September  2014, 19(7): 1955-1967. doi: 10.3934/dcdsb.2014.19.1955

An existence criterion for the $\mathcal{PT}$-symmetric phase transition

1. 

Dipartimento di Matematica, Università di Bologna, 40127 Bologna, Italy, Italy

Received  April 2013 Revised  July 2013 Published  August 2014

We consider on $L^2(\mathbb{R})$ the Schrödinger operator family $H(g)$ with domain and action defined as follows $$ D(H(g))=H^2(\mathbb{R})\cap L^2_{2M}(\mathbb{R}); \quad H(g) u=\bigg(-\frac{d^2}{dx^2}+\frac{x^{2M}}{2M}-g\,\frac{x^{M-1}}{M-1}\bigg)u $$ where $g\in\mathbb{C}$, $M=2,4,\ldots\;$. $H(g)$ is self-adjoint if $g\in\mathbb{R}$, while $H(ig)$ is $\mathcal{PT}$-symmetric. We prove that $H(ig)$ exhibits the so-called $\mathcal{PT}$-symmetric phase transition. Namely, for each eigenvalue $E_n(ig)$ of $H(ig)$, $g\in\mathbb{R}$, there exist $R_1(n)>R(n)>0$ such that $E_n(ig)\in\mathbb{R}$ for $|g| < R(n)$ and turns into a pair of complex conjugate eigenvalues for $|g| > R_1(n)$.
Citation: Emanuela Caliceti, Sandro Graffi. An existence criterion for the $\mathcal{PT}$-symmetric phase transition. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1955-1967. doi: 10.3934/dcdsb.2014.19.1955
References:
[1]

N. I. Akhiezer, The Classical Moment Problem,, Oliver & Boyd, (1965).

[2]

C. M. Bender, S. Boettcher and P. N. Meisinger, PT-Symmetric quantum mechanics,, J. Math. Phys., 40 (1999), 2201. doi: 10.1063/1.532860.

[3]

C. M. Bender, M. V. Berry and A. Mandilara, Generalized PT symmetry and real spectra,, J.Phys. A: Math. Gen., 35 (2002). doi: 10.1088/0305-4470/35/31/101.

[4]

C. M. Bender, D. C. Brody and H. F. Jones, Must a hamiltonian be hermitian?,, American Journal of Physics, 71 (2003), 1095. doi: 10.1119/1.1574043.

[5]

C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having PT symmetry,, Phys. Rev. Lett., 80 (1998), 5243. doi: 10.1103/PhysRevLett.80.5243.

[6]

P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe ansatz equations and reality properties of $\mathcal{PT}$-symmetric quantum mechanics,, J. Phys. A, 34 (2001), 5679. doi: 10.1088/0305-4470/34/28/305.

[7]

P. Dorey, C. Dunning and R. Tateo, Supersymmetry and the spontaneous breakdown of $\mathcal{PT}$ symmetry,, J. Phys. A, 34 (2001).

[8]

P. Dorey, C. Dunning, A. Lishman and R. Tateo, $\mathcal{PT}$-symmetry breaking for a class of inhomogeneous complex potentials,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/46/465302.

[9]

S. Graffi, V. Grecchi and B. Simon, Borel summability: Application to the anharmonic ocillator,, Phys. Lett., 32B (1970), 631.

[10]

I. W. Herbst, Dilation analyticity in constant electric field,, Comm. Math. Phys., 64 (1979), 279. doi: 10.1007/BF01221735.

[11]

W. Hunziker and C. A.Pillet, Degenerate asymptotic perurbation theory,, Comm. Math. Phys., 90 (1983), 219.

[12]

T. Kato, Perturbation Theory for Linear Operators,, Second edition. Grundlehren der Mathematischen Wissenschaften, (1976).

[13]

J. J. Loeffel and A. Martin, Proprietés analytiques des niveaux de l'oscillateur anharmonique et convergence des approximants de Padé,, Cargèse Lectures in Physics, 5 (1972), 415.

[14]

J. J. Loeffel, A. Martin, B. Simon and A. S.Wightman, Padé Approximants and the Anharmonic Oscillator,, Phys. Lett., 30B (1969), 656.

[15]

M. Reed and B. Simon, Methods of Modern Mathematical Physics,, IV, (1978).

[16]

K. C. Shin, On the reality of the eigenvalues for a class of $PT$-symmetric oscillators,, Comm. Math. Phys., 229 (2002), 543. doi: 10.1007/s00220-002-0706-3.

