Unavoided crossing of energy levels in PT -symmetric Natanzon-class - - PowerPoint PPT Presentation

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Unavoided crossing of energy levels in PT -symmetric Natanzon-class potentials G´ eza L´ evai

Institute for Nuclear Research, Hungarian Academy

• f Sciences (MTA Atomki), Debrecen, Hungary

AAMP19, Prague, 6-9 June 2016

Important novelties in PT QM

There can be two sets of normalizable wave functions Discriminated by the q quasi-parity quantum number These states merge when the spectrum is complexified Then they are related by PT ψ(q)

n (E) = ψ(−q) n

(E∗) They can undergo unavoided crossing Then the two wave functions become dependent This is very different from the Hermitian setting

The first examples

The PT harmonic oscillator MZ PLA 1999 V (x) = x2 − 2icx + G (x − ic)2 x ∈ (−∞, ∞) Unavoided crossing of energy levels E(q)

n n = 4n + 2 − 2qα

ψ(q)

n (x)

expressed via L(qα)

n

((x − ic)2) The PT Coulomb potential MZ, GL PLA 2000 Similar results The background: L(−j)

n

(z) ∼ zjL(j)

n−j(z)

for integer j But is α = ±j legal?

A further, more recent example

The PT Scarf II potential Ahmed et al. PLA 2015 Unavoided crossing of energy levels The background: P (−j,β)

n

(z) ∼ (1 − z)jP (j,β)

n−j (z)

for integer j Very similar results But is α = ±j legal? Are there further exactly solvable examples?

The Book∗ said...

* M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions (New York, Dover, 1970), Eq. 22.3.1

...but temptation was too strong...

...which led to expulsion from Paradise

A reminder on the exact solutions of the Schr¨

• dinger equation

An old variable transformation method Bhattacharjie and Sudarshan 1962 Schr¨

• dinger eq. =

⇒ differential equation of special function F d2ψ dx2 + (E − V (x))ψ(x) = 0 insert ψ(x) = f(x)F(z(x)) and compare with d2F dz2 + Q(z)dF dz + R(z)F(z) = 0 to get E − V (x) = z′′′(x) 2z′(x) − 3 4

z′′(x)

z′(x)

2

+ (z′(x))2

• R(z(x)) − 1

2 dQ(z) dz − 1 4Q2(z(x))

• .

Schwartzian derivatve terms E and the main potential terms

New features of PT -symmetric potentials

Unusual trajectories off the real x axis Reconsidered boundary conditions A relatively simple case: Imaginary coordinate shift (i.e integration constant in x(z)) δ = −ic Avoid singularities Note: Introducing ic does not change the spectrum ...so it cannot introduce spontaneous PT breaking In some cases the problem has to be defined on a more general trajectory The boundary conditions cannot be satisfied on y = x − ic This is the case for the Coulomb and Morse potentials Duplication of bound states: quasi-parity q = ±1 Complex energy eigenvalues: spontaneous breakdown of PT symmetry PT ψ(q)

n (x) = ψ(−q) n

(x)

Apply the method to the Jacobi polynomials: F(z) = P (α,β)

n

(z)

E − V (x) = z′′′(x) 2z′(x) − 3 4

z′′(x)

z′(x)

2

+ (z′(x))2 1 − z2(x)

• n + α + β

2 n + α + β 2 + 1

• +

(z′(x))2 (1 − z2(x))2

 1 − α + β

2

2

−

α − β

2

2 

−2z(x)(z′(x))2 (1 − z2(x))2

α + β

2

α − β

2

• .

The solutions are ψ(x) ∼ (z′(x))− 1

2(1 + z(x)) β+1 2 (1 − z(x)) α+1 2 P (α,β)

n

(z(x)) . The yet unknown z(x) can be obtained from a differential equation

• dz

dx

2 φ(z) ≡

• dz

dx

2 pI(1−z2)+pII+pIIIz

(1−z2)2

= C . by direct integration

• φ1/2(z)dz = C1/2x + ǫ .

