The bound state of weakly coupled Schrödinger operators in one and two dimensions

We study the unique bound state which $\text{(}\text{?d}{}^2\text{dx}{}^2\text{) + λV}$ and ?Δ + λV (in two dimensions) have when λ is small and V is suitable. Our main results give necessary and sufficient conditions for there to be a bound state when λ is small and we prove analyticity (resp. nonanalyticity) of the energy eigenvalue at λ = 0 in one (resp. two) dimensions.     

The bound states of weakly coupled long-range one-dimensional quantum hamiltonians

We study the small λ behavior of the ground state energy, E(λ), of the Hamiltonian $\text{?(}\text{d}{}^2\text{dx}{}^2\text{) + λV(x)}$. In particular, if V(x) ～ ?ax^?2 at infinity and if 69-1, we prove that $\text{(?E(λ))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= ?[}\text{1}\text{2}\text{λ + aλ}{}^2\text{lnλ] ∫ dxV(x) + O(λ}{}^2\text{)}$.     

Singular perturbation potentials

This is a perturbative analysis of the eigenvalues and eigenfunctions of Schrödinger operators of the form ?Δ + A + λV, defined on the Hilbert space L²(Rⁿ), where $\text{Δ = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{?}{}^2\text{?X}{}_{\mathrm{i}}{}^2$, A is a potential function and V is a positive perturbative potential function which diverges at some finite point, conventionally the origin. λ is a small real or complex parameter. The emphasis is on one-dimensional or separable problems, and in particular the typical example is the “spiked harmonic oscillator” Hamiltonian, $\text{?d}{}^2\text{dx}{}^2\text{+ x}{}^2\text{+}\text{l(l + 1)}\text{x}{}^2\text{+ λ|x|}{}^{\mathrm{?\alpha }}$, where α is a positive constant. When this kind of perturbation is very singular, the first-order Rayleigh-Schrödinger perturbative correction, (u₀, Vu₀), where u₀ is the unperturbed eigenfunction, diverges. This analysis constructs explicitly calculable terms in a modified perturbation series to a finite order, by using linear operator theory in concert with approximation methods for differential equations. Along the way a connection between a W-K-B type approximation and Bessel functions is exploited.     

Phonon heating by hot electrons in n-InSb in quantizing magnetic fields

The effect of the phonon “narrow throat” was experimentally found in n-InSb in crossed electrical and quantizing magnetic fields at temperatures 1.6—4.2°K. The phenomenon of energy relaxation by hot electrons on phonons was detected with $\text{T}\text{S}\text{?}\text{h}\text{?}\text{λ}{}^{\mathrm{?1}}$ in the case of absence of a phonon thermal tank (S is sound velocity, λ is magnetic length, T is temperature). The value of a critical electric field (E_cr) on the S-type current-voltage characteristic (CVC) was measured as a function of temperature and the magnetic field.     

Temperature and thickness dependence of low temperature transport in amorphous silicon thin films: A comparison to amorphous germanium

The temperature and thickness dependence of the low temperature electrical transport of amorphous silicon thin films measured in-situ in ultra-high vacuum are presented. As for a-Ge there is an extended range of $\text{ln ? ∝ T}{}^{\mathrm{<mtext>?1</mtext><mtext>4</mtext>}}$ for thick films with $\text{ln ? ∝ S}{}_2\text{T}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$ for thin films with $\text{S}{}_2\text{∝ d}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$ as expected from a variable range tunneling mechanism.     

Pressure dependence of dielectric constants of alums

The effect of hydrostatic pressure p on the low frequency dielectric constant ? has been investigated for 31 alums of different composition at 293 K in the pressure range between 0 and 7 kbar. The ?(p)-curves deviate from linearity. For almost all alums with symmetric ions the first-, second-, and third-order pressure derivatives of the dielectric constant obey the following relations:

$\text{(}\text{??}\text{?p}\text{)}{}_{\mathrm{T}}\text{< 0, (}\text{?}{}^2\text{?}\text{?p}{}^2\text{)}{}_{\mathrm{T}}\text{> 0, and (}\text{?}{}^3\text{?}\text{?p}{}^3\text{)}{}_{\mathrm{T}}\text{< 0}$

. The logarithmic volume derivative of the low frequency polarizability, $\text{(}\text{? ln α}\text{? ln V}\text{)}{}_{\mathrm{T}}$, is found to be quasi-invariant also in the isotypic crystal group of alums (Jarman's rule). For the cases, where piezooptical data are available, the optical and infrared contributions to the volume dependence of the polarizability are determined.     

