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Eigenvalues of complex Hamiltonians with PT -symmetry. II
21 December 1998 PHYSICS
LETTERS
A
Physics Letters A 250 (1998) 29-32
Eigenvalues of complex Hamiltonians with P7-symmetry. Eric Delabaere
‘,
II
Frt5dCricPham 2
Luboratoirede Matht%zatiques, UMR CNRS6621, Universitide Nice-SophiaAntipolis.Pan: Valrose,06108 Nice cedex 2, France Received 8 June 1998; revised manuscript received 21 September 1998; accepted for publication 13 October 1998 Communicated by P.R. Holland
Abstract Making use of our earlier work [Ann. of Phys. 261 (1997) 1801 we investigate the reality of eigenvalues of the complex ?T-symmetric quartic oscillator. In contradistiction to the Zinn-Justin-Bessis case we have studied elsewhere [Phys. Lett. A 250 (1998)], a small imaginary part in the potential function breaks the symmetry of eigenstates, leading to complex eigenvalues. @ 1998 Elsevier Science B.V. PACS: 03.65.~; 03.65.Ge; 03.65.Sq; 11.30.Er Keyworak Complex WKB method; Exact quantization conditions; Resummability of Rayleigh-Scbriidinger series
1. Introduction Consider
the Hamiltonian d2
7-ia.B =--+q4+aq2+$q. dq2
It is TV-symmetric (cf. Ref. [ 1] ) for real (Y,p 3 . For p = 0 the spectrum consists of a discrete sequence of real eigenvalues Ea( (Y) < El ( (Y) < . . . depending (analytically) on LYas shown in Fig. 1. Since these eigenvalues are simple they will also depend analytically on fi for small enough ]p] (this follows from a “regular perturbation” argument, similar to the one used by Simon in Ref. [ 31). More precisely, for every natural integer n the nth eigenvalue En continues as an analytic, real-valued function ’ E-mail: [email protected]. * E-mail: [email protected]. 3 When comparing with Ref. 121 remember that what we now denote by ip was denoted there by B (/3 in tbe first page of tbe paper).
Fig. 1. Spectral locus of tbe symmetric quartic oscillator.
E,, (a, /3) in some neighbourhood U, of /3 = 0 in the real ((Y, p) -plane ( PT-invariance implies that reality is preserved by real analytic continuation). We shall call U,, the regularity domain of E,,. Using some of the “model quantization conditions” introduced in Ref. [2], we shall now show that for
0375-9601/98/S - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00792-O
30
E. Delabaere, E PhadPhysics Letters A 250 (1998) 29-32 E
2. The “almost harmonic” well Using the same reasonings as in the first part of this note, we show here that the regularity domains U,, contain regions of the form LY> 0, p2 < ( $x)~. For a > 0, p2 < ( ;a)3 the potential function V(cu,j3) = q” + Lyq* + i/3q has three real critical values 4 , the largest of which is an analytic function of (a, /3) which we shall denote by &,.k( cy,p). Resealing the energy by setting g = (E - &,Q( LY,p) ) /& and the Hamiltonian by setting %,B = (7&p - E,,.iit(LY,p> )/J;; and resealing ,i3by putting 6 = ae3i2/3, we have the following analog of Theorem 1 in Ref. [ 11.
Fig. 2. Spectral locus of the quurtic oscillator for fixed j3 = 5.
Theorem 1. In some domain of the form LY >
C (p, 8) (C a positive continuous function, defined for -fl < B < -l-m, .??E C) the spectral locus of (7&i) consists of a disjoint union of real analyticsurfacesE=&(cu,@) (n=0,1,2,...),where the &,(a, #Q’s are read-valved analytic functions of (cY,~), and lim,+_,&(&p) = 2n + 1. Proo$ Consider the following “resealed” Schriidinger equation,
-i-i2$+$+$+$q ,...;i
Fig. 3. Spectral locus of the quartic oscillator for fixed (r = -9.
a >> 0 the regularity domains U, are fairly wide, whereas for a! < 0 they are exponentially narrow. More precisely, in the thr~-dimensional extension of Fig, I (with the p-axis perpendicular to the plane of the picture), the (real) spectral locus is a discrete union of “hoses”, each of which gets ex~nenti~ly thin as cy --+ -co The curves of Fig. 1 are nothing other than “the apparent contours” of these hoses, and Figs. 2 and 3 show respectively /3 = const and (Y= const sections of these hoses. What we called the regularity dorn~n~~ is the projection, on the (cr, /Q-plane, of the hose corresponding to the nth energy level; one has &!k = Z.&k+, , and when (cy, p) leaves this domain the 2kth and (2k + 1) th energy levels coalesce and become complex conjugate.
