2 n+1 1 2
i(1/2)Γ is measuredEas functi−onEof one slowly varyin−g sider a point (ξ ,ξ¯(i)) in parameter space away from the
1 2
5
0.08 0.08
0.04 0.04
*d-d0.00 d-d*0.00
-0.04 -0.04
n+1 C^n+1( 2) n+1 C^n+1( 3)
-0.08 C^n( 2) 0.08 -0.08 C^n( 3) 0.08
0.08 0.08
0.04 0.04
0.04 n 0.04 n
0.00 0.00
0.00 0.00
Im -0.04 -0.04 Re Im -0.04 -0.04 Re
-0.08 -0.08
-0.08 -0.08
FIG.2: ThecurvesCˆn(π2)andCˆn+1(π2)arethetrajectories FIG. 3: The curves Cˆn(π3) and Cˆn+1(π3) are the trajecto-
of the points Eˆn(ξ1,ξ2∗) and Eˆn+1(ξ1,ξ2∗) on the hypersurface riestracedbythepointsEˆn(ξ1,ξ¯2(3))andEˆn+1(ξ1,ξ¯2(3))onthe
oilEˆninnn,epntah+pre1aa(tpmξhl1ae,πtnξe2e2rs)tsh(wpRaahteceEegn.o,ξetT1hsh)eteahpnproodriuon(gtjIehmc(ξtti1Eho,,enξξs2∗e1)x)ocfmaeCrpˆoenvtsie(oeπsnc2ata)illooanpnnsogdionaCftˆtsn(ht+ξre11∗a(,isgπξuh2∗2rt))- jwhaelyicotptnhiegornξsa1us,scrotffar>acCˆiegξnh1E∗(t.πn,3lniT)n+hea1en(pξdap1ta,Chˆξt¯nh2(π3+)3π1)(g3πwoc3ihnr)eogonsntsretosthhueteghphpeoltialhninneteep((ξoLR1iRn,eξt¯.E2((3,Tξ)ξ1)1h,)cme,sξ¯oph2(vro3oe)ws)-
facesSR andSI respectively,andshowajointcrossingofen- a crossing, but the projections on the planes (ImE,ξ ) and
ergies and widths. The projections of Cˆn(π2) and Cˆn+1(π2) (ReE,ImE)do not cross. In thefigure, ξ =d−d∗. 1
on the plane (ReE,ImE) are two straight line trajectories of 1
the S−matrix poles crossing at 90◦ in the complex energy
plane. At the crossing point, the two simple poles coalesce
into one doublepole of S(E). respectively.
From eqs.(18-20), and keeping ξ =ξ¯(i), we obtain
2 2
exceptional point. To this point corresponds the pair of ∆E =En En+1 = Reˆn Reˆn+1
apnnoerdns-udErenfg+aec1ne(ξeǫˆ1ra,ξt¯e2(i)r)e(,ξsore,npξarn)ec.seeAnestneetdhrgebyypeotiiwgneotn(vpξaol,iunξ¯et(ssi))Eonnm(ξoth1v,eeξ¯s2h(oiy)n)- = σ(n2)√2h+q−(R~ ·ξ~)2+(cid:16)(I~·Eξ~)2−+(R~E·ξ~)i(cid:17)1(cid:12)(cid:12)(cid:12)/ξ22i=ξξ2¯2(=i)ξ¯2(i()37)
n,n+1 1 2 1 2
a straight line path π in parameter space, and
i
π : ξ ξ ξ , ξ =ξ¯(i) (34) ∆Γ= 1 Γ Γ =Im Im
i 1,i ≤ 1 ≤ 1,f 2 2 2(cid:16) n− n+1(cid:17) (cid:16)En+1(cid:17)−(cid:16) En(cid:17)
tthraececotrwreospcuornvdiinnggtpraojiencttso,rEiens(,ξC1ˆ,nξ¯(2(πi)1))aannddECˆnn++11((ξπ1,1ξ)¯2(oi)n) = σI(n2)√2h+q(R~ ·ξ~)2+(I~·ξ~)2−(R~ ·ξ~)i1/2(cid:12)(cid:12)ξ2=ξ¯2(i()38)
(cid:12)
the ǫˆn,n+1(ξ1,ξ2) hypersurface. Since ξ2 is kept con- Theseexpressionsallowustorelatetheterms(R~ ξ~)and
stant at the fixed value ξ¯2(i), the trajectories (sections) (I~ ξ~)directly with observablesofthe isolateddou·bletof
Cˆn(πi) and Cˆn+1(πi), may be represented as three- res·onances. Taking the product of ∆E∆Γ, and recalling
dimensional curves in a space 3 with cartesian coordi- eq.(16), we get
E
nates (Reǫ,Imǫ,ξ ), see Figs. 1,2 and 3. The projec-
1
tions of the curves Cˆn(πi) and Cˆn+1(πi) on the planes ∆E∆Γ= I~ ξ~ (39)
(Reǫ,ξ1) and (Imǫ,ξ1) are (cid:16) · (cid:17)(cid:12)(cid:12)ξ2=ξ¯2(i)
(cid:12)
Re[Cˆ (π )]=Reˆ (ξ ,ξ¯(i)) m=n,n+1 (35) and taking the differences of the squaresof the left hand
