The properties of the distribution of Gaussian packets on a spatial network
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2011
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| _version_ | 1866917601122189312 |
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| author | Chernyshev, V. L. Tolchennikov, A. A. |
| author_facet | Chernyshev, V. L. Tolchennikov, A. A. |
| contents | The article deals with the description of the statistical behavior of Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of the Cauchy problem for the Schrödinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges ("edge travel times") are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The article also discusses a situation where the times of passage over the edges are not linearly independent over the rationals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1111_3945 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | The properties of the distribution of Gaussian packets on a spatial network Chernyshev, V. L. Tolchennikov, A. A. Mathematical Physics 34B45, 35R02, 81Q20, 81Q35, 05C99 The article deals with the description of the statistical behavior of Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of the Cauchy problem for the Schrödinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges ("edge travel times") are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The article also discusses a situation where the times of passage over the edges are not linearly independent over the rationals. |
| title | The properties of the distribution of Gaussian packets on a spatial network |
| topic | Mathematical Physics 34B45, 35R02, 81Q20, 81Q35, 05C99 |
| url | https://arxiv.org/abs/1111.3945 |