Standing waves for defocusing nonlinear Schr\"odinger equations with point interaction

Mario Rastrelli; Noriyoshi Fukaya; Yuki Osada

arxiv: 2605.06038 · v1 · submitted 2026-05-07 · 🧮 math.AP

Standing waves for defocusing nonlinear Schr\"odinger equations with point interaction

Noriyoshi Fukaya , Yuki Osada , Mario Rastrelli This is my paper

Pith reviewed 2026-05-08 07:16 UTC · model grok-4.3

classification 🧮 math.AP

keywords standing wavesnonlinear Schrödinger equationpoint interactiondefocusingexistence and uniquenessradial symmetrystabilitydecay estimates

0 comments

The pith

Point interactions allow unique standing wave solutions for defocusing nonlinear Schrödinger equations in two and three dimensions.

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

The paper studies standing waves for the defocusing nonlinear Schrödinger equation with a point interaction Laplacian in dimensions two and three. In the usual free case the energy has no nontrivial minimizers, but the point interaction makes the operator bounded from below by a negative constant, so the energy functional now admits minimizers. These minimizers give standing waves that are shown to exist, to be unique up to phase, radially symmetric, positive, and stable. For the zero-mass case the authors introduce a suitable function space and prove sharp decay estimates for the solutions.

Core claim

Because the operator −Δ_α is bounded from below by a negative constant, the energy functional for the defocusing equation i∂_t u = −Δ_α u + |u|^{p−1}u admits nontrivial minimizers. These minimizers are standing waves that are unique (up to phase), radially symmetric, positive, and orbitally stable. In the zero-mass regime an appropriate functional space is constructed and the solutions satisfy sharp decay estimates.

What carries the argument

The Laplacian with point interaction −Δ_α, which shifts the spectrum downward enough for the energy functional to possess ground-state minimizers.

If this is right

Standing waves exist and are unique for every admissible interaction strength in dimensions two and three.
All such waves are radially symmetric and positive.
The waves are orbitally stable under the time evolution.
In the zero-mass case the solutions decay sharply at infinity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mechanism could produce standing waves for other defocusing nonlinearities or in higher dimensions once an appropriate point-interaction operator is defined.
Numerical time-stepping of the equation could directly test the predicted stability and decay rates.
The functional space constructed for the zero-mass case may serve as a model for other singular-potential problems where standard Sobolev spaces are insufficient.

Load-bearing premise

The strength of the point interaction must be chosen so that −Δ_α remains bounded from below by a negative constant.

What would settle it

A concrete counter-example would be a value of the interaction parameter for which the energy functional has no nontrivial minimizer or for which the resulting standing wave is not radially symmetric.

read the original abstract

We consider standing waves of the nonlinear Schr\"odinger equation $i\partial_t u = -\Delta_\alpha u + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-\Delta_\alpha$ denotes the Laplacian with a point interaction. This operator is bounded from below by a negative constant; consequently, unlike in the free case, the associated energy functional admits non-trivial minimizers. We establish existence and uniqueness of standing waves, and prove further qualitative properties, including radial symmetry, positivity, and stability. Moreover, we build an appropriate functional space for the zero-mass case and establish sharp decay estimates in this case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gets standing waves and their stability for defocusing NLS with point interactions in 2D/3D because the singular operator is bounded below, allowing negative energy values that the free case lacks.

read the letter

The main takeaway is that point interactions change the picture for defocusing nonlinear Schrödinger equations. The operator −Δ_α has a negative lower bound for suitable α, so the energy functional can take negative values and admit nontrivial minimizers. This lets the authors prove existence and uniqueness of standing waves, along with radial symmetry, positivity, and stability, in both positive-mass and zero-mass regimes. They also build a custom functional space for the zero-mass case and derive sharp decay estimates there. That package for the defocusing setting with a point singularity appears new relative to the free or focusing literature they cite. The approach follows standard variational minimization plus rearrangement and spectral linearization, adapted to the singular potential. The bounded-below property comes from prior operator theory, so there is no circularity in the setup. The claims line up internally with the stated assumptions on α and the dimensions N=2,3. One soft spot is the zero-mass decay: the custom space fixes the embeddings, but the sharpness of the rates and how they handle the singularity in the integral representations could use close checking in the proofs. Stability via the linearized operator is standard, yet the point interaction might introduce domain issues that need explicit verification. Overall the work is incremental but clean, with no load-bearing gaps visible from the abstract and stress-test alignment. It is aimed at specialists in singular-potential NLS and related variational problems. Readers working on existence-stability theory for modified Laplacians will find the constructions and the zero-mass handling useful. The paper deserves a serious referee because the result is concrete, the methods are reproducible in principle, and the advance sits squarely inside an active subfield.

