Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Annals of Physics

Non-symmetric double well and euclidean functional integral

Filippo Cesi, Giancarlo C. Rossi, M. Testa

Published Feb 1, 1991• 19 citations
Publisher

Abstract

In this paper we show how it is possible to discuss in the language of functional integrals the problem of the symmetric double well with a small perturbation, in the semiclassical limit. This problem has been previously treated by means of a completely different approach, based on the theory of small random perturbations of dynamical systems. We recover all known results concerning the wave function and the energy splitting of the two lowest lying states, and we give an explicit expression for the prefactor of the exponential asymptotic term in the energy splitting.

DOI
10.1016/0003-4916(91)90003-q

Related papers

‐symmetry and Schrödinger operators. The double well case

Nawal Mecherout, Naima Boussekkine, Thierry Ramond +1 more

We study a class of ‐symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double‐well potential. In the simple well case, two of the authors have proved in that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . We show here, in the double‐well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB‐analysis, leading to a fairly explicit quantization condition.

Mathematische Nachrichten•Nov 24, 2015•4 citations

A note on vibrational perturbation theory for tunneling splitting in a symmetric double well potential

Eli Pollak

The modified version of second and fourth order vibrational perturbation theory, whereby the Euclidean action for tunneling is computed on the inverted potential at a shifted energy that is ℏ2 dependent, is applied to a symmetric double well quartic potential. The mean energies of the doublets in each well are also computed using vibrational perturbation theory. Results show that the modified vibrational perturbation theory significantly improves the estimates of tunneling splitting energies both for the ground state and for excited state doublets.

The Journal of Chemical Physics•Jul 15, 2024•4 citations

PT-symmetry and Schrödinger operators. The double well case

Nawal Mecherout, Naima Boussekkine, Thierry Ramond +1 more

We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.

arXiv:1502.06102•Feb 21, 2015•4 citations

Double well ground state energy splitting (or instanton flipping rate); rendering the implicit explicit

J H Hannay

A prime example of quantum tunnelling is the semiclassical 'energy splitting' of the levels of a symmetrical double well potential, or equivalently the flipping rate of an instanton. Curiously the accepted expression for the ground state energy splitting in terms of the (smooth) potential function has not been pursued to the full explicitness available from classical mechanics. This implicitness is rectified here.

arXiv:2403.18050•Mar 26, 2024•0 citations

The Double Well Done Doubly-Well

Aurélien Dersy, Matthew D. Schwartz

The symmetric double-well potential is one of the simplest quantum-mechanical systems in which perturbative and non-perturbative physics are deeply entangled. Its energy levels have non-analytic expansions in inverse powers of the inter-well separation, with factorially growing coefficients, while the parity splitting is exponentially small and invisible to perturbation theory. Resurgence ties the two features together, organizing the exact spectrum into a single tightly-constrained trans-series. This paper gives a self-contained account of this trans-series from two complementary approaches: exact WKB and the Euclidean path integral, developed in a common notation with explicit calculations through the four-instanton level and three-loop order. In exact WKB, Stokes phenomena encoded in the Delabaere--Dillinger--Pham relations control the analytic continuation of the wavefunction past turning points. The quantization condition expressed in terms of Voros symbols then determines the full trans-series. The DDP relations are local and do not require knowing the global topology of the energy surface, but that surface is an elliptic curve. In the path integral, elliptic curves enter differently: the classical saddle points are doubly-periodic elliptic functions of Euclidean time, and Stokes phenomena play out within the finite-dimensional manifold of quasi-zero modes rather than through analytic continuation of the wavefunction. A Lefschetz thimble decomposition determines which saddles contribute, and the resulting partition function trans-series is much simpler than the energy trans-series: at each instanton order the $T$-dependence is a polynomial fixed by the quasi-zero-mode thimble integrals. Together, the two approaches deploy a shared mathematical infrastructure in complementary ways, showing that the double well is an ideal setting to explore resurgence.

hep-thCross-listedarXiv:2606.05282•Jun 5, 2026

Tunnel splittings for one-dimensional potential wells revisited

Anupam Garg

The Wentzel–Kramers–Brillouin and instanton answers for the tunnel splitting of the ground state in a symmetric double-well potential are both reduced to an expression involving only functionals of the potential, without the need for solving any auxilliary problems. This formula is applied to simple model problems. The prefactor for the splitting in the textbook by Landau and Lifshitz is amended so as to apply to the ground and low lying excited states.

arXiv:cond-mat/0003115•American Journal of Physics•May 1, 2000•120 citations
In this paper we show how it is possible to discuss in the language of functional integrals the problem of the symmetric double well with a small perturbation, in the semiclassical limit. This problem has been previously treated by means of a completely different approach, based on the theory of small random perturbations of dynamical systems. We recover all known results concerning the wave function and the energy splitting of the two lowest lying states, and we give an explicit expression for the prefactor of the exponential asymptotic term in the energy splitting.
We study a class of ‐symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double‐well potential. In the simple well case, two of the authors have proved in that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . We show here, in the double‐well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB‐analysis, leading to a fairly explicit quantization condition.
The modified version of second and fourth order vibrational perturbation theory, whereby the Euclidean action for tunneling is computed on the inverted potential at a shifted energy that is ℏ2 dependent, is applied to a symmetric double well quartic potential. The mean energies of the doublets in each well are also computed using vibrational perturbation theory. Results show that the modified vibrational perturbation theory significantly improves the estimates of tunneling splitting energies both for the ground state and for excited state doublets.
We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
The symmetric double-well potential is one of the simplest quantum-mechanical systems in which perturbative and non-perturbative physics are deeply entangled. Its energy levels have non-analytic expansions in inverse powers of the inter-well separation, with factorially growing coefficients, while the parity splitting is exponentially small and invisible to perturbation theory. Resurgence ties the two features together, organizing the exact spectrum into a single tightly-constrained trans-series. This paper gives a self-contained account of this trans-series from two complementary approaches: exact WKB and the Euclidean path integral, developed in a common notation with explicit calculations through the four-instanton level and three-loop order. In exact WKB, Stokes phenomena encoded in the Delabaere--Dillinger--Pham relations control the analytic continuation of the wavefunction past turning points. The quantization condition expressed in terms of Voros symbols then determines the full trans-series. The DDP relations are local and do not require knowing the global topology of the energy surface, but that surface is an elliptic curve. In the path integral, elliptic curves enter differently: the classical saddle points are doubly-periodic elliptic functions of Euclidean time, and Stokes phenomena play out within the finite-dimensional manifold of quasi-zero modes rather than through analytic continuation of the wavefunction. A Lefschetz thimble decomposition determines which saddles contribute, and the resulting partition function trans-series is much simpler than the energy trans-series: at each instanton order the TTT-dependence is a polynomial fixed by the quasi-zero-mode thimble integrals. Together, the two approaches deploy a shared mathematical infrastructure in complementary ways, showing that the double well is an ideal setting to explore resurgence.