ǫ: integration constant, coordinate shift

The Scarf II potential pII = pIII = 0

V (x) = − 1 cosh2 x

  α + β

2

2

+

α − β

2

2

− 1 4

  + 2i sinh x

cosh2 x

β + α

2

β − α

2

• Relations for the parameters:

PT symmetry: = ⇒ α, β are real or imaginary α ↔ β: = ⇒ V (x) ↔ V (−x) V (x) invariant under α ↔ −α = ⇒ qα ≡ ±α quasi-parity ψ(q)

n (x) = C(q) n (1 − i sinh(x + iǫ))

qα 2 + 1 4(1 + i sinh(x + iǫ)) β 2 + 1 4P (qα,β)

n

(i sinh(x + iǫ)) Normalizable if n(q) < −[Re(qα + β) + 1]/2 The second set corresponds to resonances in the Hermitian setting (α∗ = β) E(q)

n

= −

• n + qα + β + 1

2

2

Complex conjugate pairs if α is imaginary Spontaneous breakdown of PT symmetry “Sudden” mechanism: all the E(q)

n

turn complex at the same time

The transition to complex energy eigenvalues

Reparametrize to V (x) = −vrVR(x) + iviVI(x), vr fixed, vi varied vr = 15.1, vi = 12.7 vi = 0 → 20 Unavoided crossings! Some are at the same vi. Complexification occurs for |vi| = vr + 1/4 = 15.35 for all n Sudden mechanism of PT symmetry breaking Note: ψ(−) (x) normalizable for vi ≥ 3.886, ψ(−)

1

(x) for vi ≥ 10.858, ψ(−)

2

(x) for vi ≥ 15.803. ψ(+)

3

(x) exists for vi ≤ 12.325.

α = 1.9

Reψ Imψ ψ(−α,β) (z) ψ(α,β)

1

(z)

α = 1.5

Reψ Imψ ψ(−α,β) (z) ψ(α,β)

1

(z)

α = 1.2

Reψ Imψ ψ(−α,β) (z) ψ(α,β)

1

(z)

α = 1.0

Reψ Imψ ψ(−α,β) (z) ψ(α,β)

1

(z)

A Natanzon-class example: the generalized Ginocchio potential with 2+2 parameters

• G. L´

evai et al. J. Phys. A 36 (2003) 7611

Take pI = γ2 + 1, pII = 0, pIII = 4 γ = 1: generalized P¨

• schl–Teller limit

Implicit z(x) function: r ≡ x − iε = 1 γ2

• tanh−1

(γ2 + sinh2 u)− 1

2 sinh u

• + (γ2 − 1)

1 2 tan−1

(γ2 − 1)

1 2(γ2 + sinh2 u)− 1 2 sinh u

• .

ε = 0 to avoid singularity at x = 0 V (r) = −γ4(s(s + 1) + 1 − γ2) γ2 + sinh2 u + γ4λ(λ − 1) coth2 u γ2 + sinh2 u −3γ4(γ2 − 1)(3γ2 − 1) 4(γ2 + sinh2 u)2 + 5γ6(γ2 − 1)2 4(γ2 + sinh2 u)3 , The solutions and eigenvalues energy Enq = −γ4µ2

nq depend on the q = ±1 quasi-parity

ψnq(x) ∼ (γ2 + sinh2 u)1/4(cosh u)−2n−1−µnq−q(λ− 1

2)(sinh u) 1 2+q(λ− 1 2 )

×P

(q(λ− 1

2),−2n−1−µnq−q(λ− 1 2))

n

(cosh(2u)) . µnq = 1 γ2

 −

• 2n + 1 + q(λ − 1

2)

• +
• γ2(s + 1

2)2 + (1 − γ2)

• 2n + 1 + q(λ − 1

2)

21/2 

Unbroken PT symmetry: real µqn, s, λ

The real (left panel) and imaginary (right panel) component of the PT -symmetric generalized Ginocchio potential for ε = 0.3, γ = 1.75, s = 8.1 and λ = 1.25 (solid line) and its supersymmetric partners V (+1)

+

(x) (dashed line) and V (−1)

+

(x) (dotted line). Normalisable states are found at E0 +1 = −171.313, E1 +1 = −106.160, E2 +1 = −46.679, E3 +1 = −5.666; E0 −1 = −218.913,E1 −1 = −154.978, E2 −1 = −90.379, E3 −1 = −33.993 and E4 −1 = −1.061. The spectrum of V (q)

+ (x) is the same, with the exception of the E0 q level, which is missing from its

spectrum. The equivalence of ψ(−q)

n

(x) and ψ(q)

n−j(x) can be proven for α = j integer

Another 2+2-parameter subset of the Natanzon class

• G. L´

evai, J. Phys. A 45 (2012) 444020 pI = 1, pII = δ, pIII = 0 It contains all SI potentials as special case: δ = 0 Scarf II δ → ∞ and C/δ = const. Rosen–Morse I VR(x) and VI(x) for C = −1, δ = 0.3, Σ = 11.0624 and Λ = 1.26.

