Schrödinger equation with finitely many $δ$-interactions: closed form, integral and series representations for solutions
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866911837824483328 |
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| author | Kravchenko, Vladislav V. Vicente-Benítez, Víctor A. |
| author_facet | Kravchenko, Vladislav V. Vicente-Benítez, Víctor A. |
| contents | A closed form solution for the one-dimensional Schrödinger equation with a finite number of $δ$-interactions \[ \mathbf{L}_{q,\mathfrak{I}_{N}}y:=-y^{\prime\prime}+\left( q(x)+\sum _{k=1}^{N}α_{k}δ(x-x_{k})\right) y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] is presented in terms of the solution of the unperturbed equation \[ \mathbf{L}_{q}y:=-y^{\prime\prime}+q(x)y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] and a corresponding transmutation operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ transmutes the second derivative into the Schrödinger operator $\mathbf{L}_{q,\mathfrak{I}_{N}}$ on a Sobolev space $H^{2}$. A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_13218 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Schrödinger equation with finitely many $δ$-interactions: closed form, integral and series representations for solutions Kravchenko, Vladislav V. Vicente-Benítez, Víctor A. Classical Analysis and ODEs Mathematical Physics 34A25, 34A45, 46F10, 47G10, 81Q05 A closed form solution for the one-dimensional Schrödinger equation with a finite number of $δ$-interactions \[ \mathbf{L}_{q,\mathfrak{I}_{N}}y:=-y^{\prime\prime}+\left( q(x)+\sum _{k=1}^{N}α_{k}δ(x-x_{k})\right) y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] is presented in terms of the solution of the unperturbed equation \[ \mathbf{L}_{q}y:=-y^{\prime\prime}+q(x)y=λy,\quad0<x<b,\;λ\in\mathbb{C}% \] and a corresponding transmutation operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator $\mathbf{T}_{\mathfrak{I}_{N}}^{f}$ transmutes the second derivative into the Schrödinger operator $\mathbf{L}_{q,\mathfrak{I}_{N}}$ on a Sobolev space $H^{2}$. A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived. |
| title | Schrödinger equation with finitely many $δ$-interactions: closed form, integral and series representations for solutions |
| topic | Classical Analysis and ODEs Mathematical Physics 34A25, 34A45, 46F10, 47G10, 81Q05 |
| url | https://arxiv.org/abs/2302.13218 |