Functional derivative δTc/δα2 (Ω)F(Ω) for weak coupling superconductors

We present approximate analytic calculation of the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω)}$, where T_c is the superconducting critical temperature and $\text{α}{}^2\text{(Ω)F(Ω)}$ is the electron-phonon spectral function, within the “square-well model” for the phonon mediated electron-electron interaction and weak coupling limit $\text{ω}{}_{\mathrm{D}}\text{(2πT}{}_{\mathrm{c}}\text{)? 1 (ω}{}_{\mathrm{D}}$ is the Debye energy). It is found that $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω) = (1 + λ)}{}^{-1}\text{G(Ω)}$ where λ is the familiar electron-phonon coupling parameter and $\text{G(Ω)}$ is a universal function of the reduced frequency $\text{Ω}\text{=}\text{Ω}\text{T}{}_{\mathrm{c}}$. We compare this formula with accurate numerical results for several weak coupling superconductors. The overall agreement is good     

The ′T Hooft-Polyakov monopole near the Prasad-Sommerfield limit

A perturbative classical monopole solution for the SO(3) gauge theory is constructed in the limit of small but non-vanishing Higgs potential. This corresponds to the limit $\text{μ}{}^2\text{2M}{}_{\mathrm{W}}{}^2\text{=}\text{λ}\text{? 1}$, where μ equals the mass of the scalar particle and M_W equals the mass of the intermediate vector particles. The monopole solution and mass are found to involve non-analytic functions of λ: $\text{γ}$ and λ ln λ. The monopole mass M_m is calculated to order $\text{μ}{}^2\text{M}{}_{\mathrm{W}}$ as

$\text{M}{}_{\mathrm{m}}\text{=}\text{4π}\text{e}{}^2\text{M}{}_{\mathrm{w}}\text{1+}\text{1}\text{2}\text{μ}\text{M}{}_{\mathrm{w}}\text{+}\text{1}\text{2}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}\text{ln}\text{μ}\text{M}{}_{\mathrm{w}}\text{+0.7071}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}$

.     

Two theorems on the level order in potential models

We study the potentials of the form U(r)=?r^?1+λV(r), $\text{(}\text{d}\text{d}\text{r}\text{)(}\text{r}{}^2\text{d}\text{V}\text{d}\text{r}\text{)?0}$, and show that the energy levels satisfy the inequalities $\text{E(N}{}_{\mathrm{c}}\text{, l)?E(N}{}_{\mathrm{c}}\text{, l+1)}$ to first order in λ, where N_c denotes the coulombic principal quantum number and l the angular momentum. Similarly for potentials $\text{U(r)=r}{}^2\text{+λV(r), (}\text{d}\text{d}\text{r}{}^2\text{)}{}^2\text{V(r)?0}$, we prove to first order in λ that $\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{)?}\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{+2)}$, where NH denotes the harmonic oscillator quantum number. In the latter case, we give also quantitative restrictions on the relative positions at the lth and (l+1)th states.     

Dynamical scaling and ultrasonic attenuation in 4He near Tλ

Ultrasonic attenuation in ⁴He near T_λ has been measured at frequencies between 10.9 MHz and 163 MHz. The attenuation above T_λ is described by a scaling function as $\text{α∝ω}{}^{\mathrm{x}}\text{F(}\text{ε}\text{ω}{}^{\mathrm{Y}}\text{)}$, and which proves the dynamical scaling hypothesis.     

On some recent results concerning the superconducting transition temperature

The Eliashberg gap equations relate the transition temperature T_c of an isotropic superconductor to its electron-phonon spectral function α²F(ω) and Coulomb pseudopotential parameter $\text{μ}{}^1$. Recently the Eliashberg theory has been used to derive some supposedly rigorous results bearing on the problem of attaining higher superconducting transition temperatures: Bergmann and Rainer derived an expression for the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{F(ω)}$; Allen and Dynes showed that in the asymptotic limit of very large $\text{λ(λ?10)k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{=f(μ}{}^1\text{)(λ〈ω}{}^2\text{〉)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and Leavens proved that for any isotropic superconductor k_BT_c ?0.2309A, where A is the area under its electron-phonon spectral function. In this letter we show that the result of Allen and Dynes is not compatible with the other results and is, in fact, incorrect.     

Study by brillouin scattering of the C33 constant of thiourea submitted to an electric field

The C₃₃ constant is discontinuous for the lock-in transition at T₁ = 169 K. The variation ΔT₁ of the temperature of transition is a linear function of the applied electric field E and we find $\text{d}\text{T}{}_1\text{d}\text{E}\text{=0.82 deg.cm kV}{}^{\mathrm{?1}}$.Above a mean field E = 10 kV cm^?1 the transition observed for a first heating spreads on several degrees because damages appear in the incommensurate phase and the electric field becomes inhomogeneous.The results obtained at low fields are in very good accordance with the value of $\text{d}\text{T}{}_1\text{d}\text{E}$ calculated from the Clapeyron formula.Taking into account of the incommensurability of the phase above T₁, it can be shown that

$\text{dT}{}_1\text{dE}\text{=}\text{C}\text{2πP}{}_{\mathrm{o}}\text{3}\text{3}\text{?}\text{2}$

The knowledge of the spontaneous polarisation P₀ gives for the Curie constant C = 2.1 × 10³ K in qualitative agreement with the value deduced from measurements of the dielectric constant in the ferroelectric direction.     

