A New Treatment for Some Periodic Schrödinger Operators II: The Wave Function

Abstract

Following the approach of our previous paper we continue to study the asymptotic solution of periodic Schrödinger operators. Using the eigenvalues obtained earlier the corresponding asymptotic wave functions are derived. This gives further evidence in favor of the monodromy relations for the Floquet exponent proposed in the previous paper. In particular, the large energy asymptotic wave functions are related to the instanton partition function of N=2 supersymmetric gauge theory with surface operator. A relevant number theoretic dessert is appended. (2010): 35P20, 33E10, 34E10. Mathematics Subject Classification

A New Treatment for Some Periodic Schrödinger Operators I: The Eigenvalue

“…In this paper we demonstrate this fact by their eigenvalues, in the second paper Ref. [14] we provide further evidences by eigenfunctions of certain typical periodic Schrödinger operators. This paper is motivated by our previous works attempting to examine in detail a few simple examples of the Gauge/Bethe correspondence, proposed by Nekrasov and Shatashvili [15], where the infrared dynamics of some quantum gauge theories is related to the spectral problem of stationary Schrödinger equation with periodic potentials.…”

“…They also can be rewritten in terms of the Eisenstein series E 2 , E 4 , E 6 , or in terms of the theta constants ϑ r (q), = 1, 2, 3, 4. We denote the Floquet exponent of wave function in (14) as ν, i.e. ψ(x+2ω 1 ) = exp(i2νω 1 )ψ(x), then the asymptotical expansion for λ is…”

“…In this Appendix we normalize the convention for some elliptic function formulae useful in this paper and the subsequent paper [14]. The references are [1,3,7,10,34].…”

Quantum periods and prepotential in N = 2 $$ \mathcal{N}=2 $$ SU(2) SQCD

“…This was generalized to N = 2 SU(N ) SQCD [17]. Note that the SW curve for N = 2 * SU(2) gauge theory corresponds to the Lamé equation and the deformed period integrals also have been calculated by using the WKB analysis [18,19]. One can derive the Bohr-Sommerfeld quantization conditions which are nothing but the Baxter's T-Q relations of the integrable system [17,20,21].…”

Spectra of elliptic potentials and supersymmetric gauge theories

“…The DTV potential is a four components generalization of the Lamé potential, which in turn is the elliptic generalization of the Mathieu potential. Asymptotic solutions of the Mathieu equation were known for many years, asymptotic solutions of the Lamé equation and the ellipsoidal wave equation are also known now [11][12][13][14][15]. But asymptotic solutions of the same nature for the Heun equation become difficult to compute.…”