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Schrödinger operators periodic in octants

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We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

TidsskriftLetters in Mathematical Physics
StatusUdgivet - apr. 2021

Bibliografisk note

Funding Information:
Various parts of this paper were written during Evgeny Korotyaev’s stay as a VELUX Visiting Professor at the Department of Mathematics, Aarhus University, Denmark. He is grateful to the institute for the hospitality. In addition, our study was supported by the RFBR grant No. 19-01-00094 and the Danish Free Science Council grant No1323-00360. The authors would like to thank the referee for useful comments and Natalia Saburova for Fig. .

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.

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