m i Em 1 2 sides of (37) and (38), we get
and
2 1 2
Im[Cˆm(πi)]=ImEˆm(ξ1,ξ¯2(i)) m=n,n+1 (36) (cid:16)∆E(cid:17) − 4(cid:16)∆Γ(cid:17) =(cid:16)R~ ·ξ~(cid:17)(cid:12)(cid:12)ξ2=ξ¯2(i) (40)
(cid:12)
6
At a crossing of energies ∆E vanishes, and at a cross- V. SUMMARY AND CONCLUSIONS
ing of widths ∆Γ vanishes. Hence, the relation found in
eq.(39) means that a crossing of energies or widths can
occur if and only if (I~ ξ~) vanishes We developed the theory of the unfolding of the en-
· ξ¯(i)
For a vanishing (I~ ξ~ ) 2 =0=∆E∆Γ, we find three ergy eigenvalue surfaces close to a degeneracy point (ex-
· c ξ¯(i) ceptional point) of two unbound states of a Hamiltonian
2
cases, which are distinguished by the sign of (R~ ·ξ~c)ξ¯(i). dependingoncontrolparameters. Fromtheknowledgeof
From eqs. (37) and (38), 2 the Jost function, as function of the control parameters
of the system, we derived a 2-parameter family of func-
1. (R~ ξ~ ) > 0 implies ∆E = 0 and ∆Γ = 0, i.e. tionswhichiscontactequivalenttotheexactenergy-pole
· c ξ¯(i) 6
2 position function at the exceptional point and includes
energy anticrossing and width crossing.
all small perturbations of the degeneracy conditions. A
2. (R~ ξ~ ) = 0 implies ∆E = 0 and ∆Γ = 0, simple and explicit, but very accurate, representation of
· c ξ¯(i)
2 the eigenenergy surfaces close to the exceptional point
that is, joint energy and width crossings, which is
is obtained. In parameter space, the hypersurface rep-
also degeneracy of the two complex resonance en-
resenting the complex resonance energy eigenvalues has
ergy eigenvalues.
an algebraic branch point of rank one, and branch cuts
3. (R~ ξ~ ) < 0 implies ∆E = 0 and ∆Γ = 0, i.e. in its real and imaginary parts extending in opposite di-
· c ξ¯2(i) 6 rections in parameter space. The rich phenomenology of
energy crossing and width anticrossing.
crossings and anticrossings of the energies and widths of
This richphysicalscenarioof crossingsand anticrossings theresonancesofanisolateddoubletofunboundstatesof
for the energies and widths of the complex resonance aquantumsystem,observedwhenonecontrolparameter
energy eigenvalues, extends a theorem of von Neumann isvariedandtheotheriskeptconstant,isfullyexplained
andWigner [21] for bound states to the case ofunbound in terms of the local topology of the eigenenergy hyper-
states. surface in the vecinity of the crossing point.
The general character of the crossing-anticrossing re-
lations of the energies and widths of a mixing isolated Acknowledgments
doublet of resonances, discussed above, has been experi-
mentally established by P. von Brentano and his collab-
orators in a series of beautiful experiments [6, 7, 8]. This work was partially supported by CONACyT
A detailed account of these and other results will be M´exicoundercontractnumber40162-FandbyDGAPA-
published elsewhere [22, 23] UNAM contract No. PAPIIT:IN116202
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