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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2025-02-01
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| author | Dan Han Stanislav Molchanov Boris Vainberg |
| author_facet | Dan Han Stanislav Molchanov Boris Vainberg |
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| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-9e280f4af9814b50b4322db8cd19ccf9 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-9e280f4af9814b50b4322db8cd19ccf92025-08-20T02:59:00ZengMDPI AGMathematics2227-73902025-02-0113568510.3390/math13050685Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and IntermittencyDan Han0Stanislav Molchanov1Boris Vainberg2Department of Mathematics, University of Louisville, Louisville, KY 40292, USADepartment of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USADepartment of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USAWe 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.https://www.mdpi.com/2227-7390/13/5/685spectral analysisnonstationaryAnderson modelintermittency |
| spellingShingle | Dan Han Stanislav Molchanov Boris Vainberg Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency Mathematics spectral analysis nonstationary Anderson model intermittency |
| title | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency |
| title_full | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency |
| title_fullStr | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency |
| title_full_unstemmed | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency |
| title_short | Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency |
| title_sort | spectral analysis of lattice schrodinger type operators associated with the nonstationary anderson model and intermittency |
| topic | spectral analysis nonstationary Anderson model intermittency |
| url | https://www.mdpi.com/2227-7390/13/5/685 |
| work_keys_str_mv | AT danhan spectralanalysisoflatticeschrodingertypeoperatorsassociatedwiththenonstationaryandersonmodelandintermittency AT stanislavmolchanov spectralanalysisoflatticeschrodingertypeoperatorsassociatedwiththenonstationaryandersonmodelandintermittency AT borisvainberg spectralanalysisoflatticeschrodingertypeoperatorsassociatedwiththenonstationaryandersonmodelandintermittency |