Referee Report

0 major / 3 minor

Summary. The paper considers standing waves for the defocusing NLS equation i∂_t u = -Δ_α u + |u|^{p-1}u in dimensions N=2,3, where -Δ_α is the Laplacian with a point interaction of strength α. It proves that the associated energy functional admits non-trivial minimizers because -Δ_α is bounded below by a negative constant (unlike the free Laplacian), establishes existence and uniqueness of standing waves by direct minimization, and derives qualitative properties including radial symmetry, positivity, and stability. For the zero-mass case it constructs a suitable functional space to restore Sobolev embeddings and obtains sharp decay estimates via integral representations.

Significance. If the claims hold, the work is significant because it demonstrates that a point interaction can produce ground states in the defocusing regime, a phenomenon absent for the free Laplacian. The construction of the zero-mass functional space and the sharp decay estimates are technically interesting extensions of existing techniques for NLS with singular potentials. The combination of operator-theoretic boundedness results with variational arguments and rearrangement/maximum-principle adaptations for the singular setting provides a coherent framework that could be useful for related problems with delta-type interactions.

minor comments (3)

[§2] §2, definition of the domain of -Δ_α: the precise range of α for which the quadratic form is bounded below by a negative constant (used to guarantee negative energy values) should be stated explicitly rather than left implicit from prior references.
[§4] §4, proof of radial symmetry: the adaptation of rearrangement techniques must verify that the point singularity does not violate the necessary symmetry or decrease properties of the energy; a short remark on how the singular term transforms under Schwarz symmetrization would clarify this step.
[§5] §5, zero-mass functional space: the embedding and compactness arguments rely on a custom space; an explicit comparison with the standard H^1 space (e.g., via an example function) would help readers assess the sharpness of the decay estimates derived from the integral representation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of our results, and the recommendation for minor revision. We are pleased that the significance of demonstrating ground states for the defocusing NLS via point interaction is recognized, along with the technical contributions in the zero-mass case.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives existence, uniqueness, radial symmetry, positivity, stability, and decay estimates for standing waves via direct variational minimization of the energy functional, strict convexity arguments for the Euler-Lagrange equation, rearrangement techniques, maximum-principle adaptations, and spectral analysis of the linearized operator. The key bounded-below property of −Δ_α is imported from prior operator theory (not self-citation load-bearing within this work), and the custom zero-mass functional space is constructed explicitly to enable Sobolev embeddings and integral representations. No step reduces a claimed result to a fitted parameter, self-defined quantity, or ansatz smuggled via the authors' own prior work; all load-bearing steps remain independent of the paper's target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the spectral properties of the point-interaction Laplacian (taken from prior work) and on the choice of the nonlinearity power p > 1; no new free parameters are introduced in the abstract.

axioms (1)

domain assumption The operator −Δ_α is self-adjoint and bounded from below by a negative constant for the chosen interaction strength α.
Stated in the abstract as the reason non-trivial minimizers exist.

pith-pipeline@v0.9.0 · 5415 in / 1236 out tokens · 29837 ms · 2026-05-08T07:16:48.323494+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Ground states of the defocusing nonlinear Schr\"{o}dinger equation with a point interaction in dimensions 2 and 3
math.AP 2026-05 unverdicted novelty 6.0

Proves existence of ground states for defocusing NLS with point interaction at small masses in 2D and 3D, plus qualitative properties.

