Perturbation Theory for Time-dependent Singular Quantum Systems

Fatih Erman; Kaya G\"uven Akba\c{s}; O. Teoman Turgut

REVIEW 3 major objections 4 minor 29 references

Perturbation Theory for Time-dependent Singular Quantum Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper develops first- and second-order perturbation theory for time-dependent point interactions by expanding the renormalized Hamiltonian in the static spectral basis, avoiding the formal delta potential.

desk verdict A clean formal expansion with a good 1D sanity check, but the moving-center examples in 2D/3D are not L²-differentiable—the sphere's first-order transition probabilities diverge, so the framework is heuristic until the generator and domain questions are settled. read the letter →

arxiv 2608.08108 v1 pith:WVYTVTNK submitted 2026-08-08 math-ph math.MPquant-ph

Kaya Güven Akbaş , Fatih Erman , O. Teoman Turgut This is my paper · ORCID

classification math-phmath.MPquant-ph MSC 81Q1581Q10

keywords time-dependentpointinteractionsrenormalizedresolventheat-kernelregularizationperturbationtheorymovingdeltacenterprincipalfunctiontransitionamplitudesharmonicoscillator

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a time-dependent point interaction—a singular potential supported at a point whose renormalized strength or position varies with time—can be treated by the same spectral perturbation theory used for ordinary Hamiltonians, provided the singular interaction is encoded through its renormalized spectral data rather than through the formal delta function. If the construction is correct, first- and second-order transition amplitudes for these systems follow directly from the static eigenvalues, eigenfunctions, and the principal function $\Phi$, with no need to write down the singular potential or its time-dependent domain. The paper derives explicit pole shifts, projection corrections, and transition-amplitude equations for both a time-dependent strength and a moving support point, and illustrates them on a sphere, a two-dimensional oscillator, and a one-dimensional oscillator.

What carries the argument

The central object is the renormalized principal function $$\Phi(E,M)=\frac{1}{\alpha_R(M)}+\int_0^\infty K_s(a,a)\bigl($e^{{sM}}$-$e^{{sE}}$\bigr)\,ds,$$ the denominator of the renormalized resolvent whose zeros are the shifted bound-state energies; for a moving support, $a$ is replaced by the curve $q(t)$ and the diagonal heat kernel is evaluated at $q(t)$. The spectral representation $H=\sum_k E_k^\ast P_k$ of the static renormalized Hamiltonian turns the singular operator into ordinary spectral data, and the paper's main machinery is the Taylor expansion of $\Phi$ and of $P_k(E_k(t),q(t))$ in the small parameters. This produces the pole shifts (3.8)–(3.9) and (4.17)–(4.18), the projection derivatives (3.19)–(3.20), and the Hamiltonian corrections (3.21)–(3.22) and (4.32)–(4.34) that feed the transition-amplitude equations.

What would settle it

For the two-dimensional oscillator with a moving renormalized center, solve the time-dependent Schrödinger equation numerically with a narrow regularized potential, take the width to zero, and compare the transition probability into a first odd-oscillator state with the square of (5.127); any disagreement at order $\varepsilon^2$ that persists under regularization would show that $H(t)=\sum_k E_k(t)P_k(t)$ is not the Hamiltonian governing the singular system.

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Extended reading notes

Core claim

The central claim is that the non-autonomous Hamiltonian of a renormalized point interaction admits the instantaneous spectral representation $H(t)=\sum_k E_k(t)P_k(t)$, where the $E_k(t)$ are the zeros of the frozen principal function and the $P_k(t)$ are the corresponding rank-one projections built from the static Green function. The paper shows how to expand $E_k(t)$ and $P_k(t)$ in powers of the small time-dependent parameter $\eta(t)$ (for a varying renormalized strength) or of the support displacement $s(t)$ (for a moving center), obtaining closed formulas for the first- and second-order pole shifts and for the derivatives of the projections. These ingredients are then inserted into the time-dependent Schrödinger equation in the static eigenbasis, yielding first- and second-order transition amplitudes through the amplitude equations (3.33)–(3.36). The entire calculation runs on Green functions, heat kernels, and spectral data; the formal delta potential is never used.

