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Exercises in exact quantization | Exercices de quantification exacte

André Voros

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians $[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm$ on the half-line $\{q>0\}$, with a Dirichlet ($-$) or Neumann ($+$) condition at $q=0$. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential $V(q)=q^N$ as $N \to +\infty$ vs its (solvable) $N=\infty$ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-$N$ properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants $\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pm$ are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic $v \to +\infty$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v q^{N/2-1}$ of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with $v$ as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to $N=2$); these results probably reflect the presence of supersymmetric potentials in the family above.

Related papers

Exact quantization method for the polynomial 1D Schrödinger equation | Méthode de quantification exacte pour l'équation de Schrödinger polynômiale 1D

André Voros

The stationary 1D Schrödinger equation with a polynomial potential $V(q)$ of degree $N$ is reduced analytically to a system of (complex) exact quantization conditions of Bohr--Sommerfeld form. These reduced equations can be probed numerically for effective solvability: in test cases, they appear to generate discrete dynamical systems with contractive behavior, and converging towards the exact quantum data as their fixed point. (To date, this statement is verified only empirically and for potentials close to purely quartic or sextic ones.)

•Jan 1, 2000•0 citations

Radial power-like potentials: from the Bohr-Sommerfeld $\ell \neq 0$ state energies to the exact ones

J. C. del Valle, A. V. Turbiner

Two new Bohr-Sommerfeld (B-S) type quantization conditions for $d$-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{$V=\operatorname{sign}(m)\,r^m$} for $m=-3/2,-1,-1/2, 1,3,4,6$ and $V=\log r$ (that corresponds to $m=0$) are studied in detail for radial quantum numbers $n_r \in [0,100]$ and for angular momenta $\ell=1,2,3,4,5$ in dimensions $d=2,3$. It is shown that for large $n_r$ the B-S energies $E_{BS}$ always tend to the exact ones. For $m=2$ (the radial harmonic oscillator) and $m=-1$ (the $d$-dimensional analogue of the $3D$ Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of $E_{BS}$ with the energies $E^{(L)}_{BS}$, obtained for the radial potentials modified by the Langer correction for $d=3$, are in good agreement in general: in detail, it was checked for the quartic oscillator $V=r^4$. It is demonstrated that introducing a dimensionless {\it WKB correction} $γ$ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with $γ$ being very small for positive degrees $m \geq 0$ and sufficiently small for non-positive ones $m < 0$ for all $n_r, \ell$ studied.

quant-pharXiv:2610.36421•Oct 28, 2026•0 citations

Analysis of Quartic Potential Problems Using Fredholm Integral Operators with Airy Functions

Ori J. Ganor

Fredholm integral operators that commute with the Hamiltonians of certain quantum mechanical systems with quartic potentials are introduced in closed analytical form. The operators are expressed in terms of the Airy function, and their eigenvalues fall off exponentially. Using these operators we obtain an exact dual description in terms of infinite one-dimensional chains. The systems discussed include the anharmonic quartic oscillator as well as multivariable potentials and higher dimensional systems. We compute a few moments of the Fredholm operators, using matrix models in the higher dimensional case, and we show how those can be used to approximate the energy spectrum and obtain estimates of the asymptotic behavior of the wavefunctions.

hep-thCross-listedarXiv:2603.02374•Mar 4, 2026

Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

Yanjie Li, Sihong Shao

This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order $2s+2$ for eigenvalues and order $s+1$ for eigenfunctions in the $H^1$ norm. More importantly, the $H^1$ error with respect to the projected eigenfunction converges with a higher order $s+1+b$, where $b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0$ for arbitrarily small $\varepsilon>0$, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H^1$ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in $H^{s+4}$ together with $d+1$ asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

math.NACross-listedarXiv:2606.01718•Jun 2, 2026

Exact results for the infinite supersymmetric extensions of the infinite square well