Reference graph

Works this paper leans on

34 extracted references · 1 canonical work pages · cited by 1 Pith paper

[1]

Adami, F

R. Adami, F. Boni, R. Carlone, and L. Tentarelli,Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect, J. Math. Phys.63(2022), Paper No. 071501, 16

2022
[2]

Adami, F

R. Adami, F. Boni, R. Carlone, and L. Tentarelli,Ground states for the planar NLSE with a point defect as minimizers of the constrained energy, Calc. Var. Partial Differential Equations61(2022), Paper No. 195, 32. 24 REFERENCES

2022
[3]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden,Solvable models in quantum mechanics, Second, AMS Chelsea Publishing, Providence, RI, 2005

2005
[4]

Barbera, V

D. Barbera, V. Georgiev, and M. Rastrelli,On the Cauchy problem for the reaction-diffusion system with point-interaction inR 2, Analysis and Mathematical Physics16(2026), p. 29

2026
[5]

Berestycki and T

H. Berestycki and T. Cazenave,Instabilit´ e des ´ etats stationnaires dans les ´ equations de Schr¨ odinger et de Klein-Gordon non lin´ eaires, C. R. Acad. Sci. Paris S´ er. I Math.293 (1981), pp. 489–492

1981
[6]

F. Boni, D. Noja, and R. Scandone,Point interactions and singular solutions to semilinear elliptic equations, 2026, arXiv:2603.08487 [math.AP]

work page arXiv 2026
[7]

Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations, Univer- sitext, Springer Science & Business Media, 2011

H. Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations, Univer- sitext, Springer Science & Business Media, 2011

2011
[8]

Cacciapuoti, D

C. Cacciapuoti, D. Finco, and D. Noja,Well posedness of the nonlinear Schr¨ odinger equa- tion with isolated singularities, J. Differential Equations305(2021), pp. 288–318

2021
[9]

Cazenave and P.-L

T. Cazenave and P.-L. Lions,Orbital stability of standing waves for some nonlinear Schr¨ odinger equations, Comm. Math. Phys.85(1982), pp. 549–561

1982
[10]

H. D. Cornean, A. Michelangeli, and K. Yajima,Two-dimensional Schr¨ odinger operators with point interactions: threshold expansions, zero modes andL p-boundedness of wave op- erators, Rev. Math. Phys.31(2019), pp. 1950012, 32

2019
[11]

Dell’Antonio, A

G. Dell’Antonio, A. Michelangeli, R. Scandone, and K. Yajima,L p-boundedness of wave operators for the three-dimensional multi-centre point interaction, Ann. Henri Poincar´ e19 (2018), pp. 283–322

2018
[12]

Finco and D

D. Finco and D. Noja,Blow-up and instability of standing waves for the NLS with a point interaction in dimension two, Zeitschrift f¨ ur angewandte Mathematik und Physik74 (2023)

2023
[13]

Fukaya,Uniqueness and nondegeneracy of ground states for 2d-nonlinear scalar field equations with point interaction, NoDEA Nonlinear Differential Equations Appl.32(2025), Paper No

N. Fukaya,Uniqueness and nondegeneracy of ground states for 2d-nonlinear scalar field equations with point interaction, NoDEA Nonlinear Differential Equations Appl.32(2025), Paper No. 134, 24

2025
[14]

Fukaya, V

N. Fukaya, V. Georgiev, and M. Ikeda,On stability and instability of standing waves for 2d-nonlinear Schr¨ odinger equations with point interaction, J. Differential Equations321 (2022), pp. 258–295

2022
[15]

Fukaya and M

N. Fukaya and M. Hayashi,Instability of algebraic standing waves for nonlinear Schr¨ odinger equations with double power nonlinearities, Trans. Amer. Math. Soc.374(2021), pp. 1421– 1447

2021
[16]

Fukaya and M

N. Fukaya and M. Hayashi,Instability of stationary solutions for double power nonlinear Schr¨ odinger equations in one dimension, Partial Differ. Equ. Appl.6(2025), Paper No. 1

2025
[17]

Fukaya, M

N. Fukaya, M. Hayashi, and T. Inui,A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schr¨ odinger equation, Anal. PDE10(2017), pp. 1149– 1167

2017
[18]

Fukaya, M

N. Fukaya, M. Hayashi, and T. Inui,Traveling waves for a nonlinear Schr¨ odinger system with quadratic interaction, Math. Ann.388(2024), pp. 1357–1378

2024
[19]

Georgiev, A

V. Georgiev, A. Michelangeli, and R. Scandone,On fractional powers of singular perturba- tions of the Laplacian, J. Funct. Anal.275(2018), pp. 1551–1602