Load-bearing premise

The load-bearing assumption is that the true dynamics is generated, at every instant, by the operator obtained from the static eigenvalues and eigenprojections taken at that instant's parameter values, and the paper does not prove that this operator is the same as the one defined by the rigorous time-dependent point-interaction construction in two or three dimensions, where the class of wavefunctions on which the Hamiltonian acts depends on time.

Editorial extensions

If this is right

For any unperturbed Hamiltonian with purely discrete spectrum, the first- and second-order response to a time-dependent renormalized strength is fixed by the static principal function, its energy and coupling derivatives at the pole, and the static Green function.
For a moving support, the same response is fixed by the Green function and its first and second derivatives along the curve, including the covariant acceleration, so the formulas separate the motion of the energy levels from the motion of the eigenprojectors.
When the principal function is position-independent, as on the two-dimensional sphere by the addition theorem, the energy levels remain stationary to all orders while the first-order transition amplitudes are nonzero; the dynamics is carried entirely by the moving eigenprojectors.
Sinusoidal driving produces the standard resonance denominators $\omega_{ml}\pm\Omega$ and splits the second-order amplitude into iterated, phase, and direct terms, while a delta-kick produces a frozen transition probability after the kick.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

I would test the frozen-spectral identification directly by comparing the paper's second-order amplitudes for the two-dimensional oscillator with a numerical solution of the time-dependent Schrödinger equation for a regularized narrow potential, since (5.117), (5.127), and (5.137) give explicit predictions.
If the frozen spectral family does not coincide with the rigorously defined time-dependent point-interaction Hamiltonian in two or three dimensions, the formulas could still describe a slow-motion effective dynamics, but establishing that would require a separate limit argument.
The single-center construction should extend to several moving centers by replacing the scalar principal function with a finite-dimensional principal matrix; whether the order-by-order pole-shift structure survives with off-diagonal couplings is a natural next calculation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague:

This paper is a formal perturbation scheme for time-dependent point interactions, built from the frozen spectral data of the static renormalized Hamiltonian. The algebraic core is clean: pole shifts come from Taylor-expanding the principal function, projection derivatives come from the residue formula, and the one-dimensional moving-delta example reproduces exactly the naive δ′(x) expansion. That 1D match is a real check, and the formulas (3.8)–(3.9) and (4.17)–(4.18) hang together internally. As a heuristic toolkit it has something new: nobody has written time-dependent Dyson series directly in the static point-interaction basis.

The soft spots are not incidental. Equation (3.7) postulates that the dynamics is generated by H(t)=∑E_k(t)P_k(t); there is no proof that this equals the established time-dependent point-interaction Hamiltonians of Refs. [2–8], whose domains move with the center. In 2D/3D this is not a technicality.

The stress-test note is right, and the failure is more concrete than a missing domain proof. From (5.59)–(5.60), ∂_sψ_α has free-basis coefficients B_L/(E_L−E*_α) ~ L^{-1/2}. If you square and sum over L,M you get ∑ 1/L, divergent: the derivative of the state is not L², so s ↦ ψ_α(s) fails to be strongly differentiable. The paper uses this derivative to compute the first-order amplitudes (5.79)–(5.82), and the resulting transition probabilities are non-normalizable. The individual matrix elements are finite in a distributional sense, but the series for the first-order wavefunction does not converge in L². That is a genuine failure of the worked example, and the 3D moving-center case will be worse. So I agree with the reader's 'conditional' verdict but would move it closer to 'reject as stated': the advertised sphere calculation does not define a Hilbert-space perturbation theory.

Credit where due: the strength-modulation case (Section 5.1) and the 1D oscillator are solid; the degeneracy treatment is honest, though only by example. The ChatGPT note in the acknowledgments makes independent checking of the examples especially wise.

