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Международная молодежная конференция «Геометрия и управление»
15 апреля 2014 г. 17:00, Стендовые доклады, г. Москва, МИАН
 


Finite-Gap 2D-Schrödinger Operators with Elliptic Coefficient

Bayan Saparbayeva

Sobolev Institute of Mathematics, Novosibirsk, Russian
Материалы:
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Аннотация: In general case the potential of the finite-gap Schrödinger operator $-\frac{\partial^2}{\partial x^2}+u(x)$ is expressed in terms of theta function of the spectral curve [5]. At the same time there are examples of finite-gap operators with elliptic potentials, for example, the Lamé operators $-\frac{\partial^2}{\partial x^2}+g(g+1)\wp(x)$ or the Treibich-Verdier operator $-\frac{\partial^2}{\partial x^2}+\sum_{i=0}^3a_i(a_i+1)\wp(x+\omega_i)$, where $\omega_i$ are semi-periods. Theorems 1 and 2 show that the same phenomena are possible in two-dimensional case.
$ $
Theorem 1. The Schrödinger operator
$$ \tag{1} H=\frac{\partial^2}{\partial z\partial\bar{z}} +a(\frac{\sqrt{g_0}-\wp'(az+b\bar{z})}{2\wp(az+b\bar{z})})\frac{\partial}{\partial\bar{z}}-\frac{bg(g+1)\wp(az+b\bar{z})}{2a} $$
is finite-gap, where $\wp$ is elliptic Weierstrass function satisfying the equation
\begin{equation*}(\wp'(z))^2=\frac{2g(g+1)}{a^2}\wp(z)^3+g_2\wp(z)^2+g_1\wp(z)+g_0.\end{equation*}
The spectral curve of the operator $H$ is a hyperelliptic curve with genus $g$.
$ $
Thus for the operator $H$ theta functional formulas for the coefficients is reduced to the simpler formulas (1). Note that $H$ satisfies the identity
\begin{equation*}[H, -\frac{\partial^2}{\partial z^2}+g(g+1)\wp(az+b\bar{z})]=-2a(\frac{\partial}{\partial z}(\frac{\sqrt{g_0}-\wp'(az+b\bar{z})}{2\wp(az+b\bar{z})}))H. \end{equation*}

Theorem 2. The Schrödinger operator
\begin{equation*}H=\frac{\partial^2}{\partial z\partial\bar{z}} +\frac{7a\wp'(az+b\bar{z})}{20g_2a^2-14\wp(az+b\bar{z})}\frac{\partial}{\partial\bar{z}}+ \frac{b\wp(az+b\bar{z})}{2a}\end{equation*}
is finite-gap, where $\wp$ is elliptic Weierstrass function satisfying the equation
\begin{equation*}(\wp'(z))^2=-\frac{1}{2a^2}\wp(z)^3+g_2\wp(z)^2-(\frac{7g_0}{10g_2a^2}+\frac{20g_2^2a^2}{49})\wp(z)+g_0. \end{equation*}


Материалы: abstract.pdf (73.2 Kb)

Язык доклада: английский

Список литературы
  1. B.A. Dubrovin, I.M. Krichever, S.P. Novikov, The Schrödinger equation in a periodic field and Riemann surfaces. //Dokl.Akad.Nauk. SSSR. 1977. 229, 1. 15–18.
  2. I.A.Taimanov. Elliptic solutions of nonlinear equations. //Teoret. Mat. Fiz. 1990. 84, 1. 38–45.
  3. A.R. Its, V.B. Matveev. Schro?dinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation. //Teoret. Mat. Fiz. 1975. 23, 1. 51–68.  mathscinet


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