>
W = (&,(l,$)
+h&?u:
(2) It reduces to the equation 31,,pS = ErY by substituting ti’/3q for q and setting a! = li-2/3, p = B/E, E = &-4/3(Ec~,( 1, p) + &I?). It is a critical Schrijdinger equation, whose Stokes pattern is drawn in Ref. [ 41, Fig. 42 (middle). The rest of the proof goes completely parallel to the proof of Theorem 1 in 0 Ref. [l]. 3. The “almost symmetrical*’ double well This section provides a description of the spectrum for CY<< 0, E < 0 (the thin parts of the “hoses”). More precisely, let us rescale the energy by setting .!?= (E + @‘)/m (notice that E = -fa2 is the value of the energy corresponding to the bottom of the 4This is most easily seen by changing q into iq, thus getting the well-known case of the real quartic function 4 - a$ - pq.
E. Deiabaere, E Pham/Physics Letters A 250 (1998) 29-32
symmetrical ( p = 0) double well). We then have the following theorem. ~~eo~e~ 2. In some domain of the form LY < -C( /3,&) (C a positive continuous function of p E W, 8 E (6) the real pan of the spectral locus of ($&) consists of a disjoint union of thin “hoses” S, (n = 0, 1,2, . . .) getting exponentially close to the line(p=O,B=2n+l)asa-+-oo. More precisely, there is a real analytic change of variables (a, j3,8) k-+(W;p,s) withs=i(&-l)+ p9;il;;i;
;;~a~n~o~t~;;;)3’29
such that & is
+#f(S-n)2=&2
(3)
where E N ( W”+‘/2/fin!)e-~2. Proof. Consider the following “resealed” SchrSdinger
equation
branch points p’ (those values of /3 for which a pair of levels coalesces). 6
4. The ‘Wansient” regime What we call the “transient” regime is the intermediate regime between the “almost harmonic” case (where the energy levels are well separated as explained in Section 2) and the “almost symmetrical double well” (where the energy levels cluster pairwise as explained in Section 3). This regime holds for (large) negative (r, and E not too far from the local maximum &it (cy, p) of the double well (for cy < 0, &tit (a, p) is a real analytic function of (a, /3) which equals 0 for p = 0). Resealing p and E by setting 6 = ( -a)-3/2/3 and ,??= (E - Ec,+(a,/3))/a we have the following theorem. Theorem 3. In some domain of the form CT <
-n* -$+q4--q2+iliaq (
31
ty = (-$+@)?P.
(4)
>
It reduces to the equation 7&,p9 = E?P by substituting ti’i3q for q and setting (Y= -V2i3, E = fid4i3( -f + f&J). It is a critical Schrijdinger equation of the type considered in Section 2.3 of Ref. [ 21. The above theorem follows from the “model equations” in Ref. [ 2] (Eq. (2.10) and following), re-read with our new notations: put n+ = n_ = n and change p into $3 (and therefore x into ix in Eq. (2.11) ) ; the lattice of curves of Fig. 8 is replaced by tiny ovals centered at (/3 = 0,s = 0,1,2 ,... ): in Fig. 3 these tiny ovals fill the E CK 0 part of the figure. 0 Up to the change of /.Y+-+i/Z!,and forgetting higher order WKB corrections, the reader will recognize in Eq. (3) the well-known local (quadratic) model for avoided crossings (cf. e.g. Eq. ( 17) of Ref. [5]), which one usually gets using 2 x 2 matrix models of the Hamiltonian J . But as explained in Ref. [ 21 (see also Ref. [ 7]), our model Eq. (3) is exact and could be used to compute precise numerical values of the
s Compare also Ref. 161 for G more refined approach.