K Gutierrez, Esteban León, Mario Belloni +1 more

Abstract One-dimensional potentials defined by V ( S ) ( x ) = S ( S + 1 ) ℏ 2 π 2 / [ 2 ma 2 sin 2 ( π x / a ) ] (for integer S ) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a ). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, ψ n ( S ) ( x ) , for all V ( S ) ( x ) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions ϕ n ( S ) ( p ) for large ∣ p ∣ and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.

arXiv:1810.04701•European Journal of Physics•Aug 23, 2018•4 citations

A Power-Liouville Hermite Orbit: Common Spectral Quantization of the Singular Sextic Oscillator and Fractional-Power Potentials

Artur Ishkhanyan, Miloslav Znojil

A singular sextic oscillator in the finite-Hermite biconfluent-Heun hierarchy generates an exact four-member power-Liouville Hermite orbit. Power coordinate transformations with the Liouville amplitude map the hierarchy member $N=4$ to four half-line Schrödinger spectral charts while preserving the physical regular-recessive Wronskian zero set under energy-coupling reparametrization. Two charts are the previously studied $x^{2/3}$ and inverse-square-root conditionally integrable potentials; the orbit embeds this known power-dual pair together with the singular sextic and linear-plus-quadratic charts. On the selected physical slice, the degree-five accessory compatibility obstruction vanishes identically, while the transformed Frobenius index reproduces both published fractional-power centrifugal coefficients. The common endpoint condition yields an exact discrete sequence of quartic Sturmian eigencouplings for the sextic oscillator. Independent two-sided shooting confirms representative eigencouplings and node counts. A matched large-action analysis further derives the first matched-endpoint correction within the leading Langer action and improves the Maslov-shifted effective-level approximation. The construction thus connects exact Hermite solvability, energy quantization, coupling quantization, and power duality within a single spectral orbit.

