Semi-Classical Analysis for the Schrödinger Operator and Applications

Helffer, Bernard

doi:10.1007/bfb0078115

Cited by 307 publications

(338 citation statements)

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“…the Betti numbers of X. This way Witten proved in [38] the Morse inequalities relating the Betti numbers of X to the numbers of critical points of f of various indices (see also [21,22]). …”

Section: Recollections On Morse Theorymentioning

confidence: 95%

Instantons beyond topological theory. I

Frenkel¹,

Losev²,

Nekrasov³

2011

J. Inst. Math. Jussieu

110

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Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyse the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kähler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.In Part I we describe the class of models under study and present our results in the case of onedimensional (quantum mechanical) models, which is quite representative and at the same time simple enough to analyse explicitly. Part II will be devoted to supersymmetric two-dimensional sigma models and four-dimensional Yang-Mills theory. In Part III we will discuss non-supersymmetric models.

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Section: Recollections On Morse Theorymentioning

confidence: 95%

Instantons beyond topological theory. I

Frenkel¹,

Losev²,

Nekrasov³

2011

J. Inst. Math. Jussieu

110

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“…Let us denote by λ tδu n phq the n-th eigenvalue of the corresponding operator Ă L tδu h . Using smooth cut-off functions, the min-max principle and the Agmon estimates of Proposition 4.1, it is then standard to deduce the following proposition (see [10]). Proposition 4.2.…”

Section: The Operator Near the Boundary In Tubular Coordinatesmentioning

confidence: 99%

On the MIT Bag Model in the Non-relativistic Limit

Arrizabalaga

Treust

Raymond

2017

Commun. Math. Phys.

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“…For each couple {θ 1 , θ 2 }, Q θ1,θ2 (V) identifies with a, possibly non-selfadjoint, extension of the Hermitian operator Q 0 (V) 14) and defines an explicitly solvable model w.r.t. the selfadjoint Hamiltonian Q 0,0 (V).…”

Section: Schrödinger Operators With Non-mixed Interface Conditionsmentioning

confidence: 99%

“…Using (2.11) and (2.13)-(2.13), the linear relations (1.10) rephrase as 14) which leads to the equivalent definition…”

Section: Notationmentioning

confidence: 99%

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

Mantile

2013

Journal of Mathematical Physics

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We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schrödinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of non-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schrödinger operator with a 'physical' Hamiltonian. For small values of a deformation parameter 'θ', this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |θ| uniformly in time in the L 2 -operator norm.

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Semi-Classical Analysis for the Schrödinger Operator and Applications

Cited by 307 publications

References 0 publications

Instantons beyond topological theory. I

Instantons beyond topological theory. I

On the MIT Bag Model in the Non-relativistic Limit

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

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