Audience: people who want a quick calculus for these models, with the warning that it is a formal calculus until the generator identification and L²-regularity are settled. It deserves a serious referee—the 1D check alone is publishable and the questions raised are real—but I would not cite it in the next year and would not put it on anyone's reading list before the divergence problem is fixed or localized to the renormalized examples.

Recommendation: send to peer review with a referee instructed to pin down the domain-theoretic status of Eq. (3.7) and to redo the sphere example with explicit L² convergence checks. As written, the central example fails.

Referee Report

3 major / 4 minor

Summary. The paper develops a perturbative framework for time-dependent renormalized point interactions in systems with purely discrete spectrum. Starting from the static spectral representation H = sum_k E*_k P_k, it derives first- and second-order pole shifts and projection corrections for two types of time dependence: a time-dependent renormalized strength (Section 3) and a moving support point (Section 4). Transition amplitudes are then obtained by inserting the expanded Hamiltonian matrix elements into second-order Dyson-type equations. The formalism is illustrated by a sinusoidally modulated coupling, a two-dimensional harmonic oscillator with a moving or circularly moving delta center, a point interaction moving on a great circle of the two-sphere, and a one-dimensional oscillator with a moving delta, where the result is compared with the naive distributional expansion.

Significance. If valid, the framework would be a genuinely useful complement to the existing literature on time-dependent point interactions: it avoids writing the formal delta potential and expresses all corrections in terms of Green functions, heat kernels, and derivatives of the principal function Phi. The algebraic derivation is transparent, the one-dimensional oscillator example correctly reduces to the standard delta-perturbation result, and the selection rules obtained in the two-dimensional oscillator are explicit and physically suggestive. However, the paper's central computational claim is only conditionally established: the identification of the dynamics with the frozen spectral operator H(t) = sum_k E_k(t)P_k(t) is assumed rather than proved, and the sphere example exhibits a concrete failure of L^2 summability in the first-order transition amplitudes. The sphere failure is not a cosmetic issue; it invalidates the advertised computation of transition probabilities in that example and shows that the general moving-center expansion needs additional hypotheses.

major comments (3)

[Section 5.2] The first-order moving-center expansion in the sphere example is not L^2-valid. From Eqs. (5.59)-(5.60), B_L ~ L^{3/2}, while E_L - E*_alpha ~ L^2, so Eq. (5.77) gives <phi_{L,+/-1}|d_s psi_alpha> = +/- B_L/((E_L-E*_alpha) sqrt(N_alpha)) ~ L^{-1/2}. Hence sum_{L,M} |<phi_{L,M}|d_s psi_alpha>|^2 ~ sum_L 1/L diverges logarithmically: d_s psi_alpha is not an L^2 vector and the projector P_alpha(s) is not strongly differentiable at s=0. Consequently the first-order transition amplitudes (5.81)-(5.82) do not define a normalizable state correction; the total first-order transition probability sum_L |c^{(1)}_{L,+/-1}(t)|^2 diverges. The same ultraviolet divergence affects the second-order sums (5.86) and (5.88). This is a concrete failure of the claimed computability of transition amplitudes in a canonical compact example, not merely a missing domain proof. The authors need either to prove the relevant summability from decay of the eigenfunction derivatives or to replace the example with one where the expansion is an asymptotic expansion in L^2.
[Section 3] The paper postulates that the non-autonomous dynamics is generated by the frozen spectral Hamiltonian H(t) = sum_k E_k(t)P_k(t). This premise is load-bearing because every transition amplitude in Sections 3-5 is computed from this equation, yet the paper does not prove that this family equals the established time-dependent point-interaction Hamiltonians of Refs. [2-8], whose operator domains depend on time through the moving center and renormalized coupling. In one dimension the form domain may be fixed, but in two and three dimensions this is not automatic. The authors should either prove equivalence with the known non-autonomous point-interaction evolutions, or explicitly state and justify H(t) = sum_k E_k(t)P_k(t) as a defining model assumption. A concrete diagnostic would be to compare the resulting transition amplitudes with the exact propagators available for moving delta-type interactions in Refs. [9-11] or with the solvable three-dimensional moving-point-interaction problems of Refs. [3,4].
[Sections 3-4] The operator expansions (3.21)-(3.22) and (4.34) are presented as identities involving sums over the full spectral family, but no convergence is shown in any operator topology. The paper does not state hypotheses under which the term-by-term differentiation of the spectral representation is legitimate. The sphere example of Section 5.2 shows that these hypotheses are not automatically satisfied: the failure is not a technicality about unboundedness of individual matrix elements but a divergence of the sum of their squares. The general framework should therefore be formulated with explicit conditions on the decay of G0(x,a|E) and its derivatives, or at least with a statement that the expansions are formal at this stage.

