### Abstract

At any resolution level of wavelet expansions the physical observable of the kinetic energy is represented by an infinite matrix which is "canonically" chosen as the projection of the operator - Δ/2 onto the subspace of the given resolution. It is shown, that this canonical choice is not optimal, as the regular grid of the basis set introduces an artificial consequence of its periodicity, and it is only a particular member of possible operator representations. We present an explicit method of preparing a near optimal kinetic energy matrix which leads to more appropriate results in numerical wavelet based calculations. This construction works even in those cases, where the usual definition is unusable (i.e., the derivative of the basis functions does not exist). It is also shown, that building an effective kinetic energy matrix is equivalent to the renormalization of the kinetic energy by a momentum dependent effective mass compensating for artificial periodicity effects.

Original language | English |
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Pages (from-to) | 261-282 |

Number of pages | 22 |

Journal | Journal of Mathematical Chemistry |

Volume | 46 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 2009 |

### Fingerprint

### Keywords

- Kinetic energy operator
- Operator representation
- Wavelet analysis

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*46*(1), 261-282. https://doi.org/10.1007/s10910-008-9458-4

**The kinetic energy operator in the subspaces of wavelet analysis.** / Pipek, János; Nagy, Szilvia.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 46, no. 1, pp. 261-282. https://doi.org/10.1007/s10910-008-9458-4

}

TY - JOUR

T1 - The kinetic energy operator in the subspaces of wavelet analysis

AU - Pipek, János

AU - Nagy, Szilvia

PY - 2009/6

Y1 - 2009/6

N2 - At any resolution level of wavelet expansions the physical observable of the kinetic energy is represented by an infinite matrix which is "canonically" chosen as the projection of the operator - Δ/2 onto the subspace of the given resolution. It is shown, that this canonical choice is not optimal, as the regular grid of the basis set introduces an artificial consequence of its periodicity, and it is only a particular member of possible operator representations. We present an explicit method of preparing a near optimal kinetic energy matrix which leads to more appropriate results in numerical wavelet based calculations. This construction works even in those cases, where the usual definition is unusable (i.e., the derivative of the basis functions does not exist). It is also shown, that building an effective kinetic energy matrix is equivalent to the renormalization of the kinetic energy by a momentum dependent effective mass compensating for artificial periodicity effects.

AB - At any resolution level of wavelet expansions the physical observable of the kinetic energy is represented by an infinite matrix which is "canonically" chosen as the projection of the operator - Δ/2 onto the subspace of the given resolution. It is shown, that this canonical choice is not optimal, as the regular grid of the basis set introduces an artificial consequence of its periodicity, and it is only a particular member of possible operator representations. We present an explicit method of preparing a near optimal kinetic energy matrix which leads to more appropriate results in numerical wavelet based calculations. This construction works even in those cases, where the usual definition is unusable (i.e., the derivative of the basis functions does not exist). It is also shown, that building an effective kinetic energy matrix is equivalent to the renormalization of the kinetic energy by a momentum dependent effective mass compensating for artificial periodicity effects.

KW - Kinetic energy operator

KW - Operator representation

KW - Wavelet analysis

UR - http://www.scopus.com/inward/record.url?scp=64249137142&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64249137142&partnerID=8YFLogxK

U2 - 10.1007/s10910-008-9458-4

DO - 10.1007/s10910-008-9458-4

M3 - Article

VL - 46

SP - 261

EP - 282

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 1

ER -