[17]

B. Simon, Coupling constant analyticity for the anharmonic oscillator,, Ann. Phys. (N.Y.), 58 (1970), 76. doi: 10.1016/0003-4916(70)90240-X.

[18]

Y. Sibuya, Global Theory of Second Order Linear Differential Equations with Polynomial Coefficients,, North Holland, (1975).

[19]

J. Sjöstrand, Private, Communication., ().

[20]

, Special Issue: $\mathcal{PT}$-Symmetric Quantum Mechanics,, J. Phys. A, 39 (2006).

[21]

, Special Issue: Papers Dedicated to the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics,, J. Phys. A: Math. Theor., 21 (2008).

[22]

, Special Issue: Non-Hermitian Hamiltonians in Quantum Physics - Part I,, PRAMANA Journal of Physics, 73 (2009).

show all references

References:
[1]

N. I. Akhiezer, The Classical Moment Problem,, Oliver & Boyd, (1965).

[2]

C. M. Bender, S. Boettcher and P. N. Meisinger, PT-Symmetric quantum mechanics,, J. Math. Phys., 40 (1999), 2201. doi: 10.1063/1.532860.

[3]

C. M. Bender, M. V. Berry and A. Mandilara, Generalized PT symmetry and real spectra,, J.Phys. A: Math. Gen., 35 (2002). doi: 10.1088/0305-4470/35/31/101.

[4]

C. M. Bender, D. C. Brody and H. F. Jones, Must a hamiltonian be hermitian?,, American Journal of Physics, 71 (2003), 1095. doi: 10.1119/1.1574043.

[5]

C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having PT symmetry,, Phys. Rev. Lett., 80 (1998), 5243. doi: 10.1103/PhysRevLett.80.5243.

[6]

P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe ansatz equations and reality properties of $\mathcal{PT}$-symmetric quantum mechanics,, J. Phys. A, 34 (2001), 5679. doi: 10.1088/0305-4470/34/28/305.

[7]

P. Dorey, C. Dunning and R. Tateo, Supersymmetry and the spontaneous breakdown of $\mathcal{PT}$ symmetry,, J. Phys. A, 34 (2001).

[8]

P. Dorey, C. Dunning, A. Lishman and R. Tateo, $\mathcal{PT}$-symmetry breaking for a class of inhomogeneous complex potentials,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/46/465302.

[9]

S. Graffi, V. Grecchi and B. Simon, Borel summability: Application to the anharmonic ocillator,, Phys. Lett., 32B (1970), 631.

[10]

I. W. Herbst, Dilation analyticity in constant electric field,, Comm. Math. Phys., 64 (1979), 279. doi: 10.1007/BF01221735.

[11]

W. Hunziker and C. A.Pillet, Degenerate asymptotic perurbation theory,, Comm. Math. Phys., 90 (1983), 219.

[12]

T. Kato, Perturbation Theory for Linear Operators,, Second edition. Grundlehren der Mathematischen Wissenschaften, (1976).

[13]

J. J. Loeffel and A. Martin, Proprietés analytiques des niveaux de l'oscillateur anharmonique et convergence des approximants de Padé,, Cargèse Lectures in Physics, 5 (1972), 415.

[14]

J. J. Loeffel, A. Martin, B. Simon and A. S.Wightman, Padé Approximants and the Anharmonic Oscillator,, Phys. Lett., 30B (1969), 656.

[15]

M. Reed and B. Simon, Methods of Modern Mathematical Physics,, IV, (1978).

[16]

K. C. Shin, On the reality of the eigenvalues for a class of $PT$-symmetric oscillators,, Comm. Math. Phys., 229 (2002), 543. doi: 10.1007/s00220-002-0706-3.

[17]

B. Simon, Coupling constant analyticity for the anharmonic oscillator,, Ann. Phys. (N.Y.), 58 (1970), 76. doi: 10.1016/0003-4916(70)90240-X.

[18]

Y. Sibuya, Global Theory of Second Order Linear Differential Equations with Polynomial Coefficients,, North Holland, (1975).

[19]

J. Sjöstrand, Private, Communication., ().

[20]

, Special Issue: $\mathcal{PT}$-Symmetric Quantum Mechanics,, J. Phys. A, 39 (2006).

[21]

, Special Issue: Papers Dedicated to the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics,, J. Phys. A: Math. Theor., 21 (2008).

[22]

, Special Issue: Non-Hermitian Hamiltonians in Quantum Physics - Part I,, PRAMANA Journal of Physics, 73 (2009).

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