minor comments (4)

[Section 4] The phrase 'Starting again from (M4)' refers to an equation label that does not appear anywhere in the manuscript; it should be a reference to Eq. (4.32) or Eq. (4.34).
[Section 5.1] In the first sum of Eq. (5.31) the index l is used both for the initial state and as a summation index in V_{mn}V_{nl}J_{mnl}; the summation should run over n, as in Eqs. (5.18)-(5.19).
[Section 2] Equation (2.15) contains a stray argument and inequality: the expression '=-G^0_{k,E}(x)<0' is confusing. The quantity Phi^E_k is a number, namely -d/dE G0(a,a|E) evaluated at E*_k, and the x-dependence should be removed.
[Section 5.2] The text refers to 'Fig. 1', but the figure is not included in the manuscript; either include it or remove the reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: pole shifts come from expanding the defining pole equation, and transition amplitudes follow by standard Dyson iteration of explicitly computed matrix elements.

full rationale

The derivation is not circular. The pole shifts (3.8)-(3.9) and (4.17)-(4.18) are obtained by expanding the defining zero condition Θ(E_k(t), μ(t)) = 0 (Eq. (3.4)) and Θ(E_k^*(t), q(t)) = 0 (Eq. (4.13)) in powers of η(t) and s(t); this is an implicit-function Taylor expansion, not a fit of transition amplitudes. The Hamiltonian expansion in Proposition 3.3 is the Taylor expansion of the operator explicitly introduced in Eq. (3.7) as H(t) = Σ_k E_k(t) P_k(t), and the paper openly states that the spectral representation (2.18) is the starting point. Whether this spectral family equals the established non-autonomous point-interaction Hamiltonian of Refs. [2-8] is an unproved identification and a mathematical correctness concern, but it is not a circular reduction, because the amplitudes are subsequently computed rather than assumed. The matrix elements R^(r)_mn and the Dyson-type equations (3.33)-(3.36) are standard time-dependent perturbation theory with no fitted parameter renamed as a prediction. The concrete examples are explicit computations from known oscillator and spherical eigenfunctions, and the one-dimensional example is checked against the naive δ' expansion (5.202)-(5.204), with Remark 5.2 flagging domain issues. The self-citations [16,18,19] supply static renormalization, convergence, and completeness results; they are separate mathematical statements, not parameters fitted in this paper. The sphere example's possible failure of L^2-differentiability of ∂_s ψ_α would be a validity problem for the perturbative expansion, not an input-output equivalence, so it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameter is fitted to data or tuned to force a result. The physical inputs mu0 (binding parameter), lambda (1D coupling), epsilon, Omega (driving amplitude and frequency), and R (sphere radius) are model constants, not fit parameters. The renormalization condition 1/alpha_R(-mu0) = 0 fixes the scheme rather than fitting an output. No new physical entities are postulated.

