Spectral theory: minimizing Robin gaps, and singular limits for sign-changing weights

Kielty, Derek

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https://hdl.handle.net/2142/115344 Copy

Description

Title

Spectral theory: minimizing Robin gaps, and singular limits for sign-changing weights

Author(s)

Kielty, Derek

Issue Date

2022-03-18

Director of Research (if dissertation) or Advisor (if thesis)

Laugesen, Richard S

Doctoral Committee Chair(s)

Bronski, Jared

Committee Member(s)

• Kirr, Eduard
• Zharnitsky, Vadim

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• spectral theory
• partial differential equations
• ordinary differential equations
• spectral gaps
• indefinite weight

Abstract

The first part of this thesis considers the spectral gap of Schr\"odinger eigenvalue problems. Chapter 2 proves monotonicity results for the spectral gap with respect to the potential and the Robin parameter, for 1-dimensional Schr\"odinger operators. In particular, these results lead to sharp lower bounds on the spectral gap of operators with single-well potentials that generalize work of Cheng et al.\ and Horv\'ath. In higher dimensions, the spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter an analogous sharp lower bound on the spectral gap is conjectured. Chapter 3 shows the extension of this conjecture to each fixed negative Robin parameter fails, by constructing a family of double cone domains with an exponentially small spectral gap --- even though the first two eigenvalues themselves are polynomially large. The second part of this thesis considers the eigenvalue problem generated by a fixed differential operator with a sign-changing weight multiplying the eigenvalue term. Chapter 4 proves that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. The underlying theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. The results are applied to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type problems, and a problem where the eigenvalue appears in both the equation and the boundary condition.

Graduation Semester

2022-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Derek Kielty

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