Implicit potential:

• nly x(z) is known in closed form. . .

. . . nevertheless, everything can be evaluated exactly E − V (x) = C

• n + α + β + 1

2

2

− 3Cδ 4 (3δ + 2) (δ + 1 − z2(x))2 + 5Cδ2 4 (δ + 1) (δ + 1 − z2(x))3 − CΣ δ + 1 − z2(x) − 2CΛz(x) δ + 1 − z2(x) Σ = δ

• n + α + β + 1

2

2

− δ +

α + β

2

2

+

α − β

2

2

− 1 4 Λ = α + β 2 α − β 2 α = αn , β = βn Ψn(x) = Nn(δ + 1 − z2(x))1/4(1 − z(x))α/2(1 + z(x))β/2P (α,β)

n

(z(x)) A four-parameter (2+2) potential: C and δ control the variable transformation Σ and Λ set the coupling coefficients Λ = 0: symmetric Ginocchio case Energy eigenvalues determined from the roots of the spectral equation for ω = (α + β)/2: (δ + 1)ω4 + δ(2n + 1)ω3 +

δ

4(2n + 1)2 − δ − Σ − 1 4

• ω2 + Λ2 = 0

How does the spectrum evolve if Σ is fixed and Λ is varied?

Complexification occurs at Λcrit0 = 6.298, Λcrit1 = 7.542, Λcrit2 = 8.234 Unavoided crossings! Now at different locations when α = ±M integer. The q quasi-parity does not appear explicitly. The equivalence of ψ(−q)

n

(x) and ψ(q)

n−j(x) can be proven for α = j integer

The effect of (n, α = −j) ⇔ (n − j, α = j)

Equations determining En for Natanzon-class potentials

• n + α

2 + β + 1 2

2

− 1 4 + sI − pI E C = 0 1 − α2 2 − β2 2 + sII − pII E C = 0 −α2 2 + β2 2 + sIII − pIII E C = 0 Now take α = j The equations are invariant with respect to (n, α = −j) ⇔ (n − j, α = j)

Summary

Unavoided level crossing occurs for the PT version of the Scarf II potential Generalized Ginocchio potential 2+2 parameter subset of Natanzon potentials But it can be proven to be a general property for generic Natanzon-class potentials too

Near the Budapest station I saw a number...

...that originated from a birthday...

...of someone who once overlooked Prague...

...from a platform that still exists

.

”Perhaps we become aware of our age only at exceptional moments and most of the time we are ageless.” –Milan Kundera Happy birthday, Milos!

Thematical index of my papers on PT symmetric potentials

Solvable potentials, bound states

• Shape-invariant potentials

in general [1, 2, 27] Coulomb [3, 4, 13, 25] Scarf II [6, 7, 8, 9, 22, 24, 25] Scarf I [10, 24] Rosen-Morse I [11, 23] Rosen-Morse II [12, 25] Generalized Pschl-Teller [7]

• Natanzon-class potentials [14, 15, 26, 27]
• Conditionally exactly solvable potentials (beyond the Natanzon class) [16]
• Exactly non-solvable potentials [17]

Exact expressions for the pseudo-norm and normalization constants [8,10,11,12] Solvable potentials in higher dimensions [18, 19, 20] Spontaneous breakdown of PT symmetry

• Sudden mechanism [2, 8, 9, 13, 14, 19, 22, 24, 27]
• Gradual mechanism [15, 17, 26, 27]
• Does not happen [10, 11, 12, 27]

Scattering states [5, 6, 7, 12, 25] SUSYQM [21, 22, 23] Algebraic aspects [6, 7, 21] Results for hermitian equivalents [8, 10, 11, 24, 26]