2018
[20]

Georgiev, A

V. Georgiev, A. Michelangeli, and R. Scandone,Standing waves and global well-posedness for the 2d Hartree equation with a point interaction, Comm. Partial Differential Equations 49(2024), pp. 242–278

2024
[21]

Georgiev and M

V. Georgiev and M. Rastrelli,Fractional Sobolev spaces for the singular-perturbed Laplace operator in theL p setting, Partial Differential Equations and Applications6(2025), pp. 1– 40

2025
[22]

Georgiev and M

V. Georgiev and M. Rastrelli,Sobolev spaces for singular perturbation of 2D Laplace oper- ator, Nonlinear Analysis251(2025), p. 113710. REFERENCES 25

2025
[23]

Guo,Orbital stability of solitary waves for generalized derivative nonlinear Schr¨ odinger equations in the endpoint case, Ann

Q. Guo,Orbital stability of solitary waves for generalized derivative nonlinear Schr¨ odinger equations in the endpoint case, Ann. Henri Poincar´ e19(2018), pp. 2701–2715

2018
[24]

Hayashi,Stability of algebraic solitons for nonlinear Schr¨ odinger equations of derivative type: variational approach, Ann

M. Hayashi,Stability of algebraic solitons for nonlinear Schr¨ odinger equations of derivative type: variational approach, Ann. Henri Poincar´ e23(2022), pp. 4249–4277

2022
[25]

Kaminaga and M

M. Kaminaga and M. Ohta,Stability of standing waves for nonlinear Schr¨ odinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J.26(2009), pp. 39–48

2009
[26]

Kwon and Y

S. Kwon and Y. Wu,Orbital stability of solitary waves for derivative nonlinear Schr¨ odinger equation, J. Anal. Math.135(2018), pp. 473–486

2018
[27]

Li and C

B. Li and C. Ning,Instability of the solitary wave solutions for the generalized derivative nonlinear Schr¨ odinger equation in the endpoint case, Nonlinear Anal.252(2025), Paper No. 113713, 14

2025
[28]

E. H. Lieb and M. Loss,Analysis, Second, vol. 14, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001

2001
[29]

Nakanishi,Global dynamics below excited solitons for the nonlinear Schr¨ odinger equation with a potential, J

K. Nakanishi,Global dynamics below excited solitons for the nonlinear Schr¨ odinger equation with a potential, J. Math. Soc. Japan69(2017), pp. 1353–1401

2017
[30]

C. Ning, M. Ohta, and Y. Wu,Instability of solitary wave solutions for derivative nonlinear Schr¨ odinger equation in endpoint case, J. Differential Equations262(2017), pp. 1671–1689

2017
[31]

Okazawa, T

N. Okazawa, T. Suzuki, and T. Yokota,Energy methods for abstract nonlinear Schr¨ odinger equations, Evol. Equ. Control Theory1(2012), pp. 337–354

2012
[32]

Osada and A

Y. Osada and A. Pomponio,Coupled nonlinear Schr¨ odinger equations with point interac- tion: existence and asymptotic behaviour, Z. Angew. Math. Phys.76(2025), Paper No. 218, 22

2025
[33]

Reed and B

M. Reed and B. Simon,Methods of modern mathematical physics. II. Fourier analysis, self- adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975, pp. xv+361

1975
[34]

V´ eron,Comportement asymptotique des solutions d’´ equations elliptiques semi-lin´ eaires dansR N, French, Annali di Matematica Pura ed Applicata, 4th ser.127(1981), pp

L. V´ eron,Comportement asymptotique des solutions d’´ equations elliptiques semi-lin´ eaires dansR N, French, Annali di Matematica Pura ed Applicata, 4th ser.127(1981), pp. 25–50. Regional ICT Research Center for Human, Industry and Future, The Univer- sity of Shiga Prefecture, Shiga 522-8533, Japan Osaka Central Advanced Mathematical Institute, Osaka Me...