assumptions (4)

standard math H0 has purely discrete spectrum with heat-kernel eigenfunction expansion (2.11)-(2.12).
Section 2, used to write G0 and Phi as convergent series; standard spectral and heat-kernel theory.
domain assumption The static renormalized Hamiltonian has complete eigenfunctions {psi_k} with spectral representation (2.18).
Section 2, Eq. (2.18), imported from the authors' Ref. [19]; load-bearing for all expansions.
ad hoc to paper The non-autonomous dynamics is generated by H(t) = sum_k E_k(t) P_k(t).
Eq. (3.7) and analogous Eq. (4.19); asserted, not derived or cited to the moving-domain literature.
domain assumption Simple eigenvalues and off-nodal support points; degenerate cases via adapted bases.
Introduction and Section 5 examples; restricts the general theorem, with degenerate sectors handled case-by-case.

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Cite this review

Pith. "Pith review of Perturbation Theory for Time-dependent Singular Quantum Systems." pith.science (2026). https://pith.science/paper/WVYTVTNK

@misc{pith2026260808108,
  author       = {Pith},
  title        = {Pith review of: Perturbation Theory for Time-dependent Singular Quantum Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVYTVTNK}},
  note         = {Machine review of arXiv:2608.08108}
}

read the original abstract

We develop a perturbative framework for time-dependent point interactions in quantum systems with purely discrete unperturbed spectrum. In two and three dimensions the point interaction is described through the renormalized resolvent obtained from heat-kernel regularization, while in one dimension the diagonal Green function is finite and no renormalization is required. Time dependence is introduced either through the interaction parameter or through the motion of the support point. In both cases we expand the non-autonomous Hamiltonian in the spectral basis of the corresponding static point-interaction problem and derive the first- and second-order pole shifts, projection corrections, and transition amplitudes. The method is illustrated by explicit examples: point interactions with time-dependent coupling, harmonic oscillators perturbed by moving point interactions, and a particle on a sphere with a moving interaction center. We also discuss the one-dimensional harmonic oscillator with a moving delta potential, showing that in the non-renormalized case the present formulation reduces to the standard time-dependent perturbation theory.

Figures

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Works this paper leans on

29 extracted references · 28 canonical work pages

[1]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden,Solvable Models in Quan- tum Mechanics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. 43

work page 2005
[2]

A limit evolution problem for time- dependent point interactions,

G. F. Dell’Antonio, R. Figari, and A. Teta, “A limit evolution problem for time- dependent point interactions,”J. Funct. Anal.142, 249–275 (1996)

work page 1996
[3]

Schr¨ odinger equation with moving point interactions in three dimensions,

G. F. Dell’Antonio, R. Figari, and A. Teta, “Schr¨ odinger equation with moving point interactions in three dimensions,” inStochastic Processes, Physics and Geometry: New Interplays, I, edited by F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. R¨ ockner, and S. Scarlatti, CMS Conference Proceedings Vol. 28, American Mathematical So- ciety, Providence, RI, ...

work page 2000
[4]

The Schr¨ odinger equation with a moving point interaction in three dimensions,

A. Posilicano, “The Schr¨ odinger equation with a moving point interaction in three dimensions,”Proc. Am. Math. Soc.135, 1785–1793 (2007)

work page 2007
[5]

Ionization for three di- mensional time-dependent point interactions,

M. Correggi, G. Dell’Antonio, R. Figari, and A. Mantile, “Ionization for three di- mensional time-dependent point interactions,”Commun. Math. Phys.257, 169–192 (2005)

work page 2005
[6]

Time-dependent delta-interactions for 1D Schr¨ odinger Hamiltonians,

T. Hmidi, A. Mantile, and F. Nier, “Time-dependent delta-interactions for 1D Schr¨ odinger Hamiltonians,”Math. Phys. Anal. Geom.13, 83–103 (2010)

work page 2010
[7]