quant-pharXiv:2610.11533•Oct 9, 2026
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians [−d2/dq2+V(q)]±[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm[−d2/dq2+V(q)]± on the half-line {q>0}\{q>0\}{q>0}, with a Dirichlet (−-−) or Neumann (+++) condition at q=0q=0q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential V(q)=qNV(q)=q^NV(q)=qN as N→+∞N \to +\inftyN→+∞ vs its (solvable) N=∞N=\inftyN=∞ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-NNN properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants det⁡(−d2/dq2+q4+vq2)±\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pmdet(−d2/dq2+q4+vq2)± are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic v→+∞v \to +\inftyv→+∞ properties of Airy type. The third study addresses the potentials V(q)=qN+vqN/2−1V(q)=q^N+v q^{N/2-1}V(q)=qN+vqN/2−1 of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with vvv as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2N=2N=2); these results probably reflect the presence of supersymmetric potentials in the family above.
The stationary 1D Schrödinger equation with a polynomial potential V(q)V(q)V(q) of degree NNN is reduced analytically to a system of (complex) exact quantization conditions of Bohr--Sommerfeld form. These reduced equations can be probed numerically for effective solvability: in test cases, they appear to generate discrete dynamical systems with contractive behavior, and converging towards the exact quantum data as their fixed point. (To date, this statement is verified only empirically and for potentials close to purely quartic or sextic ones.)
Radial power-like potentials: from the Bohr-Sommerfeld ℓ≠0\ell \neq 0ℓ=0 state energies to the exact ones
Two new Bohr-Sommerfeld (B-S) type quantization conditions for ddd-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{V=sign⁡(m) rmV=\operatorname{sign}(m)\,r^mV=sign(m)rm} for m=−3/2,−1,−1/2,1,3,4,6m=-3/2,-1,-1/2, 1,3,4,6m=−3/2,−1,−1/2,1,3,4,6 and V=log⁡rV=\log rV=logr (that corresponds to m=0m=0m=0) are studied in detail for radial quantum numbers nr∈[0,100]n_r \in [0,100]nr​∈[0,100] and for angular momenta ℓ=1,2,3,4,5\ell=1,2,3,4,5ℓ=1,2,3,4,5 in dimensions d=2,3d=2,3d=2,3. It is shown that for large nrn_rnr​ the B-S energies EBSE_{BS}EBS​ always tend to the exact ones. For m=2m=2m=2 (the radial harmonic oscillator) and m=−1m=-1m=−1 (the ddd-dimensional analogue of the 3D3D3D Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of EBSE_{BS}EBS​ with the energies EBS(L)E^{(L)}_{BS}EBS(L)​, obtained for the radial potentials modified by the Langer correction for d=3d=3d=3, are in good agreement in general: in detail, it was checked for the quartic oscillator V=r4V=r^4V=r4. It is demonstrated that introducing a dimensionless {\it WKB correction} γγγ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with γγγ being very small for positive degrees m≥0m \geq 0m≥0 and sufficiently small for non-positive ones m<0m < 0m<0 for all nr,ℓn_r, \ellnr​,ℓ studied.
Fredholm integral operators that commute with the Hamiltonians of certain quantum mechanical systems with quartic potentials are introduced in closed analytical form. The operators are expressed in terms of the Airy function, and their eigenvalues fall off exponentially. Using these operators we obtain an exact dual description in terms of infinite one-dimensional chains. The systems discussed include the anharmonic quartic oscillator as well as multivariable potentials and higher dimensional systems. We compute a few moments of the Fredholm operators, using matrix models in the higher dimensional case, and we show how those can be used to approximate the energy spectrum and obtain estimates of the asymptotic behavior of the wavefunctions.
This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials V∈HsV \in H^{s}V∈Hs, where s>max⁡{d/2−2,−1}s > \max\{d/2-2,-1\}s>max{d/2−2,−1} and ddd denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order 2s+22s+22s+2 for eigenvalues and order s+1s+1s+1 for eigenfunctions in the H1H^1H1 norm. More importantly, the H1H^1H1 error with respect to the projected eigenfunction converges with a higher order s+1+bs+1+bs+1+b, where b=min⁡{s+2−d/2−ε,  s+1,  2}>0b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0b=min{s+2−d/2−ε,s+1,2}>0 for arbitrarily small ε>0\varepsilon>0ε>0, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders 2s+2+2b2s+2+2b2s+2+2b for eigenvalues and s+1+bs+1+bs+1+b for eigenfunctions in the H1H^1H1 norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in Hs+4H^{s+4}Hs+4 together with d+1d+1d+1 asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.
Abstract One-dimensional potentials defined by V ( S ) ( x ) = S ( S + 1 ) ℏ 2 π 2 / [ 2 ma 2 sin 2 ( π x / a ) ] (for integer S ) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a ). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, ψ n ( S ) ( x ) , for all V ( S ) ( x ) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions ϕ n ( S ) ( p ) for large ∣ p ∣ and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.
A singular sextic oscillator in the finite-Hermite biconfluent-Heun hierarchy generates an exact four-member power-Liouville Hermite orbit. Power coordinate transformations with the Liouville amplitude map the hierarchy member N=4N=4N=4 to four half-line Schrödinger spectral charts while preserving the physical regular-recessive Wronskian zero set under energy-coupling reparametrization. Two charts are the previously studied x2/3x^{2/3}x2/3 and inverse-square-root conditionally integrable potentials; the orbit embeds this known power-dual pair together with the singular sextic and linear-plus-quadratic charts. On the selected physical slice, the degree-five accessory compatibility obstruction vanishes identically, while the transformed Frobenius index reproduces both published fractional-power centrifugal coefficients. The common endpoint condition yields an exact discrete sequence of quartic Sturmian eigencouplings for the sextic oscillator. Independent two-sided shooting confirms representative eigencouplings and node counts. A matched large-action analysis further derives the first matched-endpoint correction within the leading Langer action and improves the Maslov-shifted effective-level approximation. The construction thus connects exact Hermite solvability, energy quantization, coupling quantization, and power duality within a single spectral orbit.