The Grüneisen parameter γ of KBr,RbCl and Bi through high pressure phase transitions

High pressure values for the adiabatic pressure derivative of temperature $\text{(}\text{?T}\text{?P}\text{)}{}_{\mathrm{s}}$ have been obtained by measuring the temperature change caused by a small rapid increase in pressure. Values for KBr and RbCl in phases B1 and B2 and for Bi in phases I, II and III are given for T = 295 K. The Grüneisen parameter γ is given by $\text{γ =}\text{B}{}_{\mathrm{s}}\text{(}\text{?T}\text{?P}\text{)}{}_{\mathrm{s}}\text{T}$ where B_s is the adiabatic bulk modulus. Ultrasonic and statk compressibility data are used to estimate the pressure and phase dependence of B_s. Dramatic increases in both γ and (?T/?P)_s are observed as the pressure increases through a phase transition. Values for the logarithmic derivate $\text{q ≡ (? ln}\text{γ}\text{?}\text{ln V}{}_{\mathrm{T}}$ are given.     

Experimental detection and investigation of muon catalyzed fusion of deuterium and tritium

Measurement of the neutron yield of the reaction $\text{dt}\text{μ →}{}^4\text{He}\text{+ μ}{}^{\mathrm{?}}\text{+ 17.6}\text{MeV}$, induced by negative muons in a mixture of gaseous D₂ and T₂, has shown that the rate of muon transfer from deuterium to tritium is λ_dt = (2.7 ± 0.9) × 10⁸s^?1 and that the lower limit of the formation rate of dt_μ molecules is λ_dtμ > 10⁸s^?1.     

Effects of composition on the upper critical field of V-Ga

Measurements of the temperature dependence of the upper critical field, H_c2(T), for a series of V_100?xGa_x materials are presented for 20.5 ≤ × ≤ 29.6. Fits of the data to conventional theory for a paramagnetically limited, dirty, type II superconductor show: 1) a maximum in T_c and H_c2(0) for x ? 25; 2) a constant ( for x ≤ 25; 3) a slowly increasing value of λ_so with increasing x up to x ～ 25; and 4) good agreement with stoichiometric ordered and thermally disordered V₃Ga. Above $\text{x ? 25}$ broader transitions are observed. For x = 25, T_c = 15.3 K, (, λ_so = 0.3 and H_c2(0) = 23.4 tesla. The effects of inclusion of strong-coupling in the theory are discussed briefly.     

The kinetics of the photolysis of thin films of lead iodide

The kinetics of the photolysis of thin films of PbI₂ have been followed using a quartz crystal microbalance method. Rate as a function of temperature, light intensity and wavelength has been determined. The influence of grain size and gaseous ambient has also been investigated. Three ranges of kinetics were identified. Initial; associated with nucleation and surface chemical reactions. Intermediate; associated with photolysis without the intervention of optically absorbing product, gives $\text{(}\text{?θ}\text{?t}\text{)}{}_{\mathrm{T}}\text{,λ,I ∝ T}$. Terminal; attenuation due to lead particles, gives $\text{(}\text{?θ}\text{?t}\text{)}{}_{\mathrm{T}}\text{,λ,I ∝ t}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$.Results show that iodine is generated at the solid/vacuum interface and lead is generated in the bulk. Photolysis is described by phenomenological equations involving bulk and surface recombination of hole/electron pairs plus photolysis. A mechanism is proposed in which the rate determining step is anion vacancy diffusion from the surface, the activation energy for photolysis and anion vacancy diffusion being the same.     

Conductivity of a quasi-one-dimensional metal at finite temperatures

The longitudinal conductivity of a quasi-one-dimensional metal is calculated for the case T?ω_D (ω_D being the limiting phonon frequency) and ω_Dl¹/v?1 where l¹ is the effective mean free path determined by impurity and phonon scattering: $\text{l}{}^1\text{= (l}{}^{\mathrm{?1}}{}_{\mathrm{ph}}\text{+ l}{}^{\mathrm{?1}}{}_{\mathrm{i}}\text{)}{}^{\mathrm{?1}}\text{, l}{}_{\mathrm{ph}}\text{= v/λT, l}{}_{\mathrm{i}}$ is the impurity mean free path. The conductivity is σ = (c₁e²/πS)l³_iv^?2ω_DλT for l_i?l_ph, σ = (c₂e²/πS)vω_D(λT)^?2 for l_i?l_ph, λ being the dimensionless electron-phonon interaction constant, c₁, c₂ ～ 1, S = a_xa_y is the (xy) area per one chain.     

Higher order correction to the sine-Gordon soliton mass

We calculate in the 2-dimensional sine-Gordon theory

$\text{L}{}_{\mathrm{GC}}\text{=}\text{1}\text{2}\text{(?}{}^{\mathrm{\mu }}\text{φ)}{}^2\text{?(}\text{m}{}_4\text{λ}\text{)[1?cos(}\text{λ}\text{φ}\text{m}\text{)]}$

the soliton mass M_sol to order λ and compare it to the meson mass. We find

$\text{M}{}_{\mathrm{sol}}\text{m}{}_{\mathrm{menson}}\text{=}\text{8m}{}^2\text{λ}\text{?}\text{1}\text{π}\text{+}\text{λ}\text{192m}{}^2\text{+O(λ}{}^2\text{)}$

This supports the belief that the field-theoretic WKB approximation of Dashen, Hasslacher and Neveu gives the correct answer for the sine-Gordon problem.