Two-dimensional time-dependent point interactions,

R. Carlone, M. Correggi, and R. Figari, “Two-dimensional time-dependent point interactions,” inFunctional Analysis and Operator Theory for Quantum Physics, edited by J. Dittrich, H. Kovaˇ r ´ ık, and A. Laptev, EMS Series of Congress Reports, European Mathematical Society, Z¨ urich, 2017, pp. 189–211

work page 2017
[8]

Complete ionization for a non- autonomous point interaction model in d = 2,

W. Borrelli, R. Carlone, and L. Tentarelli, “Complete ionization for a non- autonomous point interaction model in d = 2,”Commun. Math. Phys.395, 963–1005 (2022)

work page 2022

Show all 29 references

[9]

Exact propagators for time- dependent Coulomb, delta and other potentials,

V. V. Dodonov, V. I. Man’ko, and D. E. Nikonov, “Exact propagators for time- dependent Coulomb, delta and other potentials,”Phys. Lett. A162, 359–364 (1992)

1992
[10]

The propagators forδandδ ′ potentials with time-dependent strengths,

F. Erman, M. Gadella, and H. Uncu, “The propagators forδandδ ′ potentials with time-dependent strengths,”Front. Phys.8, 65 (2020)

2020
[11]

Propagator calculations for time dependent Dirac delta potentials and corresponding two state models,

S. Mudra and A. Chakraborty, “Propagator calculations for time dependent Dirac delta potentials and corresponding two state models,”Phys. Lett. A418, 127725 (2021)

2021
[12]

Rosenberg,The Laplacian on a Riemannian Manifold, Cambridge University Press, Cambridge, 1998

S. Rosenberg,The Laplacian on a Riemannian Manifold, Cambridge University Press, Cambridge, 1998

1998
[13]

Chavel,Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, Vol

I. Chavel,Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, Vol. 115, Academic Press, Orlando, 1984

1984
[14]

Renormalization of multi-delta-function point scat- terers in two and three dimensions, the coincidence-limit problem, and its resolu- tion,

F. Loran and A. Mostafazadeh, “Renormalization of multi-delta-function point scat- terers in two and three dimensions, the coincidence-limit problem, and its resolu- tion,”Ann. Phys.443, 168966 (2022). doi: 10.1016/j.aop.2022.168966

2022
[15]

Albeverio and P

S. Albeverio and P. Kurasov,Singular Perturbations of Differential Operators, Cam- bridge University Press, Cambridge, 2000. 44

2000
[16]

Point interactions in two- and three-dimensional Rie- mannian manifolds,

F. Erman and O. T. Turgut, “Point interactions in two- and three-dimensional Rie- mannian manifolds,”J. Phys. A: Math. Theor.43, 335204 (2010). doi: 10.1088/1751- 8113/43/33/335204

2010 doi
[17]

E. B. Davies,Heat Kernels and Spectral Theory, Cambridge University Press, Cam- bridge, 1989

1989
[18]

On Schr¨ odinger operators modified by δinteractions,

K. G. Akbas, F. Erman, and O. T. Turgut, “On Schr¨ odinger operators modified by δinteractions,”Ann. Phys.458, 169468 (2023)

2023
[19]

Completeness relation in renormalized quantum sys- tems,

F. Erman and O. T. Turgut, “Completeness relation in renormalized quantum sys- tems,”Front. Phys.13, 1695365 (2025). doi: 10.3389/fphy.2025.1695365

2025
[20]

Reed and B

M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, 1980

1980
[21]

Hassani,Mathematical Physics: A Modern Introduction to Its Foundations, 2nd ed., Springer, Cham, 2013

S. Hassani,Mathematical Physics: A Modern Introduction to Its Foundations, 2nd ed., Springer, Cham, 2013

2013
[22]

Solution of the one-dimensional harmonic oscillator with a delta- function potential,

S. H. Patil, “Solution of the one-dimensional harmonic oscillator with a delta- function potential,”Eur. J. Phys.27(4), 899–906 (2006)

2006
[23]