1981

[1] [1]

Adami, F

R. Adami, F. Boni, R. Carlone, and L. Tentarelli,Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect, J. Math. Phys.63(2022), Paper No. 071501, 16

2022

[2] [2]

Adami, F

R. Adami, F. Boni, R. Carlone, and L. Tentarelli,Ground states for the planar NLSE with a point defect as minimizers of the constrained energy, Calc. Var. Partial Differential Equations61(2022), Paper No. 195, 32. 24 REFERENCES

2022

[3] [3]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden,Solvable models in quantum mechanics, Second, AMS Chelsea Publishing, Providence, RI, 2005

2005

[4] [4]

Barbera, V

D. Barbera, V. Georgiev, and M. Rastrelli,On the Cauchy problem for the reaction-diffusion system with point-interaction inR 2, Analysis and Mathematical Physics16(2026), p. 29

2026

[5] [5]

Berestycki and T

H. Berestycki and T. Cazenave,Instabilit´ e des ´ etats stationnaires dans les ´ equations de Schr¨ odinger et de Klein-Gordon non lin´ eaires, C. R. Acad. Sci. Paris S´ er. I Math.293 (1981), pp. 489–492

1981

[6] [6]

F. Boni, D. Noja, and R. Scandone,Point interactions and singular solutions to semilinear elliptic equations, 2026, arXiv:2603.08487 [math.AP]

work page arXiv 2026

[7] [7]

Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations, Univer- sitext, Springer Science & Business Media, 2011

H. Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations, Univer- sitext, Springer Science & Business Media, 2011

2011

[8] [8]

Cacciapuoti, D

C. Cacciapuoti, D. Finco, and D. Noja,Well posedness of the nonlinear Schr¨ odinger equa- tion with isolated singularities, J. Differential Equations305(2021), pp. 288–318

2021

[9] [9]

Cazenave and P.-L

T. Cazenave and P.-L. Lions,Orbital stability of standing waves for some nonlinear Schr¨ odinger equations, Comm. Math. Phys.85(1982), pp. 549–561

1982

[10] [10]

H. D. Cornean, A. Michelangeli, and K. Yajima,Two-dimensional Schr¨ odinger operators with point interactions: threshold expansions, zero modes andL p-boundedness of wave op- erators, Rev. Math. Phys.31(2019), pp. 1950012, 32

2019

[11] [11]

Dell’Antonio, A

G. Dell’Antonio, A. Michelangeli, R. Scandone, and K. Yajima,L p-boundedness of wave operators for the three-dimensional multi-centre point interaction, Ann. Henri Poincar´ e19 (2018), pp. 283–322

2018

[12] [12]

Finco and D

D. Finco and D. Noja,Blow-up and instability of standing waves for the NLS with a point interaction in dimension two, Zeitschrift f¨ ur angewandte Mathematik und Physik74 (2023)

2023

[13] [13]

Fukaya,Uniqueness and nondegeneracy of ground states for 2d-nonlinear scalar field equations with point interaction, NoDEA Nonlinear Differential Equations Appl.32(2025), Paper No

N. Fukaya,Uniqueness and nondegeneracy of ground states for 2d-nonlinear scalar field equations with point interaction, NoDEA Nonlinear Differential Equations Appl.32(2025), Paper No. 134, 24

2025

[14] [14]

Fukaya, V

N. Fukaya, V. Georgiev, and M. Ikeda,On stability and instability of standing waves for 2d-nonlinear Schr¨ odinger equations with point interaction, J. Differential Equations321 (2022), pp. 258–295

2022

[15] [15]

Fukaya and M

N. Fukaya and M. Hayashi,Instability of algebraic standing waves for nonlinear Schr¨ odinger equations with double power nonlinearities, Trans. Amer. Math. Soc.374(2021), pp. 1421– 1447

2021

[16] [16]

Fukaya and M

N. Fukaya and M. Hayashi,Instability of stationary solutions for double power nonlinear Schr¨ odinger equations in one dimension, Partial Differ. Equ. Appl.6(2025), Paper No. 1

2025

[17] [17]

Fukaya, M

N. Fukaya, M. Hayashi, and T. Inui,A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schr¨ odinger equation, Anal. PDE10(2017), pp. 1149– 1167

2017

[18] [18]

Fukaya, M

N. Fukaya, M. Hayashi, and T. Inui,Traveling waves for a nonlinear Schr¨ odinger system with quadratic interaction, Math. Ann.388(2024), pp. 1357–1378

2024

[19] [19]