-C(B, 8) (C a positive continuous function of p E R, & E C), there is a real analytic change of variables (a, j3, E) t-+ (X, j3, V) with X N (-cr)-3/2w,ti,(-l,& and U N T(&&, such that the spectral locus is given by cos
[ix+
ElnX-argT
(~+i~)]
Proofi Rewriting the resealed Schriidinger equation as in (2) (with our new meaning of &k(a,p)), we recognize the situation of Ref. [ 25, Section 3, (the unstable layer model), up to the change of notation (our
ip is the fi of Ref. [2]). Eq. (5) above is Eq. (3.5) of Ref. [2]. Cardan’s formulae give w,liit(-LB)
= ;JiT?$
- 2barcsinh( 2p/&?$) and r(B) = 1f ds,
with
6The referee informed us that such precise numerid values can also be obtained by the hyperviriai perturbationmethod of Ki&gbeck [ 51.
E. Delabaere, E Pham/Physics Letters A 250 (1998) 29-32
32
p(~)=;
-
K B+
JE
113
curacy of such a method is commented on in the last subsection of Ref. [ 21.
>
(-p+JF-T-ijG)li3].
References
q Figs. 2 and 3 show how the real sbectral locus looks like in this “transient regime” between the regular regime of Section 2 and the regime of Section 3. To make these pictures, we have used the same method as in Ref. [2 ] : we solved IQ. (5) numerically in the X, p, U variables (using Maple), and reexpressed the result in terms of the (Y,p, E variables using the lowest order WKB approximation. The ac-
[ I] E. Delabaere, F. Pham, Phys. Len. A 250 ( 1998) [2] E. Delabaere, F. Pham, A&. of Phys. 261 ( 1997) 180. [3] B. Simon, Ann. of Phys. 58 ( 1970) 76. [4] E. Delabaere, H. Dillinger, F. Pham, J. Math. Phys. 38 (1997) 6126. [5] J. Killingbeck, J. Phys. A 21 (1988) 3399. [6] S.Yu. Slavyanov, N.A. Veshev, J. Phys. A 30 (1997) 673. [7] E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. de l’lnst. Henri Poincan? - Physique Th&xique, to appear ( 1998).
LETTERS
A
Physics Letters A 250 (1998) 29-32
Eigenvalues of complex Hamiltonians with P7-symmetry. Eric Delabaere
‘,
II
Frt5dCricPham 2
Luboratoirede Matht%zatiques, UMR CNRS6621, Universitide Nice-SophiaAntipolis.Pan: Valrose,06108 Nice cedex 2, France Received 8 June 1998; revised manuscript received 21 September 1998; accepted for publication 13 October 1998 Communicated by P.R. Holland
Abstract Making use of our earlier work [Ann. of Phys. 261 (1997) 1801 we investigate the reality of eigenvalues of the complex ?T-symmetric quartic oscillator. In contradistiction to the Zinn-Justin-Bessis case we have studied elsewhere [Phys. Lett. A 250 (1998)], a small imaginary part in the potential function breaks the symmetry of eigenstates, leading to complex eigenvalues. @ 1998 Elsevier Science B.V. PACS: 03.65.~; 03.65.Ge; 03.65.Sq; 11.30.Er Keyworak Complex WKB method; Exact quantization conditions; Resummability of Rayleigh-Scbriidinger series
1. Introduction Consider
the Hamiltonian d2
7-ia.B =--+q4+aq2+$q. dq2
It is TV-symmetric (cf. Ref. [ 1] ) for real (Y,p 3 . For p = 0 the spectrum consists of a discrete sequence of real eigenvalues Ea( (Y) < El ( (Y) < . . . depending (analytically) on LYas shown in Fig. 1. Since these eigenvalues are simple they will also depend analytically on fi for small enough ]p] (this follows from a “regular perturbation” argument, similar to the one used by Simon in Ref. [ 31). More precisely, for every natural integer n the nth eigenvalue En continues as an analytic, real-valued function ’ E-mail: [email protected]. * E-mail: [email protected]. 3 When comparing with Ref. 121 remember that what we now denote by ip was denoted there by B (/3 in tbe first page of tbe paper).
Fig. 1. Spectral locus of tbe symmetric quartic oscillator.