Janke, and B

W. Janke, and B. K. Cheng, Statistical properties of a harmonic plus a delta- potential,Physics Letters A129, 3, 140-144 (1988)

1988
[24]

L. C. Ioriatti, S. G. Rosa, and O. Hip´ olito, Bose-Einstein condensation in a one- dimensional system due to an attractive-δimpurity center,American Journal of Physics44, 8, 744-748 (1976)

1976
[25]

H. Uncu, D. Tarhan, E. Demiralp, ¨O. E. M¨ ustecaplıo˘ glu, Bose-Einstein condensate in a harmonic trap decorated with Diracδfunctions,Physical Review A,761, 013618 (2007)

2007
[26]

Grosche, Path integrals for potential problems withδ-function perturbation,Jour- nal of Physics A: Mathematical and General,23, 22, 5205 (1990)

C. Grosche, Path integrals for potential problems withδ-function perturbation,Jour- nal of Physics A: Mathematical and General,23, 22, 5205 (1990)

1990
[27]

Fassari, G

S. Fassari, G. Inglese, On the spectrum of the harmonic oscillator with aδtype perturbation,Helv. Physica Acta67, 6, 650-659 (1994)

1994
[28]

Thangavelu,Lectures on Hermite and Laguerre Expansions, Mathematical Notes, Vol

S. Thangavelu,Lectures on Hermite and Laguerre Expansions, Mathematical Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993

1993
[29]

Konishi and G

K. Konishi and G. Paffuti,Quantum Mechanics: A New Introduction, Oxford Uni- versity Press, Oxford, 2009. 45

2009

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden,Solvable Models in Quan- tum Mechanics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. 43

work page 2005

[2] [2]

A limit evolution problem for time- dependent point interactions,

G. F. Dell’Antonio, R. Figari, and A. Teta, “A limit evolution problem for time- dependent point interactions,”J. Funct. Anal.142, 249–275 (1996)

work page 1996

[3] [3]

Schr¨ odinger equation with moving point interactions in three dimensions,

G. F. Dell’Antonio, R. Figari, and A. Teta, “Schr¨ odinger equation with moving point interactions in three dimensions,” inStochastic Processes, Physics and Geometry: New Interplays, I, edited by F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. R¨ ockner, and S. Scarlatti, CMS Conference Proceedings Vol. 28, American Mathematical So- ciety, Providence, RI, ...

work page 2000

[4] [4]

The Schr¨ odinger equation with a moving point interaction in three dimensions,

A. Posilicano, “The Schr¨ odinger equation with a moving point interaction in three dimensions,”Proc. Am. Math. Soc.135, 1785–1793 (2007)

work page 2007

[5] [5]

Ionization for three di- mensional time-dependent point interactions,

M. Correggi, G. Dell’Antonio, R. Figari, and A. Mantile, “Ionization for three di- mensional time-dependent point interactions,”Commun. Math. Phys.257, 169–192 (2005)

work page 2005

[6] [6]

Time-dependent delta-interactions for 1D Schr¨ odinger Hamiltonians,

T. Hmidi, A. Mantile, and F. Nier, “Time-dependent delta-interactions for 1D Schr¨ odinger Hamiltonians,”Math. Phys. Anal. Geom.13, 83–103 (2010)

work page 2010

[7] [7]

Two-dimensional time-dependent point interactions,

R. Carlone, M. Correggi, and R. Figari, “Two-dimensional time-dependent point interactions,” inFunctional Analysis and Operator Theory for Quantum Physics, edited by J. Dittrich, H. Kovaˇ r ´ ık, and A. Laptev, EMS Series of Congress Reports, European Mathematical Society, Z¨ urich, 2017, pp. 189–211

work page 2017

[8] [8]

Complete ionization for a non- autonomous point interaction model in d = 2,

W. Borrelli, R. Carlone, and L. Tentarelli, “Complete ionization for a non- autonomous point interaction model in d = 2,”Commun. Math. Phys.395, 963–1005 (2022)

work page 2022