###
North-Holland Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear *Schr*6*dinger* equation*

, 1991

"... The nonlinear Schr6dinger (NLS) equation i ~ t + V2 ~ + al~lS ~ = 0 is a canonical and universal equation which is of major importance in continuum echanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension an ..."

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The nonlinear

*Schr*6*dinger*(NLS) equation i ~ t + V2 ~ + al~lS ~ = 0 is a canonical and universal equation which is of major importance in continuum echanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension###
On the *Discrete* Spectrum of *Schr*\"odinger *Operators* with Perturbed Magnetic Fields

"... In this note we study a Schr\"odinger operator with a magnetic field: (1.1) $H=(-i\nabla-b(x))^{2}+V(x)$ defined on $C_{o}^{\infty}(R^{3}) $ , where $V\in L_{loc}^{2}(R^{3}) $ is a scalar potential and $b\in C^{1}(R^{3})^{3} $ is a vector potential, both of which are real-valued, $andarrow B(x) ..."

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In this note we study a

*Schr*\"odinger*operator*with a magnetic field: (1.1) $H=(-i\nabla-b(x))^{2}+V(x)$ defined on $C_{o}^{\infty}(R^{3}) $ , where $V\in L_{loc}^{2}(R^{3}) $ is a scalar potential and $b\in C^{1}(R^{3})^{3} $ is a vector potential, both of which are real-valued, $andarrow B###
*Schr*\"Odinger *Operators* with Periodic Potentials

"... (Kazushi Yositomi) 1 Introduction and main results Schr\"odinger $H(\lambda)=(D_{x_{1}}+bx_{2})^{2}+(D_{x_{2}}-bx_{1})2+\lambda^{2}V(x) $ in $L^{2}(\mathrm{R}^{2})$ $D_{x_{j}}=-i\partial/\partial x_{j}(j=1,2), $ $\lambda $ $b\in \mathrm{R}$ $H(\lambda) $ B $=-2bdx_{1}\wedge dx_{2}$ ..."

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(Kazushi Yositomi) 1 Introduction and main results

*Schr*\"odinger $H(\lambda)=(D_{x_{1}}+bx_{2})^{2}+(D_{x_{2}}-bx_{1})2+\lambda^{2}V(x) $ in $L^{2}(\mathrm{R}^{2})$ $D_{x_{j}}=-i\partial/\partial x_{j}(j=1,2), $ $\lambda $ $b\in \mathrm{R}$ $H(\lambda) $ B $=-2bdx_{1}\wedge dx_{2}$###
A THEOREM ON "LOCALIZED" SELF-ADJOINTNESS OF $CHRDINGER *OPERATORS* WITH L [oc-POTENTIALS

, 1982

"... ABSTRACT. We prove a result which concludes the self-adjointness of a Schr6"dinger operator from the self-adjointness of the associated "localized" Schr6"dinger oper- ..."

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ABSTRACT. We prove a result which concludes the self-adjointness of a

*Schr*6"*dinger**operator*from the self-adjointness of the associated "localized"*Schr*6"*dinger**oper*-###
Random *Schr*\"odinger *Operator* Lifschitz Tail (TADASHI SHIMA)

"... nested fractal random Schr\"odinger operator Lifschitz tail ([7]) $\circ $ ( 2 Sierpinski gasket-$1_{-}\wedge $ Paluba [6] ) [7] $pre- nes\{ed $ fractal Anderson model (Anderson model $Z^{d} $ random potential $Schrodinge\cdot|_{-}^{\backslash} $ operator ‘ $Z^{d} $ (infinite) pre-nested fract ..."

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nested fractal random

*Schr*\"odinger*operator*Lifschitz tail ([7]) $\circ $ ( 2 Sierpinski gasket-$1_{-}\wedge $ Paluba [6] ) [7] $pre- nes\{ed $ fractal Anderson model (Anderson model $Z^{d} $ random potential $Schrodinge\cdot|_{-}^{\backslash} $*operator*‘ $Z^{d} $ (infinite) pre###
QUASI-CLASSICAL VERSUS NON-CLASSICALSPECTRAL ASYMPTOTICS FOR MAGNETIC SCHRö*DINGER* *OPERATORS* WITH DECREASING POTENTIALS

"... We consider the Schrödinger operator H (V) on L²(R³) or L³(R¼) withconstant magnetic field, and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V) near the boundary points of it ..."

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We consider the Schrödinger

*operator*H (V) on L²(R³) or L³(R¼) withconstant magnetic field, and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the*discrete*spectrum of H(V) near the boundary points###
QUASI-CLASSICAL VERSUS NON-CLASSICALSPECTRAL ASYMPTOTICS FOR MAGNETIC SCHRö*DINGER* *OPERATORS* WITH DECREASING ELECTRIC POTENTIALS

, 2002

"... We consider the Schrödinger operator H V) on L²(R²) or L³(R³), withconstant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate theasymptotic behaviour of the discrete spectrum of H(V) near the boundarypoints of its e ..."

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We consider the Schrödinger

*operator*H V) on L²(R²) or L³(R³), withconstant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate theasymptotic behaviour of the*discrete*spectrum of H(V) near the boundarypoints of its