E,, (a, /3) in some neighbourhood U, of /3 = 0 in the real ((Y, p) -plane ( PT-invariance implies that reality is preserved by real analytic continuation). We shall call U,, the regularity domain of E,,. Using some of the “model quantization conditions” introduced in Ref. [2], we shall now show that for
0375-9601/98/S - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00792-O
30
E. Delabaere, E PhadPhysics Letters A 250 (1998) 29-32 E
2. The “almost harmonic” well Using the same reasonings as in the first part of this note, we show here that the regularity domains U,, contain regions of the form LY> 0, p2 < ( $x)~. For a > 0, p2 < ( ;a)3 the potential function V(cu,j3) = q” + Lyq* + i/3q has three real critical values 4 , the largest of which is an analytic function of (a, /3) which we shall denote by &,.k( cy,p). Resealing the energy by setting g = (E - &,Q( LY,p) ) /& and the Hamiltonian by setting %,B = (7&p - E,,.iit(LY,p> )/J;; and resealing ,i3by putting 6 = ae3i2/3, we have the following analog of Theorem 1 in Ref. [ 11.
Fig. 2. Spectral locus of the quurtic oscillator for fixed j3 = 5.
Theorem 1. In some domain of the form LY >
C (p, 8) (C a positive continuous function, defined for -fl < B < -l-m, .??E C) the spectral locus of (7&i) consists of a disjoint union of real analyticsurfacesE=&(cu,@) (n=0,1,2,...),where the &,(a, #Q’s are read-valved analytic functions of (cY,~), and lim,+_,&(&p) = 2n + 1. Proo$ Consider the following “resealed” Schriidinger equation,
-i-i2$+$+$+$q ,...;i
Fig. 3. Spectral locus of the quartic oscillator for fixed (r = -9.
a >> 0 the regularity domains U, are fairly wide, whereas for a! < 0 they are exponentially narrow. More precisely, in the thr~-dimensional extension of Fig, I (with the p-axis perpendicular to the plane of the picture), the (real) spectral locus is a discrete union of “hoses”, each of which gets ex~nenti~ly thin as cy --+ -co The curves of Fig. 1 are nothing other than “the apparent contours” of these hoses, and Figs. 2 and 3 show respectively /3 = const and (Y= const sections of these hoses. What we called the regularity dorn~n~~ is the projection, on the (cr, /Q-plane, of the hose corresponding to the nth energy level; one has &!k = Z.&k+, , and when (cy, p) leaves this domain the 2kth and (2k + 1) th energy levels coalesce and become complex conjugate.
>
W = (&,(l,$)
+h&?u:
(2) It reduces to the equation 31,,pS = ErY by substituting ti’/3q for q and setting a! = li-2/3, p = B/E, E = &-4/3(Ec~,( 1, p) + &I?). It is a critical Schrijdinger equation, whose Stokes pattern is drawn in Ref. [ 41, Fig. 42 (middle). The rest of the proof goes completely parallel to the proof of Theorem 1 in 0 Ref. [l]. 3. The “almost symmetrical*’ double well This section provides a description of the spectrum for CY<< 0, E < 0 (the thin parts of the “hoses”). More precisely, let us rescale the energy by setting .!?= (E + @‘)/m (notice that E = -fa2 is the value of the energy corresponding to the bottom of the 4This is most easily seen by changing q into iq, thus getting the well-known case of the real quartic function 4 - a$ - pq.
