Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency
We investigate the nonstationary parabolic Anderson problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow>&l...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-02-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/5/685 |
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| Summary: | We investigate the nonstationary parabolic Anderson problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>ϰ</mi><mi mathvariant="script">L</mi><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><msub><mi>ξ</mi><mi>t</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mspace width="1.em"></mspace><mi>u</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>≡</mo><mn>1</mn><mo>,</mo><mspace width="1.em"></mspace><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow><mo>×</mo><msup><mi>Z</mi><mi>d</mi></msup></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϰ</mi><mi mathvariant="script">L</mi></mrow></semantics></math></inline-formula> denotes a nonlocal Laplacian and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mi>t</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is a correlated white-noise potential. The irregularity of the solution is linked to the upper spectrum of certain multiparticle Schrödinger operators that govern the moment functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>p</mi></msub><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>p</mi></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo>⟨</mo><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>⋯</mo><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><msub><mi>x</mi><mi>p</mi></msub><mo stretchy="false">)</mo></mrow><mo>⟩</mo></mrow></mrow></semantics></math></inline-formula>. First, we establish a weak form of intermittency under broad assumptions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">L</mi></semantics></math></inline-formula> and on a positive-definite noise correlator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We then examine strong intermittency, which emerges from the existence of a positive eigenvalue in a related lattice Schrödinger-type operator with potential <i>B</i>. Here, <i>B</i> does not have to be positive definite but must satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∑</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The presence of such an eigenvalue intensifies the growth properties of the second moment <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mn>2</mn></msub></semantics></math></inline-formula>, revealing a more pronounced intermittent regime. |
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| ISSN: | 2227-7390 |