# Semiclassical spectral analysis of the Bochner–Schrödinger operator on symplectic manifolds of bounded geometry

@article{Kordyukov2021SemiclassicalSA, title={Semiclassical spectral analysis of the Bochner–Schr{\"o}dinger operator on symplectic manifolds of bounded geometry}, author={Yuri A. Kordyukov}, journal={Analysis and Mathematical Physics}, year={2021} }

We study the Bochner-Schrödinger operatorHp = 1 p ∆ p + V on high tensor powers of a positive line bundle L on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are…

## 7 Citations

### Semiclassical asymptotic expansions for functions of the Bochner-Schr\"odinger operator

- Mathematics
- 2022

. The Bochner-Schr¨odinger operator H p = 1 p ∆ L p ⊗ E + V on tensor powers L p of a Hermitian line bundle L twisted by a Hermitian vector bundle E on a Riemannian manifold of bounded geometry is…

### On the spectrum of non degenerate magnetic Laplacian

- Mathematics
- 2021

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non degenerate. Under a general condition, the Laplacian acting on high tensor powers of the bundle exhibits…

### Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

- MathematicsJournal of Spectral Theory
- 2022

In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner…

### Landau levels on a compact manifold

- Mathematics
- 2020

We consider a magnetic Laplacian on a compact manifold, with a constant non-degenerate magnetic ﬁeld. In the large ﬁeld limit, it is known that the eigenvalues are grouped in clusters, the…

### Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry

- Mathematics
- 2021

We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner…

### Квантование по Березину-Теплицу на симплектических многообразиях ограниченной геометрии

- Matematicheskie Zametki
- 2022

Мы развиваем теорию квантования по Березину-Теплицу
на симплектических многообразиях ограниченной геометрии.
Пространством квантования является
подходяшее собственное подпространство…

### Trace formula for the magnetic Laplacian at zero energy level

- Mathematics
- 2022

. The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of the semiclassical…

## References

SHOWING 1-10 OF 30 REFERENCES

### Generalized Bergman kernels on symplectic manifolds of bounded geometry

- MathematicsCommunications in Partial Differential Equations
- 2019

Abstract We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded…

### On the spectrum of non degenerate magnetic Laplacian

- Mathematics
- 2021

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non degenerate. Under a general condition, the Laplacian acting on high tensor powers of the bundle exhibits…

### Proper uniform pseudodifferential operators on unimodular Lie groups

- Mathematics
- 1989

An algebra of proper pseudodifferential operators on an arbitrary unimodular Lie group is constructed. This algebra is a generalization of a well-known algebra of operators with uniform estimates of…

### Lp-Theory of elliptic differential operators on manifolds of bounded geometry

- Mathematics
- 1991

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in Lpspaces. We prove the coincidence of…

### Berezin–Toeplitz Quantization for Eigenstates of the Bochner Laplacian on Symplectic Manifolds

- Mathematics
- 2017

We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic…

### Semiclassical Spectral Asymptotics for a Magnetic Schrödinger Operator with Non-vanishing Magnetic Field

- Mathematics
- 2014

We consider a magnetic Schrodinger operator Hh on a compact Riemannian manifold, depending on the semiclassical parameter h > 0. We assume that there is no electric field. We suppose that the minimal…

### Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

- MathematicsJournal of Spectral Theory
- 2022

In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner…

### A semiclassical Birkhoff normal form for symplectic magnetic wells

- MathematicsJournal of Spectral Theory
- 2022

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{o}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional…

### On asymptotic expansions of generalized Bergman kernels on symplectic manifolds

- MathematicsSt. Petersburg Mathematical Journal
- 2019

A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a…

### Landau levels on a compact manifold

- Mathematics
- 2020

We consider a magnetic Laplacian on a compact manifold, with a constant non-degenerate magnetic ﬁeld. In the large ﬁeld limit, it is known that the eigenvalues are grouped in clusters, the…