E. Deiabaere, E Pham/Physics Letters A 250 (1998) 29-32
symmetrical ( p = 0) double well). We then have the following theorem. ~~eo~e~ 2. In some domain of the form LY < -C( /3,&) (C a positive continuous function of p E W, 8 E (6) the real pan of the spectral locus of ($&) consists of a disjoint union of thin “hoses” S, (n = 0, 1,2, . . .) getting exponentially close to the line(p=O,B=2n+l)asa-+-oo. More precisely, there is a real analytic change of variables (a, j3,8) k-+(W;p,s) withs=i(&-l)+ p9;il;;i;
;;~a~n~o~t~;;;)3’29
such that & is
+#f(S-n)2=&2
(3)
where E N ( W”+‘/2/fin!)e-~2. Proof. Consider the following “resealed” SchrSdinger
equation
branch points p’ (those values of /3 for which a pair of levels coalesces). 6
4. The ‘Wansient” regime What we call the “transient” regime is the intermediate regime between the “almost harmonic” case (where the energy levels are well separated as explained in Section 2) and the “almost symmetrical double well” (where the energy levels cluster pairwise as explained in Section 3). This regime holds for (large) negative (r, and E not too far from the local maximum &it (cy, p) of the double well (for cy < 0, &tit (a, p) is a real analytic function of (a, /3) which equals 0 for p = 0). Resealing p and E by setting 6 = ( -a)-3/2/3 and ,??= (E - Ec,+(a,/3))/a we have the following theorem. Theorem 3. In some domain of the form CT <
-n* -$+q4--q2+iliaq (
31
ty = (-$+@)?P.
(4)
>
It reduces to the equation 7&,p9 = E?P by substituting ti’i3q for q and setting (Y= -V2i3, E = fid4i3( -f + f&J). It is a critical Schrijdinger equation of the type considered in Section 2.3 of Ref. [ 21. The above theorem follows from the “model equations” in Ref. [ 2] (Eq. (2.10) and following), re-read with our new notations: put n+ = n_ = n and change p into $3 (and therefore x into ix in Eq. (2.11) ) ; the lattice of curves of Fig. 8 is replaced by tiny ovals centered at (/3 = 0,s = 0,1,2 ,... ): in Fig. 3 these tiny ovals fill the E CK 0 part of the figure. 0 Up to the change of /.Y+-+i/Z!,and forgetting higher order WKB corrections, the reader will recognize in Eq. (3) the well-known local (quadratic) model for avoided crossings (cf. e.g. Eq. ( 17) of Ref. [5]), which one usually gets using 2 x 2 matrix models of the Hamiltonian J . But as explained in Ref. [ 21 (see also Ref. [ 7]), our model Eq. (3) is exact and could be used to compute precise numerical values of the
s Compare also Ref. 161 for G more refined approach.
-C(B, 8) (C a positive continuous function of p E R, & E C), there is a real analytic change of variables (a, j3, E) t-+ (X, j3, V) with X N (-cr)-3/2w,ti,(-l,& and U N T(&&, such that the spectral locus is given by cos
[ix+
ElnX-argT
(~+i~)]
Proofi Rewriting the resealed Schriidinger equation as in (2) (with our new meaning of &k(a,p)), we recognize the situation of Ref. [ 25, Section 3, (the unstable layer model), up to the change of notation (our
ip is the fi of Ref. [2]). Eq. (5) above is Eq. (3.5) of Ref. [2]. Cardan’s formulae give w,liit(-LB)
= ;JiT?$
- 2barcsinh( 2p/&?$) and r(B) = 1f ds,
with
6The referee informed us that such precise numerid values can also be obtained by the hyperviriai perturbationmethod of Ki&gbeck [ 51.
E. Delabaere, E Pham/Physics Letters A 250 (1998) 29-32
32
p(~)=;
-
K B+
JE
113
curacy of such a method is commented on in the last subsection of Ref. [ 21.
>
(-p+JF-T-ijG)li3].
References
q Figs. 2 and 3 show how the real sbectral locus looks like in this “transient regime” between the regular regime of Section 2 and the regime of Section 3. To make these pictures, we have used the same method as in Ref. [2 ] : we solved IQ. (5) numerically in the X, p, U variables (using Maple), and reexpressed the result in terms of the (Y,p, E variables using the lowest order WKB approximation. The ac-
[ I] E. Delabaere, F. Pham, Phys. Len. A 250 ( 1998) [2] E. Delabaere, F. Pham, A&. of Phys. 261 ( 1997) 180. [3] B. Simon, Ann. of Phys. 58 ( 1970) 76. [4] E. Delabaere, H. Dillinger, F. Pham, J. Math. Phys. 38 (1997) 6126. [5] J. Killingbeck, J. Phys. A 21 (1988) 3399. [6] S.Yu. Slavyanov, N.A. Veshev, J. Phys. A 30 (1997) 673. [7] E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. de l’lnst. Henri Poincan? - Physique Th&xique, to appear ( 1998).