1 G. L´ evai, M. Znojil: Systematic search for PT -symmetric potentials with real energy spectra, J. Phys. A 33 (2000) 7165 2 G. L´ evai, M. Znojil: Conditions for complex spectra in a class of PT -symmetric potentials, Mod. Phys. Lett. A 16 (2001) 1973 3 M. Znojil, G. L´ evai: The Coulomb - harmonic oscillator correspondence in PT - symmetric quantum mechanics, Phys. Lett. A 271 (2000) 327 4 Znojil M., Siegl P., L´ evai G.: Asymptotically vanishing PT -symmetric potentials and negative-mass Schrdinger equations, Phys. Lett. A 373 (2009) 1921 5 L´ evai G., Siegl P., Znojil M.: Scattering in the PT -symmetric Coulomb potential, J.

• Phys. A 42 (2009) 29:5201

6 G. L´ evai, F. Cannata, A. Ventura: Algebraic and scattering aspects of a PT -symmetric solvable potential, J. Phys. A 34 (2001) 839 7 G. L´ evai, F. Cannata, A. Ventura: PT -symmetric potentials and the so(2,2) algebra,

• J. Phys. A 35 (2002) 5041

8 G. L´ evai, F. Cannata, A. Ventura: PT symmetry breaking and explicit expressions for the pseudo-norm in the Scarf II potential, Phys. Lett. A 300 (2002) 271 9 G. L´ evai: Exact analytic study of the PT -symmetry-breaking mechanism, Czech. J.

• Phys. 54 (2004) 77

10 L´ evai G.: On the pseudo-norm and admissible solutions of the PT -symmetric Scarf I potential, J. Phys. A 39 (2006) 10161

11 L´ evai G.: On the normalization constant of PT -symmetric and real Rosen-Morse I potentials, Phys. Lett. A 372 (2008) 6484 12 L´ evai G., Magyari E.: The PT -symmetric Rosen-Morse II potential: effects of the asymptotically non-vanishing imaginary potential component, J. Phys. A 42 (2009) 19:5302 13 L´ evai G.: Spontaneous breakdown of PT symmetry in the complex Coulomb potential, Pramana J. Phys. 73 (2009) 2:329 14 G. L´ evai, A. Sinha, P. Roy: An exactly solvabte PT -symmetric potential from the Natanzon class, J. Phys. A 36 (2003) 7611 15 M. Znojil, G. L´ evai, P. Roy, R. Roychoudhury: Anomalous doublets of states in a PT -symmetric quantum model, Phys. Lett. A 290 (2001) 249 16 A. Sinha, G. L´ evai, P. Roy: PT symmetry of a conditionally exactly solvable poten- ital, Phys. Lett. A 322 (2004) 78 17 M. Znojil, G. L´ evai: Spontaneous breakdown of PT symmetry in the solvable square- well model, Mod. Phys. Lett. A 16 (2001) 2273 18 L´ evai G.: Solvable PT -symmetric potentials in higher dimensions, J. Phys. A 40 (2007) F273 19 L´ evai G.: PT symmetry and its spontaneous breakdown in three dimensions, J. Phys. A 41 (2008) 24:4015 20 L´ evai G.: Solvable PT -symmetric potentials in 2 and 3 dimensions, J. Phys. Conf.

• Ser. 128 (2008) 1:2045(12)

21 G. L´ evai: SUSYQM and other symmetries in quantum mechanics, J. Phys. A 37 (2004) 10179 22 G. L´ evai, M. Znojil: The interplay of supersymmetry and PT symmetry in quantum mechanics: a case study for the Scarf II potential, J. Phys. A 35 (2002) 8793 23 G. L´ evai: Symmetries without hermiticity, Czech. J. Phys. 54 (2004) 1121 24 G. L´ evai: Comparative analysis of real and PT -symmetric Scarf potentials, Czech.

• J. Phys. 56 (2006) 953

25 G. L´ evai: Asymptotic properties of solvable PT -symmetric potentials Int. J. Theor.

• Phys. 50 (2011) 4:997

26 G. L´ evai: Gradual spontaneous breakdown of PT symmetry in a solvable potential,

• J. Phys. A 45 (2012) 44:4020

27 G. L´ evai: PT symmetry in Natanzon-class potentials, Int.

• J. Theor.

Phys. 54 (2015) 54:2724