Georgiev, A

V. Georgiev, A. Michelangeli, and R. Scandone,On fractional powers of singular perturba- tions of the Laplacian, J. Funct. Anal.275(2018), pp. 1551–1602

2018

[20] [20]

Georgiev, A

V. Georgiev, A. Michelangeli, and R. Scandone,Standing waves and global well-posedness for the 2d Hartree equation with a point interaction, Comm. Partial Differential Equations 49(2024), pp. 242–278

2024

[21] [21]

Georgiev and M

V. Georgiev and M. Rastrelli,Fractional Sobolev spaces for the singular-perturbed Laplace operator in theL p setting, Partial Differential Equations and Applications6(2025), pp. 1– 40

2025

[22] [22]

Georgiev and M

V. Georgiev and M. Rastrelli,Sobolev spaces for singular perturbation of 2D Laplace oper- ator, Nonlinear Analysis251(2025), p. 113710. REFERENCES 25

2025

[23] [23]

Guo,Orbital stability of solitary waves for generalized derivative nonlinear Schr¨ odinger equations in the endpoint case, Ann

Q. Guo,Orbital stability of solitary waves for generalized derivative nonlinear Schr¨ odinger equations in the endpoint case, Ann. Henri Poincar´ e19(2018), pp. 2701–2715

2018

[24] [24]

Hayashi,Stability of algebraic solitons for nonlinear Schr¨ odinger equations of derivative type: variational approach, Ann

M. Hayashi,Stability of algebraic solitons for nonlinear Schr¨ odinger equations of derivative type: variational approach, Ann. Henri Poincar´ e23(2022), pp. 4249–4277

2022

[25] [25]

Kaminaga and M

M. Kaminaga and M. Ohta,Stability of standing waves for nonlinear Schr¨ odinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J.26(2009), pp. 39–48

2009

[26] [26]

Kwon and Y

S. Kwon and Y. Wu,Orbital stability of solitary waves for derivative nonlinear Schr¨ odinger equation, J. Anal. Math.135(2018), pp. 473–486

2018

[27] [27]

Li and C

B. Li and C. Ning,Instability of the solitary wave solutions for the generalized derivative nonlinear Schr¨ odinger equation in the endpoint case, Nonlinear Anal.252(2025), Paper No. 113713, 14

2025

[28] [28]

E. H. Lieb and M. Loss,Analysis, Second, vol. 14, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001

2001

[29] [29]

Nakanishi,Global dynamics below excited solitons for the nonlinear Schr¨ odinger equation with a potential, J

K. Nakanishi,Global dynamics below excited solitons for the nonlinear Schr¨ odinger equation with a potential, J. Math. Soc. Japan69(2017), pp. 1353–1401

2017

[30] [30]

C. Ning, M. Ohta, and Y. Wu,Instability of solitary wave solutions for derivative nonlinear Schr¨ odinger equation in endpoint case, J. Differential Equations262(2017), pp. 1671–1689

2017

[31] [31]

Okazawa, T

N. Okazawa, T. Suzuki, and T. Yokota,Energy methods for abstract nonlinear Schr¨ odinger equations, Evol. Equ. Control Theory1(2012), pp. 337–354

2012

[32] [32]

Osada and A

Y. Osada and A. Pomponio,Coupled nonlinear Schr¨ odinger equations with point interac- tion: existence and asymptotic behaviour, Z. Angew. Math. Phys.76(2025), Paper No. 218, 22

2025

[33] [33]

Reed and B

M. Reed and B. Simon,Methods of modern mathematical physics. II. Fourier analysis, self- adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975, pp. xv+361

1975

[34] [34]

V´ eron,Comportement asymptotique des solutions d’´ equations elliptiques semi-lin´ eaires dansR N, French, Annali di Matematica Pura ed Applicata, 4th ser.127(1981), pp

L. V´ eron,Comportement asymptotique des solutions d’´ equations elliptiques semi-lin´ eaires dansR N, French, Annali di Matematica Pura ed Applicata, 4th ser.127(1981), pp. 25–50. Regional ICT Research Center for Human, Industry and Future, The Univer- sity of Shiga Prefecture, Shiga 522-8533, Japan Osaka Central Advanced Mathematical Institute, Osaka Me...

1981